Further Topics in Actuarial Mathematics: Premium Reserves. Matthew Mikola

Transcription

1 Further Topics in Actuarial Mathematics: Premium Reserves Matthew Mikola April 26, 2007

2 Contents 1 Introduction Expected Loss An Overview of the Project Net Premium Reserves Introduction The Net Premium Reserve for Various Life Insurance Policies Whole Life Insurance Policy Term Insurance Endowment Examples Other Forms of the Net Premium Reserve Equation Recursive Formulae Savings and Risk Premium Retrospective Reserves The Expected Value and Variance of Loss Variables Prospective Loss The Insurer s Net Cash Loss Allocation of the Risk Expense Loadings Expense-Loaded Premium Reserves Uses of Reserves Introduction Alterations and Conversions With-profits policies The Net Premium Method The Bonus Reserve Method Asset Share Method Applications of Reserves Beyond Life Insurance Insurance Actuaries Uses of Reserves The Balance Sheet The Income Statement Loss Reserves Adequacy Testing Reinsurance Conclusion 36 i

3 A Notation i A.1 Lifetime Models i A.2 Life Insurance ii A.3 Endowments iii A.4 Life Annuities iii B Useful Equations iv B.1 Proofs of Equations used in Section iv B.2 Proofs of Equations used in Section v B.3 Proofs of Equations used in Section v C Life Tables vi ii

4 Chapter 1 Introduction Actuarial Mathematics is a very interesting area of mathematics for me, as it has such a large impact in the real world. The concepts that have been developed are used daily by insurance companies in the calculations of their insurance policies and also when a monetary value is required for any choice that involves risk. Insurance mathematics relies heavily on probability theory, as the lifetime of a person or the occurrence of an event is unpredictable. Insurers calculate the expected probability that these events occur, so they can price their insurance policies correspondingly and also so no large unexpected losses occur. In this project, I am going to explore the topic of net premium reserves with regards to life insurance policies. This project assumes that the reader has understood the concepts covered in the 2nd year mathematics course Actuarial Mathematics II, Appendix A contains explanations and formulae for most of the theory covered for the reader s benefit. I am only going to consider discrete life insurance policies and the corresponding reserve formulae. Variants of the formulae for the continual policies can be used in most cases, this simply involves using the appropriate continuous net single premium formulae and lifetime probability distributions covered in the 2nd year course. Premium reserves, also known as benefit reserves, are of great interest to insurers as it enables them to work out their liability, this is how much money they should put aside, reserve, each year to cover the benefit payment to the insured in case of their death. The insurers take into account the premiums already paid by the insured and the investment return they can receive on these payments and the insured s expected future lifetime. Premium reserves look at the obligations of the insurer and the insured, at a time later than policy issue, as either party may still have financial duties that need settling at that time. In the next section I will look at one of the most useful concepts for calculating premium reserves, the expected loss of the insurer, which was covered in the course Actuarial Mathematics. This has particular significance for the formulation of net premium reserves as the expected loss to the insurer is used in its definition. 1

5 1.1 Expected Loss In this section I have used my notes from the 2nd year mathematics course Actuarial Mathematics II which, were based on concepts covered in [Gerber]. In this section I am going to recap the relevant actuarial concepts covered in the 2nd year mathematics course Actuarial Mathematics II. One of the important ideas developed near the end of the course was calculating the expected loss to the insurer. For an insurance policy we define the prospective loss to the insurer, L, to be the difference between the present value of future premium payments (paid by the insured) and the present value of future benefit payments (paid by the insurer). The prospective loss is a really important concept for the derivation of the formulae for the net premium reserve, as we shall see later. Premiums can be paid in a variety of different ways to suit the preferences of the insured, three of these ways are: 1. One single premium. 2. Periodic premiums of a constant amount, also known as level premiums. 3. Periodic premiums of varying amounts. In the case of the periodic premiums the frequency and the duration of the payments needs to be specified as well as the premium amount, commonly the premiums are paid annually and while the insured is still alive. A premium is called a net premium if it satisfies the following equivalence principle: E(L) = 0 (1.1.1) If the policy is financed by a single premium, then the premium used is the net single premium, the formulae for the different policies can be found in Appendix A.2. As the net single premium is used to calculate the benefit payment as well, then the premium payment and the benefit payment are equal in value so they satisfy (1.1.1). If periodic premiums of constant amount are used to finance the policy, then we use (1.1.1) to calculate the net premiums. We cannot use (1.1.1) to calculate the net premiums for periodic premiums of varying amounts though. (1.1.1) is called an equivalence relation, this is because the present value of the benefit payments exactly equals the present value of the premium payments, so the expected loss to the insurer is zero. In practice this is rarely true because the insurance company would not make any money, but we use this assumption in our calculations for simplicity. The prospective loss is very important in the formulation of premium reserves as we shall see in the next chapter. If we now consider the more realistic case where the insurer wants to make sure a profit is made, we can change (1.1.1) so that the difference between the premium payments and the benefit payments is a positive value. If the insurer wants to make a profit of θ then in (1.1.1) we would want the loss to the insurer to be θ: E(L) = θ The insurer could then use this variation, instead of (1.1.1), in their calculation of the net premiums. We shall look later at another way the insurer can factor profit and the costs of setting-up and running an insurance policy, by using expense loadings in the premiums. 2

6 1.2 An Overview of the Project In chapter 2, I shall derive the formula for the net premium reserve for various life insurance policies. I will then proceed to look at some of the interesting properties of reserves and why they are useful to insurers. Near the end of the chapter I look at expected value and variance of the loss to the insurer, which is very interesting as it gives an indication of the risk associated with the policy. I also consider expense loadings, which are extra amounts added to the premium payments, to cover various expenses of the insurer. This part gives an insight into how, in the real world, insurance companies have to factor their overheads and profit margins into their policies. In chapter 3, I will look at some of the uses of reserves for life insurance policies. I examine how the reserves are used when the insured may want to make alterations to his existing policy and I also look at how reserves can be used to determine the level of extra bonuses, added to the benefit payment, in with-profit policies. In chapter 4, I look at why insurance is so important to society as a whole and how specialists known as actuaries work in insurance companies to ensure financial stability. I also look at some of the uses of reserves outside of life insurance. In writing this chapter I looked at papers written by researchers, who are looking at ways to apply reserves and various actuarial methods to real life situations and problems. They look at ways of using different types of reserves to try to estimate the amount of money an insurance company sets aside to cover their overall losses and expenses. Other forms of reserves are used by the Financial Services Authority (F.S.A.), who regulate all providers of financial services to make sure they are making reasonable levels of profit and have enough reserves to cover their policies and actions. Finally I look at when reserves are used by insurers to see if large risky policies are worth reinsuring, this is when an insurer looks to another insurance company to cover all or part of the risk of one of their policies or group of policies. In this project I have used information from the sources listed in the bibliography. At the beginning of each section I have noted where I have directly used equations or text from a source and what is my own work. All examples demonstrated in this project are my own work and have been calculated by myself. 3

7 Chapter 2 Net Premium Reserves 2.1 Introduction As we found in section 1.1, the expected loss to the insurer at the time of policy issue is zero (1.1.1), this is because the expected value of future premium payments equals the expected value of future benefit payments. In this chapter we look at the expected loss at a later time than policy issue. We therefore define a random variable t L as the difference at time t between the present value of future premium payments and the present value of future benefit payments. According to [Gerber, p.59] the net premium reserve at time t is defined as the conditional expectation of t L, given that T > t. This is the conditional expectation of the difference between the present value of future benefit payments and the present value of future premium payments at time t, given that the insured survives to t. It is denoted by the symbol t V. The standard convention for reserves is to calculate them just before the payment of the premium due at time t. The net premium reserve can be seen as the present value of the liability for the insurer at time t, i.e. the expected extra amount of money the insurer needs to set aside each year, to cover the benefit payment taking into account the premium payments already paid by the insured. More specifically, they are reserves that are calculated without the allowance for expenses and where the reserve basis and the premium basis agree. The reserve basis is the mortality, interest and expense assumptions used to calculate t V and the premium basis is a set of similar assumptions applied to the premium payments. The net premium reserve is often positive to give the insured some incentive to stay in the scheme, this also means that the insured will not have any additional payments to pay to the insurer, if they leave the scheme. Typically the premiums that the insured pays each year in the early years of the policy, are more than the insurer needs in order to cover the benefit payment, but this margin decreases as time progresses. At some point later in time, the future premiums will not be sufficient to cover the remaining benefit payment, therefore the excess collected in the early years, which is invested to gain interest, will be used to cover this deficit; this is another reason why net premium reserves are often positive. Negative reserves can occur on policies where the benefit payment decreases each year or the premium payments increase each year. Some of the uses of reserves are for when the insured wants to leave the insurance policy before a claim is made and so would want some compensation amount, this is known as the surrender value of the policy. Also the insured, if they were part of their employer s pension scheme, may change jobs and so may want to transfer their existing pension to their new employer s pension scheme. Most modern insurance policies are quite flexible and allow the insured to make changes to their policy (within reasonable boundaries, so the insurer still makes a profit), knowing the reserve makes this possible. Reserves can also be used to determine the bonus rates of with-profits policies and for the distribution of profits to shareholders. With-profits contracts are life insurance 4

8 policies where the insured has the right to share in the profits of the company, usually in the form of bonuses added to the benefit payment. 2.2 The Net Premium Reserve for Various Life Insurance Policies In this section equations (2.2.3) and (2.2.4) are based on equations found in [Bowers, p.206], but the derivation of (2.2.4) has been extended by me to provide clearer explanation. Using the definition of the net premium reserve given in the Introduction I am now going to find the net premium reserve for various life insurance policies Whole Life Insurance Policy The present value of a whole life insurance policy for a life aged x, which covers the insured until death, with a benefit payment of 1 unit payable at the end of year of death, is given by: Z = v K+1 K 0 (2.2.1) Where K(x) is the curtate future lifetime of the insured and v is the discount factor, with v = 1 1+i where i is the Annual Equivalent Rate of interest (see the end of A.1 for further details). A whole life insurance is financed by level annual premiums P x, payable at the beginning of each year whilst the insured is still alive. An equivalent way of describing this mathematically is setting up a whole life annuity with benefit payment P x, this is because the payments are made at the start of every year whilst the insured is still alive. The present value of this is given by: Y = P x ä K+1 K 0 (2.2.2) The prospective loss to the insurer, as described in section 1.1, is the difference between the present value of future premium payments and future present value of the benefit payments. For the whole life insurance policy we just appropriately combine the two present value formulae (2.2.1) and (2.2.2) to get: L = v K+1 P x ä K+1 K 0 If we now consider the prospective loss k years later than policy issue, this is given by the symbol k L and is defined in exactly the same way as before but we have just adjusted which point in time we are looking at. If the insured s curtate future lifetime is K(x) years, then if we look k years later than policy issue, we only need to consider the remaining K k years of the insured s life, so the index K in (2.2.1) and (2.2.2) changes to K k. For example if the insured was 50 when the policy was issued and we expect them to die at 80 (in reality this is a random event as we don t know when someone is going to die, but for simplicity assume we can for this case), then K = 30 and if we now look at the person k = 10 years later when they are 60, we now only need to consider the remaining K k = = 20 years. Therefore the present value of the prospective loss for a whole life insurance is: kl = v K+1 k P x ä K+1 k K(x) k (2.2.3) For the formulation of the net premium reserve we need the expected value of (2.2.1) and (2.2.2) and these are shown by : E(Z) = A x = v k+1 kp x q x+k k=0 5

9 and E(Y ) = ä x = ä k+1 kp x q x+k. k=0 as shown by (A.2.3) and (A.4.2) respectively. The net premium reserve; the conditional expectation of the prospective loss, for a whole life insurance policy, is given by: kv = E[ k L K(x) > k] = E[v K+1 k K(x) > k] P x E[ä K+1 k K(x) > k] = v j+1 Pr[v K+1 k K(x) > k] P x ä j+1 Pr[ä K+1 k K(x) > k] = j=0 v j+1 jp x+k q x+j+k P x j=0 j=0 j=0 ä j+1 jp x+k q x+j+k = A x+k P x ä x+k (2.2.4) Term Insurance Similarly the net premium reserve at the end of year k of a term insurance is denoted by k V 1 x:n. A term life insurance pays 1 unit benefit payment at the end of year of death, if the insured dies within n years and is financed by premiums payable at the beginning of each year the insured is alive, up to the start of year n. Using the expected values for the term life insurance and the n year temporary life annuity due, which can be found in (A.2.4) and (A.4.4), its net premium reserve is given by: Endowment kv 1 x:n = A 1 x+k:n k P 1 x:n ä x+k:n k k = 0, 1,..., n 1 (2.2.5) The net premium reserve for an n year endowment is denoted by k V x:n. An n year endowment pays 1 unit benefit payment at the end of the year of death if the insured dies within n years, or if the insured survives n years the benefit payment is at the end of the n th year. Using the expected values for the n year endowment and the n year temporary life annuity due can be found in (A.3.3) and (A.4.4), it is given by: Examples kv x:n = A x+k:n k P x:n ä x+k:n k k = 0, 1,..., n 1 (2.2.6) I will now look at a numerical illustration of the reserves for a term insurance and a n year endowment. I will assume a sum insured of 100 units, the initial age of the insured as x = 50 years old and a duration of n = 10 years. We will use the life tables found in appendix C and i = 5%. As a first step we work out the net annual premiums for each policy. As the premiums are constant we can use the equivalence relation. This is calculated in a very similar way to the method used in the whole life insurance policy example above. This process is shown below: 6

10 E(L) = 0 P 1 = 100 A 1 x:n P 1 x:n ä x:n ä 50:10 50:10 = 100 A 1 50:10 = 100(M 50 M 60 ) N 50 N 60 = Similarly: P 50:10 = Using (2.2.5), (A.2.5) and (A.4.5): kv 1 50:10 = 100(M 50+k M 60 ) P 1 50:10 50+k N 60 ) D 50+k k = 0, 1,..., 9 (2.2.7) and using (2.2.6), (A.3.4) and (A.4.5): kv 50:10 = 100(M 50+k M 60 + D 60 ) P 50:10 (N 50+k N 60 ) D 50+k k = 0, 1,..., 9 (2.2.8) Using these equations we can produce the following table which will enable us to directly compare the net premium reserves for the term insurance with the endowment. Table 2.1: The net premium reserves for a term insurance and an endowment k 100(A 1 50+k:10 k 50+k:10 k kv 1 100(A 50:10 50+k:10 k ) kv 50: Things to notice from these results are: The net premium reserve for the term insurance and the endowment at k = 0, i.e policy issue, is 0 as expected. This follows directly from the net premium equivalence principle that E(L) = 0 at policy issue. The net single premium of the term insurance decreases (1st column), this is because the probability of the insured surviving to the end of the term increases (see Table 2.2), and this out weighs the decreasing time for interest to accumulate on the premium payment. 7

11 The net single premium of the n year temporary life annuity due decreases (2nd column), this is because fewer benefit payments have to be paid and therefore there is less time for interest to accumulate, this out weighs the probability of the insured surviving to the end of the term, which increases. The net single premium of the endowment increases (4th column), this is because the decreasing time for interest to accumulate on the premium payment out weighs the decreasing probability of the insured dying before the end of the term. The net premium reserve of the term insurance is very small and does not change by much (3rd column). It grows to start with since the premiums slightly exceed that of a corresponding 1 year term insurance. Near the end of the period the net premium reserve drops again as the insurer does not pay the benefit payment if the insured survives and the probability of that occurring increases. The net premium reserve of the endowment increases over time (5th column), this reflects the increasing net single premium of the endowment and this is due to the decreasing time for the interest to accumulate. Note that the net premium reserve of at the end of the 9th year, plus the last premium payment of , plus interest of 5% on both, is sufficient to cover the benefit payment of 100 at the end of the 10 years = = Table 2.2: The probability of a life aged 50 surviving to 60 given they have survived k years. k l x 10 k P 50+k Other Forms of the Net Premium Reserve Equation In this section equations (2.3.1) and (2.3.2) can be found in [Gerber, p.63,64], but the derivations of these have been extended by me for clearer explanation. So far I have only used the prospective method to write formulae for the net premium reserve, stating that the net premium reserve is defined as the difference between the present value of future benefit payments and of future premium payments. We can use this method to develop other formulae for the net premium reserve, in terms of the premiums paid by the insured. These other formulae are of interest because they give us different interpretations of the net premium reserve and therefore give us greater insight into their use to actuaries. This is useful because the actuary can now consider the reserve in terms of the amount paid by the insured each 8

12 year and therefore if they wanted to reserve only a certain proportion of the premium, then they can adjust other factors accordingly to take this into account. Also by having an alternative way to look at the concept of the net premium reserve I hope it will aid the reader in their understanding. If we consider the net premium reserve of a whole life insurance policy as defined in section 2.2.1: kv = A x+k P x ä x+k Using results derived in [Gerber, p.64], the premium difference formula for k V can be found by using A x+k = 1 d ä x+k (as shown in (B.1.3)) and 1 d ä x+k = P x+k ä x+k (from (B.1.5)), giving: kv = A x+k P x ä x+k = (1 d ä x+k ) P x ä x+k = 1 (P x + d)ä x+k = (P x+k P x )ä x+k (2.3.1) As we can see from (2.3.1), if a whole life insurance, of 1 unit, was bought k years later than the original, now costing P x+k ä x+k, the net premium reserve is the expected present value of the deficit of the premiums. Another way to look at this is to think of P x+k as the amount that should be charged, for a policy with the same benefits if issued at time k to a person aged x + k, in order that the premium equals the benefit payment after k years. Then the quantity P x+k P x is the difference between what should be charged and what is actually charged. This means that the reserve at time k is the yearly deficit between these premium amounts. Also using results derived in [Gerber, p.64], the paid up insurance formula for k V can be found by using ä x+k = (1 A x+k )/d (as shown in (B.1.3)) and d = (P x+k (1 A x+k ))/A x+k (as shown in (B.1.6)), giving: kv = A x+k P x ä x+k = A x+k P x ( 1 A x+k ) d = A x+k P x d (1 A x+k) = A x+k A x+kp x (1 A x+k ) P x+k (1 A x+k ) = A x+k A x+kp x P x+k = (1 P x P x+k )A x+k (2.3.2) (2.3.2) shows the net premium reserve as the present value of a portion of the remaining future benefit payments, the portion which is not funded by future premium payments. P x+k is the premium required if the future benefit payments were to be funded from only the future premium payments, but P x is the benefit payment actually paid. Therefore P x /P x+k is the portion of future benefit payments actually funded by future premium payments. Alternatively, as mentioned before in the early years of the policy the premium payments exceed what should be paid for the benefit payment, but this changes later in the policy, so the amount (1 Px P x+k ) must be the that portion of future benefits that has already been provided for by the excess past premiums. 9

13 Example If we now calculate the net premium reserve for our example in section using the premium difference formula and the paid up insurance formula. We can calculate the values of P x+k from the equivalence relation: P 1 = A 1 x+k:n k (2.3.3) x+k:n k ä x+k:n k for the term insurance and P x+k:n k = A x+k:n k ä x+k:n k for the endowment. The results are shown below (see Table 2.1 for the unchanged values of 100(A 1 ), 50+k:10 k ä 50+k:10 k and 100(A 50+k:10 k )): Table 2.3: The net premium reserve for the term life insurance and endowment calculated via the premium difference and paid up formulae k P 1 50+k:10 k kv 1 P 50:10 50+k:10 k kv 50: Unsurprisingly we see that the values for the net premium reserve are the same as they were when we performed the calculation the first time. 2.4 Recursive Formulae In this section I have used the theory covered in [Gerber, p.61], but the derivations of all the equations have been extended by me for greater clarity. In this section I am going to develop a relation between k V and k+h V which is the net premium reserve h years after k. If we consider the net premium reserve for a whole life insurance, with varying benefit payment c j, being the amount insured in the jth year after policy issued, financed by varying premiums Π 0, Π 1,..., Π k, being the premium due at time k. Then the net premium reserve at the end of year k is: If we re-write ä j+1 kv = c k+j+1 v j+1 jp x+k q x+k+j Π k+j ä j+1 jp x+k q x+k+j (2.4.1) j=0 as: j=0 ä j+1 = 1 + v + + v j = 10 v j I {J=j} k=0

14 Where I {J=j} is an indicator function: Then: So: j=0 Now using (2.4.2) in (2.4.1) we get: I {J=j} = { 1 if J = j 0 elsewhere E(I {J=j} ) = P r(j = j) = j p x ä j+1 jp x+k q x+k+j = ä x = E(ä j+1 ) = v j jp x (2.4.2) kv = c k+j+1 v j+1 jp x+k q x+k+j Π k+j v j jp x+k (2.4.3) j=0 If we then use (A.1.5) with s = h, t = j h and x = x + k and rearrange we have: j=0 j=0 jp x+k = h p x+k j h p x+k+h (2.4.4) Then substitute (2.4.4) in all except the first h terms of (2.4.3) then we have: kv = h 1 h 1 c k+j+1 v j+1 jp x+k q x+k+j Π k+j v j jp x+k j=0 + j=0 c k+j+1 v j+1 hp x+k j h p x+k+h q x+k+j j=h Π k+j v j hp x+k j h p x+k+h j=h (2.4.5) If we then use j = j h as a summation index in the second and third lines of (2.4.5) then this part becomes: c k+j +h+1 v j +h+1 h p x+k j p x+k+h q x+k+j +h Π k+j +h v j +h h p x+k j p x+k+h (2.4.6) j =0 There is a common term of v h hp x+k in (2.4.6), if this is taken out as a factor we are left with: j =0 j =0 c k+j +h+1v j +1 j p x+k+h q x+k+j +h Π k+j +h v j j p x+k+h Which is the net single premium of k+h V. If we then combine (2.4.6) and the first line of (2.4.5) and rearrange, we get: h 1 kv + Π k+j v j jp x+k = j=0 j =0 h 1 c k+j+1 v j+1 jp x+k q x+k+j + h p x+k v h k+hv (2.4.7) j=0 If we look at (2.4.7) we can see that if the insured survived to the end of year k, then the net premium reserve plus the expected present value of the premium payments for the next h years is 11

15 just sufficient to cover a life insurance for those years, plus a pure endowment of k+h V, at the end of year k + h. A recursive equation for the net premium reserve can be achieved by letting h = 1: kv + Π k = v[c k+1 q x+k + k+1 V p x+k ] (2.4.8) We can therefore calculate the net premium reserve recursively in 2 ways: 1. We can calculate 1 V, 2 V,... successively by starting with the initial value 0 V = 0, by rearranging (2.4.8) to make k+1 V the subject. 2. Or if the duration of the insurance is of finite duration n, then we may calculate n 1 V, n 2 V,... from a known value of n V, by letting k + 1 = n The recursive formula is really useful for actuaries as it enables the insured to have greater flexibility with their insurance policies. By being able to work out the net premium reserve each year from the previous value can allow the insured to change the policy and use the current value of the net premium reserve to begin a new policy. An example may be when the insured wants to convert the insurance policy to a policy which has no further premium payments, know as a paid-up insurance policy. The actuary can then use the value of the last net premium reserve to calculate the net single premium for the paid up insurance. There is a type of insurance known as universal life or flexible life, which offers the maximum degree of flexibility to the insured. The insured can change any two of the following parameters: Π k, the next premium to be paid, c k+1, the benefit paid in case of death in the next year, k+1 V, the net premium reserve for next year, this can be seen as the target value of the insured s saving s in the next year (see the savings premium in Section 2.5 for further details on the consideration of the net premium reserve as a saving). We calculate the value of k+1 V from the recursive formula (2.4.8), so we can see how useful it is for actuaries when they are helping the insured modify their insurance policy. There are usually some restrictions applied as to how much these values can be changed, so as not to leave the insurer out of pocket, but yet still allow flexibility. 2.5 Savings and Risk Premium All the equations from this section are from [Gerber, p.61,62] If we examine (2.4.8) more closely we can see that the net premium reserve at time k plus the premium payment equals the expected present value of the funds needed by the insurer at the end of that year, c k+1 if the insured dies, or k+1 V otherwise. As p x+k = 1 q x+k then (2.4.8) can be rewritten as: kv + Π k = v[ k+1 V + (c k+1 k+1 V ) q x+k ] (2.5.1) (2.5.1) can be interpreted as; k+1 V is needed whatever happens to the insured and an additional amount of c k+1 k+1 V if the insured dies. According to [Gerber, p.61], c k+1 k+1 V is known as the net amount at risk. (2.5.1) can be rearranged to decompose the premium into two components, Π k = Π s k + Πr k, where Π s k = k+1 V v k V (2.5.2) 12

16 and Π r k = (c k+1 k+1 V )v q x+k (2.5.3) Π s k is known as the savings premium and Πr k is known as the risk premium. We can see in (2.5.2) that the savings premium is the part of the premium used to increase the net premium reserve and cover the change in liability for the insurer. It is known as the savings premium as it is the extra amount the insurer has to save each year to be able to cover any future benefit payment. This part can also be seen as a savings account, with the premium accumulating interest only, which can be used by the insurer in later years to cover the deficit between the premium payments and the benefit payment. Some financial advisers suggest that people should not buy whole life or endowment policies, but instead should purchase term life insurance. That way the premiums are less and the insured can choose where to invest this extra money and so has greater control of a portion of the savings premium and can make money from this. This led to the insurance companies introducing the universal life policy, mentioned at the end of section 2.4, in an attempt to offer the policyholder more flexibility with regards to the levels of the risk and savings premium. In (2.5.3) we see that the risk premium is the part used to fund a one-year term insurance, of amount c k+1 k+1 V, to cover the net amount at risk to the insurer, which is the benefit payment. This part of the policy has no reserves, since the risk premium is just sufficient to purchase appropriate coverage for the net amount at risk for that 1 year. If we multiply (2.5.2) by (1 + i) j k and just consider the first j years, then sum over k = 0, 1,..., j 1, we get: j 1 jv = (1 + i) j k Π s k k=0 We can see from this equation that the net premium reserve is the total value of the savings premiums since the beginning of the policy. Also we can re-write (2.4.8) by using d = i/(1 + i), where d is the discount rate covered in the Actuarial Mathematics II course, which gives us v = 1 d, resulting in: Π k + d k+1 V = ( k+1 V k V ) + Π r k We can see that the premium plus the interest received on the net premium reserve, is used to change (increase or decrease) the net premium reserve and to cover the risk premium. Example The decomposition of the net annual premium into the savings premiums and the risk premiums for the same example of section is tabulated below: We can see from these results that: The savings premium for the term insurance decreases steadily and becomes negative from year 5 onwards, this is because the net premium reserve decreases from year 6 and once the net single premium in year 6 has been discounted it is less than the value for year 5. The risk premium for the term insurance increases steadily this is because the probability of the insured dying each year increases (See Table 2.5), which out weighs the increasing net premium reserve during the first five years. After that the net premium reserve decreases and the probability of the insured dying each year continues to increase, so an increasing risk premium is expected. The savings premium for the endowment increases slowly, this is because the net premium reserve increases at roughly the same rate. 13

17 Table 2.4: Decomposition into savings premium and risk premium Term insurance Endowment k Π s k Π r k Π s k Π r k The risk premium for the endowment decreases to zero, this is because the net premium reserve increases as does the probability of the insured dying each year. The risk premium in year 9 is zero because the net premium reserve in year 10 is 100 which is equal to the benefit payment in year 10. Table 2.5: The probability of the insured dying each year k l x q x+k Retrospective Reserves In this section the definition of retrospective reserves is from [Scott, p.102]. Equations (2.6.1) and (2.6.2) can be found in [Scott, p.103], but their derivations have been extended by me for further clarity. As mentioned in section 2.1 one of the uses of reserves is calculating the surrender value of the policy, in this section I will find formulae to achieve this. If the insured decides to surrender the contract before a claim is made, he will probably expect in the early of the years of the policy, to receive a surrender value related to the accumulation of the premiums he has paid, less expenses and the cost of the life insurance cover, since he hasn t received any benefit payment. Such a surrender value is related to the retrospective reserve of the contract, which is defined to be the total of the premiums paid, less expenses and the cost of the benefit payment, of a hypothetical large group of identical policies whose mortality exactly follows a set life table and then dividing the hypothetical funds amongst the survivors. 14

18 If we now consider the whole life insurance policy defined in section 2.2.1, the retrospective reserve is found by collecting together the funds of say l x identical policies until time k and then sharing the accumulated money out among the survivors. If we use standard notation used for life tables, there are d x deaths in the first policy year, d x+1 deaths in the second policy year and l x+k survivors at time k. The total accumulated premiums paid by the survivors is given by: l x P x (1 + i) k + l x+1 P x (1 + i) k l x+k 1 P x (1 + i) = l x P x (1 + i) k [ i (1+i) k 1 ] = l x P x (1 + i) k [1 + v + + v k 1 ] = l x P x (1 + i) k ä x:k The first line can be simply explained: all the l x people pay premium P x at policy issue (k = 0), which gains k years of interest, then at the start of the next year the surviving l x+1 people again pay premium P x, which can gain k 1 years of interest, until at the start of the k th year the surviving L x+k 1 people pay premium P x which only has that one year to accumulate interest. The total of the benefit payments, of 1 unit, that need to be paid out to the people that died each year is given by (we use the fact that d x+k = l x k p x q x+k which is proved in section B.2): d x (1 + i) k 1 + d x+1 (1 + i) k d x+k 1 = l x 0 p x q x+0 v(1 + i) k + l x 1 p x q x+1 v 2 (1 + i) k + + l x k 1 p x q x+k 1 v t (1 + i) k = l x (1 + i) k [v 0 p x q x+0 + v 2 1p x q x v k k 1p x q x+k 1 ] = l x (1 + i) k [ k 1 j=0 vj+1 jp x q x+j ] = l x (1 + i) k A 1 x:k The first line can be simply explained: at the end of the first year the d x people that died that year need to get the benefit payment of 1 unit, but they receive it at the end of year k, so it has k 1 years to gain interest. This carries on till the end of year k when the d x+k 1 people that died receive the benefit payment of 1 unit, but with no years of interest as this is the end of the final year. We now need to work out the difference between these quantities to work out the loss to the insurer, which can simply be written as: l x (1 + i) k [A 1 x:k P x ä x:k ] (2.6.1) As we said before we now share this total accumulated amount amongst the l x+k survivors at time k by dividing (2.6.1) by l x+k to find the retrospective reserve at time k, which is given by the v symbol k V R (we also use the fact that x l x v x+k l x+k = Dx D x+k ). kv x R = = = = l x (1 + i) k [A 1 l P x:k x ä x:k ] x+k l x 1 l x+k v k [A1 P x:k x ä x:k ] l x v x l x+k v x+k [A1 P x:k x ä x:k ] D x [A 1 D P x:k x ä x:k ] x+k (2.6.2) 15

19 Example I shall now find the retrospective reserves for the term insurance and the endowment policies we considered in section The equations for the respective reserves for the term insurance and the endowment policies are respectively given by: kv 1 x:n R = V x:n R = D x D x+k [A 1 x:k P 1 x:n ä x:k ] D x D x+k [A 1 x:k P x:n ä x:k ] The retrospective reserves for each year are shown below, included is the prospective reserves calculated earlier for comparison. We can see from these results that: Table 2.6: Retrospective Reserves k kvx:n 1 R kvx:n 1 V x:n R V x:n The retrospective reserve for the term insurance (1st column) matches quite closely to the prospective reserve (2nd column) for the first 3 years, which is what the retrospective reserve set out to accomplish. After that the retrospective reserve increases faster than the prospective reserve and the difference between the two widens. This shows that the retrospective reserve is not beneficial to the insurer later into the contract and so surrendering the contract would probably not be possible. A similar trend can be seen between the retrospective reserve (3rd column) and prospective reserve (4th column) for the endowment policy. 16

20 2.7 The Expected Value and Variance of Loss Variables In this section equation (2.7.1) is from [Bowers, p.230]. Equations (2.7.2), (2.7.3), (2.7.4), (2.7.5) are from [Bowers, p ], but the derivations of (2.7.3), (2.7.4) have been extended by me for greater clarity. Table 2.7 is from [Bowers, p.233]. Equations (2.7.6), (2.7.8), (2.7.9) and (2.7.10) are from [Bowers, p ], but (2.7.8), (2.7.9) and (2.7.10) have been extended by me for greater clarity. Table 2.8 is from [Bowers, p.242]. In this section I am going to find formulae for the expected value and variance of different loss variables Prospective Loss The first loss variable I am going to look at is the prospective loss, touched on in section 1.1. I shall consider a general discrete whole life insurance where: 1. The benefit payment is payable at the end of year of death. 2. The benefit payment in the jth year is c j, j = 1, 2, The premium payments are paid annually at the beginning of the year. 4. The premium payment in the jth year is Π j 1, j = 1, 2,.... Similar to before, the prospective loss in year k, k L, is the present value of future benefit payments less the present value of future premium payments. Expressed as a function of k this is: { 0 K = 0, 1,..., k 1 kl = c K+1 v K+1 k K j=k Π j v j k (2.7.1) K = k, k + 1,... The expected value of the prospective loss, E( k L K k) is by definition the net premium reserve k V. I will leave Var( k L K k) until later when we have seen the other loss variables The Insurer s Net Cash Loss The next loss variable I am going to look at is the insurer s net cash loss within each insurance year for the general discrete life insurance defined above. Table 2.7 is a time diagram that illustrates the annual cash income and expenses for the insurer. Table 2.7: The Insurer s Annual Cash Income and Expenses Year K 1 K K + 1 K + 2 K + 3 Expenses c K Income Π 0 Π 1 Π 2 Π K 1 Π K etc... Let C k be the present value, at the beginning of year k, of the net cash loss during the year (k, k + 1). If (k, k + 1) is after the year of death (i.e. K(x) < k) then, the insured has already died and so no premium payments are made and the benefit payment has already been paid, so C k = 0. If (k, k + 1) is the year of death (i.e. K(x) = k) then the premium payment, Π k, was paid at the start of the year and the benefit payment, c k+1, is paid by the insurer at the end of the year, so C k = c k+1 v Π k. If (k, k + 1) is before the year of death (i.e. K(x) > k) then, the insured is still alive and has to pay the premium Π k at the start of the year, so C k = Π k. The explicit function of K is shown below: 17

21 0 K = 0, 1,..., k 1 C k = c k+1 v Π k K = k Π k K = k + 1, k + 2,... For the conditional distribution of C k, given that K k, we see that: (2.7.2) C k = v c k+1 I Π k where I is again an indicator function but this time: { 1 with probability qx+k I = 0 with probability p x+k This is the case since C k only takes the value v c k+1 in year k when the insured dies and that occurs with probability q x+k. Therefore the expectation of C k is: E(C k K k) = v c k+1 q x+k Π k (2.7.3) Using that: then the variance is: Var(x) = E(x 2 ) [E(x)] 2 Var(C k K k) = E[(C k ) 2 K k] [E(C k K k)] 2 = E[(v c k+1 I Π k ) 2 ] (v c k+1 q x+k Π k ) 2 = E(v 2 c 2 k+1 I 2 2v Π k c k+1 I + Π 2 k) v 2 c 2 k+1 q 2 x+k + 2v Π k c k+1 q x+k Π 2 k = E(v 2 c 2 k+1 I) 2v Π k c k+1 q x+k + Π 2 k v 2 c 2 k+1 q 2 x+k + 2v Π k c k+1 q x+k Π 2 k = v 2 c 2 k+1 q x+k v 2 c 2 k+1 q 2 x+k = v 2 c 2 k+1 q x+k v 2 c 2 k+1 q x+k (1 p x+k ) = v 2 c 2 k+1 q x+k v 2 c 2 k+1 q x+k + v 2 c 2 k+1 q x+k p x+k = v 2 c 2 k+1 q x+k p x+k (2.7.4) By rearranging the terms in the definition of the prospective loss in (2.7.1), we find an equivalent one that states k L as the sum of the present value of the insurer s future net annual losses at year k. This gives us: kl = v j k C j (2.7.5) j=k We can verify (2.7.5) for K k by substituting in (2.7.2), this is shown explicitly below: kl = v j k C j j=k Obviously we get (2.7.1) as expected. K 1 = v K k (v c K+1 Π k ) v j k Π j = c K+1 v K+1 k j=k K Π j v j k j=k 18

22 2.7.3 Allocation of the Risk The next loss variable I am going to look at explores the allocation of the risk within each year of the insurance policy. It is very similar to the net cash loss looked at previously, but now it also includes the change in liability for the insurer each year. The change in liability, as mentioned before, is basically the change in the net premium reserve over each year, this can be thought of as the extra money the insurer needs to set aside each year to cover the premium payment. I will now look at the risk to the insurer within each insurance year for the general discrete life insurance defined above, Table 2.8 is a time diagram that illustrates the annual cash income, expenses and change in liability for the insurer. Table 2.8: The Insurer s Annual Cash Income, Expenses and Change in Liability Year K K + 1 K + 2 Expenses c K+1 0 Income Π 0 Π 1 Π 2 Π K 0 0 etc... Liability 1V 2V (1 + i) 1 V KV (1 + i) K 1 V K V 0 Let Λ k be the present value, at the beginning of year k, of the net cash loss and change in liability during the year (k, k + 1). If (k, k + 1) is after the year of death (i.e. K(x) < k) then, the insured has already died and the benefit payment has already been paid, so the insurer has no liability and as before C k = 0 so Λ k = 0. If (k, k + 1) is the year of death (i.e. K(x) = k) then the insurer has the reserve from the beginning of the year, k V, as the premium payment was paid by the insured, but there is no reserve needed at the end of the year for next year as the benefit payment has already been made, so Liability = k V. C k = c k+1 v Π k for the reasons described previously. The change in liability has to be discounted back to the start of the year so Λ k = c k+1 v Π k v k V. If (k, k + 1) is before the year of death (i.e. K(x) > k) then, the insured is still alive and the insurer has the reserve k V at the beginning of the year, as the premium payment was paid by the insured. It is also paid at the beginning of next year as the insured will survive to at least that point so the insurer now has reserve k+1 V. The insurer s change in liability for the year (k, k + 1), with discounting, is therefore Liability= v k+1 V k V. C k = Π k for the reasons described previously. This gives us Λ k = v k+1 V k V Π k. The explicit function of K is shown below: 0 K = 0, 1,..., k 1 Λ k = (c k+1 v Π k ) + ( k V ) K = k (2.7.6) ( Π k ) + (v k+1 V k V ) K = k + 1, k + 2,... We can write (2.7.6) in terms of the risk and savings premiums from (2.5.3) and (2.5.2): 0 K = 0, 1,..., k 1 Λ k = Π r k + (c k+1 k+1 V )v K = k Π r k K = k + 1, k + 2,... The conditional distribution of Λ k, given that K k, we see that: Λ k = v c k+1 I + v k+1 V J Π k k V (2.7.7) where I is again an indicator function : { 1 with probability qx+k I = 0 with probability p x+k 19

23 As is J but this is: This is the case as Λ k takes: { 1 with probability px+k J = 0 with probability q x+k v c k+1 in year k when the insured dies and that occurs with probability q x+k v k+1 V if the insured survives year k and that occurs with probability p x+k. The expectation of Λ k is given by: E(Λ k K k) = v c k+1 q x+k + v k+1 V p x+k Π k k V = 0 from (2.4.8) (2.7.8) The variance of Λ k is given by: Var(Λ k K k) = E[(Λ k ) 2 K k] [E(Λ k K k)] 2 = E[(v c k+1 I + v k+1 V J Π k k V ) 2 ] (v c k+1 q x+k + v k+1 V p x+k Π k k V ) 2 = E[v 2 c 2 k+1 I + 2v 2 c k+1 k + 1V IJ 2v c k+1 Π k I 2v c k+1 k V I + v 2 k+1v 2 J 2v k+1 V Π k J 2v k+1 V k V J + Π 2 k + 2Π k k V + k V 2 ] [E(Λ k K k)] 2 = v 2 c 2 k+1 q x+k + 0 2v c k+1 Π k q x+k 2v c k+1 k V q x+k + v 2 k+1v 2 p x+k 2v k+1 V Π k p x+k 2v k+1 V k V p x+k + Π 2 k + 2Π k k V + k V 2 v 2 c 2 k+1 q 2 x+k 2v 2 c k+1 k+1 V p x+k q x+k + 2v c k+1 Π k q x+k + 2v c k+1 k V q x+k v 2 k+1v 2 p 2 x+k + 2v k+1 V Π k p x+k + 2v k+1 V k V p x+k Π 2 k 2Π k k V k V 2 = v 2 c 2 k+1 q x+k + v 2 k+1v 2 p x+k v 2 c 2 k+1 q 2 x+k 2v 2 c k+1 k+1 V p x+k q x+k v 2 k+1v 2 p 2 x+k = v 2 c 2 k+1 q x+k v 2 c 2 k+1 q x+k + v 2 c 2 k+1 q x+k p x+k + v 2 k+1v 2 p x+k v 2 k+1v 2 p x+k + v 2 k+1v 2 p x+k q x+k 2v 2 c k+1 k+1 V p x+k q x+k = v 2 c 2 k+1 p x+k q x+k + v 2 k+1v 2 p x+k q x+k 2v 2 c k+1 k+1 V p x+k q x+k = v 2 (c k+1 k+1 V ) 2 p x+k q x+k (2.7.9) We can now express the prospective loss variable k L in terms of Λ k using the definition of the Λ k s and (2.7.5): v j k Λ k = v j k [C k + v Liability(k, k + 1)] j=k j=k = k L + v j k+1 Liability(k, k + 1) j=k If we look at Table 2.8 we can see that for the sum of all the discounted change in liabilities, most of the terms cancel each other out and we are simply left with 0 k V for the year (k, k + 1) 20

24 with all the years after this obviously having no change in liability. rearranging the above formula we have the relationship: Using this deduction and kl = { 0 K < k j=k vj k Λ j + k V K k We can now find Var( k L) using the simple identity: Var(Λ k K j) = Var(Λ k K k) k j p x+j for j k this can be simply explained as k j p x+j is just the probability that the insured survives to the year k, given that he has survived j years. We also need: Var(aX + b) = a 2 Var(X) Using these identities, the arrangement of k L above and (2.7.9) we get: Var( k L K k) = Var[ v j k Λ j + k V K k] = = = = j=k Var[v j k Λ j + k V K k] j=k v 2(j k) Var[Λ j K k] j=k v 2(j k) j kp x+k Var[Λ k K k] j=k v 2(j k) j kp x+k [v 2 (c k+1 k+1 V ) 2 p x+k q x+k ] j=k (2.7.10) Example These quantities for the term insurance and endowment example from section are shown in table 2.9 below: We can see from these results that: The expected net cash loss for the insurer for the term insurance (1st column) is negative, showing that the insurer expects to make money each year (as a negative loss is a profit), but the amount increases each year as more needs to be set aside just in case the insured dies. The variance of the net cash loss is the same for both the term insurance (2nd column) and the endowment (6th column) as it only depends on the sum insured and the probabilities of surviving the period, which are the same for each. We can see that the variance increases over time showing that the risk of the insurer losing money each year increases. The variance of the allocation of the risk (3rd column) increases each year showing that the term insurance becomes more risky each year as there is more at stake should the insured die, as the insurer would lose a lot of the money he could have been able to keep, in order to pay the benefit payment. The variance of the prospective loss (4th column) increases each year again showing that the term insurance becomes more risky. 21

25 Table 2.9: The Expected Value and Variance of Loss Variables for the Term Insurance and Endowment Term Insurance Endowment k E(C k ) Var(C k ) Var(Λ k ) Var( k L) E(C k ) Var(C k ) Var(Λ k ) Var( k L) The expected net cash loss for the insurer for the endowment (5th column) is negative and increases, showing that the insurer expects to make money each year (as a negative loss is a profit), and the amount decreases each year, showing that more profit is expected to be made. The variance of the allocation of risk (7th column) decreases each year showing that the endowment becomes less risky for the insurer each year because the insurer is more likely to have to make the benefit payment at the end of the period and so get the profit he was expecting. The variance of the prospective cash loss (8th column) decreases each year again showing that the endowment becomes less risky for the insurer. The expected value is of use to the insurer as it enables them to see statistically how much profit they are likely to make. The variance is probably more useful as it is a measure of the risk of the policy for the insurer as it shows the variability in the expected value. The insurer may not want to take on a policy with a high variance because, if something were to go wrong, i.e. the insured dies early into the contract, then there could be large losses for the insurer. We can therefore see that from the insurer s point of view, being able to calculate the expected value and variance each year for the different loss variables is really useful. 2.8 Expense Loadings In this section I have used my notes from Actuarial Mathematics II which are based on [Gerber, p ]. As discussed in chapter 1.1, if the insurer wants to make a profit and cover the setup and running costs of the policy, they will have to take this into account when calculating the premium amounts for the insured to pay. The reason for looking at the expense loadings is that the resulting premiums and reserves will closely resemble the net premiums and reserves calculated earlier, so this is a basic extension of what has already been looked at. It also enables us to see what happens more often in the real world, where the insurer is looking to make a profit, rather than allowing the premiums to be net premiums, but also how they strive to be competitive with other insurance companies. 22

26 There are 3 main groups of expenses: Acquisition Expenses This expense takes into account all the costs associated with the setup of the policy e.g. policy writing, agents commission, medical exam and advertising. The expenses are proportional to the sum insured and are charged as a single amount. The equivalent premium rate, due every year, is given by α. Collection Expenses This expense is charged at the beginning of every year in which a premium is to be collected, it can be thought of as the expenses for an agent to travel to the insured for the annual payments, or in case of an electronic transfer a handling fee. The expenses are proportional to the expense-loaded premium, which is the net annual premium, when all the expenses have been taken into account. The equivalent premium rate, due every year, is given by β. Administration Expenses All other expenses are included in this group, such as data processing costs, wages, taxes and license fees. These expenses are charges at the beginning of each year for the entire length of the contract while the insured is still alive and is usually proportional to the sum insured. The equivalent premium rate, due every year, is given by γ. The expense-loaded premium (or adequate premium, hence the a in the superscript in the symbol, see later) is the overall amount of premium to be paid at the beginning of each year, taking into account both the sum insured and the expense loadings. Its expected present value should be just sufficient to cover the total of these amounts. The symbol P a is used to denote the expense-loaded premium and we denote the expected present value of the net annual premium and the premiums for acquisition expenses, collection expenses and administration expenses by P, P α, P β, P γ respectively. As the total of P, P α, P β, P γ equals the expense-loaded premium by definition then we get: P a = P + P α + P β + P γ (2.8.1) Example If we now try and find the expense loaded premiums for the term insurance and endowment from section As the expense-loaded premium is payable at the start of each year the insured is alive for the n year period, then it must have a factor of an n year temporary life annuity due, like the original premium payments did. The original premium for the term insurance,p x:n 1, can be written as P x:n 1 = A1 x:n /ä x:n by using the equivalence relation, see (2.3.3). The acquisition expense is a one off payment and as mentioned before the premium rate due each year is α. The collection expenses as mentioned before are proportional to P a and are due at the start of every year like the original premium so it also must have the factor of an n year temporary life annuity due. The administration expenses are proportional to the sum insured and is also due at the start of every year the insured is alive and as the contract ends after n years it is capped here too, so also has a factor of an n year temporary life annuity due. Taking all this into account we can write the expense-loaded premium for the term insurance as (N.b. I have written this as P a (Ax:n 1 ) purely for aesthetic purposes): P a (A 1 x:n ) ä x:n = A 1 x:n + α + β P a (A 1 x:n ) ä x:n + γ ä x:n 23

27 giving us: similarly for the endowment: P a (A 1 x:n ) = A1 x:n + α + γ ä x:n (1 β)ä x:n P a x:n = A x:n + α + γ ä x:n (1 β)ä x:n Let the insured be 50 years old, the duration of the policy be n = 10 years and the sum insured be 100 units as before. The net annual premiums were calculated before and are P 1 = :10 and P 50:10 = for the term insurance and endowment respectively. If we have an acquisition expense of 0.5% of the 100 units, a collection expense of 10% of each expense-loaded premium and an administration expense of 0.8% of the 100 units, for both policies then we get for the term insurance. P a (A 1 50:10 ) = 100(A 50:10 ) (ä 50:10 ) (1 0.1)ä 50:10 = = 1.86 Similarly for the endowment: P a x:n = = 9.79 We can see that the expense-loaded premiums are higher than the standard premiums for both policies, as expected, since the insurer is charging more to take into account the expenses Expense-Loaded Premium Reserves In this section (2.8.2) is from [Gerber, p.105] but its derivation has been extended by me for greater clarity. The expense-loaded premium reserve is similarly given by k V a. Unsurprisingly it is defined similarly to how the net premium reserve was, but it also takes into account the expense-loaded premiums. Its definition is therefore given by: the difference between the expected present value of future benefits and the expected present value of future expense-loaded premium reserves. Using this definition we can find the expense-loaded premium reserve for the whole life insurance policy used in section and substitute (2.8.1) for the expense-loaded premium P a x, which replaces the normal premium P x : kvx a = A x+k Px a ä x+k = A x+k (P x + Px α + Px β + Px γ ) ä x+k = A x+k P x ä x+k P α x ä x+k P β x ä x+k P γ x ä x+k = k V x + k V α x + k V β x + k V γ x Where k Vx α is the reserve for the acquisition expense, k Vx β is the reserve for the collection expense and k Vx γ is the reserve for the administration expense. 24

28 We can find expressions for each of the expense-loaded premium reserves in terms of the expense-loaded premiums: Proof of äx+k ä x kv α x = P α x ä x+k = 1 k V x can be found in (B.3.1). = α äx+k ä x = α(1 k V x ) kvx β = Px β ä x+k = β Px a ä x+k (2.8.2) kvx γ = Px γ ä x+k = γ ä x+k Example Using the above formulae we can now find the expense-loaded premium reserves for the term insurance and endowment from section As before, if we have an acquisition expense of 0.5% of the 100 units, a collection expense of 10% of each expense-loaded premium and an administration expense of 0.8%, for both policies. The results are shown in table 2.10 below: k kv 1 x:n kv 1 α x:n Table 2.10: Expense-Loaded Premium Reserves Term Insurance kv 1 β x:n kv 1 γ x:n Endowment kvx:n 1 a kv x:n k Vx:n α kv β x:n kv γ x:n kvx:n a We can see from the results that: The expense-loaded premium reserve for the term insurance (5th column) is less than the standard premium reserve (1st column) as expected, since the insurer is now receiving more money from the insured, so consequently needs to reserve less. What this example fails to take into account though, is that the insurer will not be able to keep all of this extra premium amount purely as profit, he will have to use some of it to pay the expenses incurred. This means that the reserves will need to be slightly higher than indicated by the expense-loaded premium reserve, to take this into account. The expense-loaded premium reserve for the endowment (10th column) is also less than the standard premium reserve (6th column) for the same reasons. 25

29 Chapter 3 Uses of Reserves 3.1 Introduction At the end of section 2.1 I gave some of the uses of reserves. I have already covered the surrender value, which is related to the retrospective reserves in section 2.6. I will now look at altering one part of an existing policy or converting a policy from one form to another. Also I will look at the reserves for with-profits policies. 3.2 Alterations and Conversions In this section the general principle is based on information from [Scott, p ], but the illustrating example is my own. If insurance companies want to stay competitive they need to offer policies that are flexible and allow the insured person to make changes to the policy, if their financial standing changes. Such alterations that are possible include: altering the amount of the premium payments, the duration of the premium payments, maturity time of endowment policies and completely changing from one form of insurance to another. Such changes often involve reducing the benefit payment in order to make sure that the remains fair for both the insurer and the insured. According to [Scott, p.126] the usual rule for carrying out these calculations is to equate the reserves before and after the conversion. If V 1 and V 2 are the reserves before and after the conversion respectively, then we get the following equality: V 1 = V 2 If there are expenses of F for the conversion itself, then the formula becomes: V 1 F = V 2 This is indeed correct because at the date of conversion, the insured s policy is worth V 1 on its immediate surrender. Then imagine that the amount V 1 is used as a net single premium to purchase the new contract, thus the equation of value of the new contract is: V 1 + expected value of future premiums = expected value of future benefits + F Rearranging this gives: V 1 F = expected value of future benefits expected value of future premiums 26

30 We notice that the right-hand side of this equation is in fact the net premium reserve formula, so is equal to V 2 as required. We can use this equation to solve for the unknown quantity, which is usually the new benefit payment or premium payments. Example Suppose a man aged 40 takes out a whole life insurance policy, with benefit payment of 1000 units paid at the end of year of death, and paid for by level annual premiums. After 10 years he wants to change the policy to a 10 year endowment policy with the same annual premium payments. We shall now work out the benefit payment, C, of the endowment policy. First we work out the annual premium payment for the whole life insurance policy, P. Using that the expected loss to the insurer at policy issue is 0 we get: 0 = 1000 A 40 P ä 40 Now rearranging this, using the commutation functions for the whole life and insurance and whole life annuity, which are given by (A.2.2) and (A.4.3) respectively and using the life tables found in appendix C we get: P = 1000 A 40 ä 40 = 1000 M 40 D 40 D 40 N 40 = 1000 M 40 N 40 = Now we work out the reserve just before the conversion: V 1 = 10 V 40 = 1000 A ä 50 = 1000 M50 D 50 = N50 D 50 We now use this amount as the reserve for the 10 year endowment policy, which has the same premium payments as before and benefit payment C, which we have to calculate. Again we use the commutation functions for the endowment policy and the n year temporary annuity due, which are given by (A.3.4) and (A.4.5) respectively: = C A 50: ä 50:10 = C(M 50 M 60 + D 60 ) (N 50 N 60 ) D 50 C = D (N 50 N 60 ) M 50 M 60 + D 60 =

31 3.3 With-profits policies In this section the general principle is based on information from [Scott, p ]. With-profit policies are life insurance policies where the insured has the right to share in the profits of the company, usually in the form of bonuses added to the benefit payment. As the benefit payments from these policies are usually higher than normal non-profit policies, especially if the company makes a lot of money from its investments, then the premium payments for these policies are also higher. According to [Scott, p.46], the bonuses come typically in two forms, reversionary bonuses, which are usually added annually and are not subsequently reduced, or terminal bonuses, which are paid only upon death and are not guaranteed to continue. Here I will only deal with reversionary bonuses only, as terminal bonuses are a lot more complicated to calculate since the bonuses are not guaranteed. Let us consider the whole life insurance policy used in section 2.2.1, but with benefit payment of C, which is paid with the total bonuses accumulated at the end of year of death. Again the policy is paid for by level annual premiums P x due at the start of each year. Each year reversionary bonuses are added to the benefit payment when the corresponding premium has been paid and we denote the total bonuses added to date by B. This gives a total benefit payment of C + B at the end of each year. According to [Scott, p.114], there are 3 main ways used to calculate the reserves for with-profit policies are: 1. The net premium method. 2. The bonus reserve (or gross premium) method. 3. The asset share method The Net Premium Method According to [Scott, p.115], the reserve is taken to be the net premium reserve for the corresponding non-profit policy, plus the mean present value of the bonuses already declared. For our whole life insurance policy considered above, the reserve is given by the formula: kv W P = C k V x + B A x+k = (C + B)A x+k C P x ä x+k The explanation for this method is that the additional premium amount (relative to the corresponding non-profit policy) is considered to have earned the bonus B so far, so an extra reserve is required to cover the value of these bonuses The Bonus Reserve Method In this section I have also used information from [Scott, p.48]. For this method the benefits and bonuses gain interest each year at a bonus rate b, which is known as compound bonuses, instead of the usual Annual Equivalent Rate of i. This means that the benefits, for our whole life insurance policy considered above, will be: 28

32 and (C + B)(1 + b) in year 1 (C + B)(1 + b) 2 in year 2 (C + B)(1 + b) k in year k This means that the net single premium is given by: ( ) k b (C + B) (1 + b) k+1 v k+1 kp x q x+k = (C + B) kp x q x+k 1 + i k=0 k=0 = (C + B)A x (3.3.1) Where A x is at rate of interest: This can be shown by using i = i b 1+b in v = 1 1+i : i b 1+b = i b 1 + b 1 1+b+i b 1+b = 1 + b 1 + i As required in (3.3.1) The reserve for our whole life insurance policy is now found exactly as in the net premium method, but obviously using the bonus rate of interest, this gives us: Asset Share Method kv W P = (C + B)A x+k C P x ä x+k The asset share of a life insurance according to [Scott, p.106] is a retrospective gross premium reserve calculated on the basis of the mortality, interest and expenses actually experienced by the office. A retrospective gross premium reserve is a reserve calculated by a retrospective accumulation of premiums (as shown in Section 2.6) less expenses and the cost of the benefit payment. We also allow for the possibility of expenses in the premium and reserving bases. The calculation of the asset share is quite complicated and so we can simplify this method by using the retrospective reserve without expenses, this was given by: kv x R = D x D x+k [P ä x:k A 1 x:k ] The with-profit reserve can now be simply calculated as before, by using the retrospective reserve and the mean present value of the bonuses already declared: kv W P = C k V x R + B A x+k These three methods vary in their approach to calculating the reserves for the with-profit policies, to match the variety of ways there are of adding the different types of bonuses, giving the insured the widest scope of choice for their policy. 29

33 Chapter 4 Applications of Reserves Beyond Life Insurance 4.1 Insurance Insurance has always been important to a developing society, as communal interest is crucial to the furthering of civilization and insurance basically involves financially sharing life s risks. The basic principle for insurance is that the company, known as the insurer agrees to pay out money, which is known as benefits, upon the occurrence of a specified event, which usually results in financial loss for a person. The person buying the insurance is known as the insured and they agree to pay amounts of money known as premiums for this cover. The contract between the two parties is known as an insurance policy. The purpose of insurance is to reduce the financial cost to individuals that arise from the occurrence of, usually negative, events. An insurance company will cover the insured for all or part of the financial losses that may occur due to these events. The pooling of liabilities by the insurance company makes the total losses more predictable, than they would be for each of the insured individuals separately, so reducing the overall risk. Also, as the insurance company has lots of resources and money to draw upon, they are able to take on more risky operations, which increases creativity, competition and efficiency in the marketplace. Insurance has been required by a variety of different people throughout time, for example: travelling merchants, who had the risk of having their goods stolen, needed insurance. Property owners needed insurance as they have the risk of fire and burglary. The main provider in any family always has the risk of dying prematurely, disability or infirmity leaving their dependents without a source of income. Also people sometimes live too long, use up all their savings and then become a burden to their family or society. 4.2 Actuaries As society develops, people and companies take more complicated risks and so businesses and people need a greater variety of insurance policies to protect their money. Professionals called actuaries now work with insurance companies to help them manage the financial implications of risk and uncertainty. They help evaluate the probability of events occurring and try to quantify the outcomes in order to minimise the financial losses associated with certain undesirable events. Since many events, such as death cannot totally be avoided, it is helpful to take measures to minimise their financial impact when they do occur. They also need to make sure that the premiums and investment earnings of these policies are adequate to provide for the payment of the benefits. The difficulty for actuaries is that there are many areas of uncertainty, the amount and timing of when the benefits need to be paid, along with the investment earnings, are unknown and are subject to 30

34 random fluctuations. Another complication is that the actuaries need to make sure that the rates they charge for a policy are perceived as being fair in order to attract business, but also make a profit for the insurer. Actuaries working in the life insurance sector deal with life insurance, health insurance and pensions. They look at mortality and try to construct models that accurately predict the life expectancy of a person given certain lifestyle details about them. They also have to take into account medical advances, inflation, politics and living costs when designing their models and the insurance policies, as these also have an influence on the risk. Casualty actuaries work in the general insurance field and deal with catastrophic, unnatural risks that can occur to people or their property. They design policies for car insurance, house insurance and liability insurance. Their analysis involves calculating the probability of a loss, called the frequency and the size of the event, called the severity. Also it is useful for them to calculate the expected time of the loss, as it will give the insurer a good indication of how long they have to generate the required benefit payment. Actuaries don t just work in insurance, their understanding of uncertainty and finance is very valuable to companies and so they are also employed by professional services companies to give advice on investments, mergers and pension schemes. They also are involved in reporting the companies assets and liabilities to the government to make sure the company is making a profit, but is doing so fairly. For both life and casualty actuaries their main role is to calculate the premiums and reserves for insurance policies so that they cover the various risks. Premiums, as mentioned before, are the amount of money the insurer needs to collect from the insured to cover the expected loss, expenses and a margin for profit. 4.3 Uses of Reserves In this section I have used information from [Wiser], any direct quotation from the text is given in quotation marks. Reserves are very important to actuaries as they indicate how much money should be set aside to cover future benefit payments. Actuaries have lots of different types of reserves to calculate, depending on the nature of the insurance dealt with by the insurer. Before exploring some of these types of reserves, I am going to cover some basic accounting principles to illustrate the importance of reserves and how they show the financial strength and stability of an insurance company The Balance Sheet The balance sheet reports on the financial position of the company at a specific point in time. It shows the level of assets and liabilities and the status of the shareholders equity. Assets are any economic resource that is held by the company, they could be cash, stocks, bonds or real estate for example. Liabilities are claims on the resources of the company and that need to be paid to satisfy the obligations to their clients. They could be mortgages, bank debt and benefits that need paying for example. Owners equity is the owners claim on the assets of the firm and they are always lower than all other liabilities of the company. Owners equity is also called the surplus of the insurer. For a publicly owned company the stockholders are the owners, but for a mutual insurer owned by its members, the policyholders are the owners and this surplus belongs to them. These three are related by the following equation, which is given on the balance sheet: Assets = Liabilities + Owners Equity (4.3.1) 31

35 If liabilities exceed the assets available then the value of the owner s equity is negative and the company is in a state known as insolvency The Income Statement The income statement measures changes in the owners equity during a stated period of time. The income statement measures the company s performance in this period as follows: Income = Revenue Expense Revenue measures the gaining of assets from the company s products or services. The expense measures the loss of assets that used up in order to provide those products or services. The owners equity can be divided into 2 parts; the capital contributed by the owners and any earnings retained by the company from past periods. This gives us: Owners Equity = Contributed Capital + Retained Earnings (4.3.2) The income can be used either to increase the owners equity (i.e. increase retained earnings) or be given out to the owners as dividends. This can be written as: Income = Change in Retained Earnings + Dividends to Owners (4.3.3) By rearranging and combining (4.3.1), (4.3.2) and (4.3.3) we get: Income = Change in Assets Change in Liabilities Change in Contributed Capital + Dividends to Owners Loss Reserves One of the types of reserves actuaries need to be able to calculate is called loss reserves. Loss reserves are the actuarial process of estimating an insurance company s liabilities for loss and their loss adjustment expenses, which are used in the balance sheet and income statements above. Loss reserves are challenging for causality actuaries to calculate because, the process involves complex technical calculations and important judgment too, as no single formula will provide the correct answer. According to [Wiser], there are five parts to a loss reserve: 1. Case reserves assigned to specific claims. 2. A provision for future development on known claims. 3. A provision for claims that re-open after they have been closed. 4. A provision for claims that have occurred but have not yet been reported to the insurer. 5. A provision for claims that have been reported to the insurer but have not yet been recorded. One of the reasons why loss reserves need to be estimated is due to the delay that occurs from when a loss occurs, to when it is reported and when it is finally settled. This has not been accounted for in the calculation of my net premiums reserves for simplicity, but the reader should be aware this happens. Dates therefore are very important in keeping track of the loss reserve estimation process. 32

36 According to [Wiser], five key dates are: 1. Accident date: the date on which the loss occurred. 2. Report date: the date on which the loss is first reported to an insurer. 3. Recorded date: the date on which the loss is first recorded in the insurer s statistical information. 4. Accounting date: the date used to define when the group of claims is to be included in the liability estimate. A loss reserve is an estimate of the liability for unpaid claims as of a given date, called the accounting date. An accounting date may be any date and is generally a date for which a financial statement is prepared, such as a month end, quarter end or year-end. 5. Valuation date: the date as of which the evaluation of the loss liability is made. The valuation date defines the point in time through which all the transactions are to be included for a group of claims. The valuation date can be before, after or at the same time as the accounting date. As the loss reserve is only an estimate and as the date of its calculation is important, since its value will change depending when it is calculated, there is some conventional terminology used to distinguish between the results of the loss reserve process, which according to [Wiser] is: The required loss reserve as of a given accounting date is the amount that must ultimately be paid to settle all claim liabilities. The value of the required loss reserve can only be known when all claims have been settled. Thus, the required loss reserve as of a given accounting date is a fixed number that does not change at a different valuation date. However, the value of the required loss reserve is generally unknown for an extremely long period of time. The indicated loss reserve is the result of the actuarial analysis of a reserve inventory as of a given accounting date conducted as of a certain valuation date. The indicated loss reserve is the analyst s opinion of the amount of the required loss reserve. This estimate will change with successive valuation dates and will converge to the required loss reserve as the time between valuation date and the accounting date of the inventory increases. The carried loss reserve is the amount of unpaid claim liability shown on external or internal financial statements. The loss reserve margin is the difference between the carried reserve and the required loss reserve. Since the required reserve is an unknown quantity we only have an indicated margin. The indicated loss reserve margin is defined to be the carried loss reserve minus the indicated loss reserve. One should not generally expect the margin to be zero, since for any subset of an entity s business it is unlikely that the carries loss reserve will be identical to either the indicated or required loss reserve. As the process of estimating the loss reserve is complicated and involves using lots of actuarial and accounting methods that have not been explored in this project, the details of how the estimate can be found will be excluded. For further details see [Wiser]. 33

37 4.3.4 Adequacy Testing In this section I have used information from [Fisher], any direct quotation from the text is given in quotation marks. Following on from loss reserves, another requirement of causality actuaries in particular, as their reserves are not usually given by set actuarial formulae but more often established by claims adjusters on an individual case basis, is to test the adequacy of the reserves. These tests are very important for assessing the financial solvency of the company and are carried out on a statistical basis, reviewing a whole portfolio of cases at the same time. Government organizations like the Financial Services Authority (F.S.A.), who regulate all providers of financial services, make sure companies like insurers are making reasonable levels of profit and have enough reserves to cover their policies and actions. Using the results of these tests the insurers can decide whether it may be necessary to increase or decrease existing reserves, add special bulk reserves, or issue new instructions redirecting the claims adjusters in the setting of reserves. One method now used, called the report year approach, was developed more than 30 years ago and was based on an approach known as the accident year method. It was designed to reveal whether the reserve is adequate and to measure: The extent of any redundancy or inadequacy. The slippage or strengthening of the equity position of the reserve since last evaluation. The contribution of various report years to the overall position. The first two results are significant in the financial position of the company in that the first one deals with solvency while the second deals with the possible distortions in the income statement. The third result is of value in the administration of the claim department, in that it tells us whether any redundancy is due to old or new cases. In other words, it indicates where corrective action is needed and in latter evaluations it monitors that corrective action Reinsurance In this section I have used information from [Patrik], any direct quotation from the text is given in quotation marks. Another important use for reserves is in the calculation of reinsurance. Reinsurance is a form of insurance, where the an insurer looks to another insurance company to cover them for part or all of the insurance policy they are offering their client. The terminology used is that the reinsurer assumes the liability surrendered on the subject policies. The cession, or share of claims to be paid by the reinsurer, may be defined on a proportional basis (a specified percentage of each claim) or on an excess basis (part of each claim above some specified monetary amount). As mentioned before for insurance, reinsurance further helps reduce the financial costs to insurance companies arising from the possible claims and also helps to spread the risk. This promotes more innovation, competition and efficiency in the marketplace. More specific reasons for the insurer to take out reinsurance are to increase their capacity, stability, financial management and for management advice. Capacity: By having reinsurance cover, an insurer can offer insurance policies with larger benefit payments while still maintaining a manageable risk level. By doing this the total net loss exposure can be kept in line with the insurer s surplus. This enables smaller insurance companies to compete with larger insurers. 34

38 Stabilisation: Reinsurance can help stabilise the insurer s financial results over time and shield them from large unexpected losses. Reinsurance can be written such that the insurer has to pay the smaller predictable claims, but shares the cost of the larger more infrequent claims, thus decreasing the insurer s chance of financial ruin. Financial results management: Reinsurance can be used to make a company look like it is performing better, by altering the timing of income, enhancing surplus and improving the various financial ratios by which insurers are judged. Insurers are able to use reinsurance to give up part of their liability and make use of the reinsurer s surplus. This can be seen as a loan of surplus from the reinsurer to the insurer, which they could use to take on new business or finance existing policies. Management advice: When reinsurance is taken out, the reinsurer will have to critically review the insurer s operation to make sure that they are taking on something that they can cope with. In this process the reinsurer may give the insurer advice and assistance on pricing, loss prevention, claims handling, reserving and investment, to make sure that the contract they make is viable for themselves. This enables smaller insurers to greatly benefit from knowledge and advice from more experienced insurers. 35

39 Chapter 5 Conclusion Reserves have many interesting properties and play a vital role in modern insurance. They are one of the factors that ensure that insurance companies rarely go bankrupt and they also make the whole process of insurance feasible. This is because they enable insurance companies to keep track of extra amount of money they need to set aside, so that they can afford to pay the benefit payment. We also saw that they enable insurance companies to offer a wide range of insurance policies, they are important for assessing the financial position of a company and in reinsurance. In chapter 2 I derived the formula for the net premium reserve and we saw that the net premium reserve was basically the extra money the insurer needs that year, to ensure that the benefit payment can be paid. I then looked at some of the interesting properties of reserves and why they are useful to insurers. I studied various ways to express the reserve in terms of the premiums, this was useful because the actuary could consider the reserve in terms of the amount paid by the insured each year and therefore if they wanted to reserve only a certain proportion of the premium, then they can adjust other factors accordingly to take this into account. It also enabled us to see the portion of the benefit payment paid for by the premium payment that year and the surplus/deficit between the premium payment and the benefit payment at that point in time. I then looked at the recursive formula for the net premium reserve, which enables the insurer to offer policies that are flexible and that can be changed from one form to another. After that I examined the decomposition of the reserve and corresponding premium into the savings and risk premium. The savings premium was the part of the premium that accumulated interest only and is used by the insurer in later years to cover the deficit between the premium payments and the benefit payment. The risk premium the part used to fund a one-year term insurance to cover the net amount at risk, the benefit payment. This enables insurers to know how much of the premium that they receive can be invested and how much needs to cover the benefit payment. I next looked at retrospective reserves, which involved dividing the gains of the insurer each year between the survivors of a hypothetical large group. This had uses in surrender values, which is when the insured wants to leave an insurance policy and expects some compensation if they are in the early years of the policy. I then studied the expected value and variance of various loss variables, which enables insurers to have an indication of their expected loss/gain and the risk associated with that amount. Finally I looked at a way of factoring the insurer s expenses into the cost of the policy, in the form of expense loading. I also examined the expense-loaded premium reserves for completeness. This part gave an insight into how, in the real world, insurance companies have to factor their overheads and profit margins into their policies. In chapter 3 I looked at some of the uses of reserves for life insurance policies. I examined how the reserves are used when the insured may want to make alterations to his existing policy and I also looked at how reserves can be used to determine the level of extra bonuses, added to the benefit payment, in with-profit policies. This chapter showed the variety of life insurance policies that are available to people and the ways they are tailored to meet the requirements of a variety of different customers. 36

40 In chapter 4 I looked at why insurance is so important to society as a whole, as it encourages creativity, competition and efficiency in the marketplace. Also by pooling resources an insurance company is able to reduce the risk relative to an individual. I then looked at how specialists known as actuaries work in insurance companies to ensure financial stability. I also looked at some of the uses of reserves outside of life insurance. We saw how loss reserves are the actuarial process of estimating the amount of money an insurance company should set aside to cover their overall losses and expenses. Adequacy testing, which uses reserves, are used to assess whether an insurance company is making a reasonable level of profit and that they have enough reserves to cover their policies and actions. Finally I looked at when reserves are used by insurers to see if large risky policies are worth reinsuring, this is when an insurer looks to another insurance company to cover all or part of one of their policies or group of policies. In this project in order to make it accessible to 3H students and to explore the topic of reserves in enough detail, I have had to use some simplified assumptions that could be extended for greater accuracy and a closer fit to the actual events and occurrences of the real world. One such area is in the use of the life tables. These are very useful for a quick estimation of the number of people dying each year, but obviously these need to be updated each year, due to medical advances and the fact that people are living longer. Also a continuous distribution of the survival of a population is more realistic than the discrete distribution I have used, since a person can die at any time. There are studies and research into methods of modeling the future lifetime as this is very important for insurance and pension companies, because if people live longer than predicted then the company is very likely to make a loss. Other models that are used for many random quantities are known as deterministic and stochastic. In this project I have used deterministic models, which are ones where the insurer pretends it knows exactly how much they will have to pay out in benefit payments and then sets its premiums accordingly, to match this amount. This method is justified by the concept of the law of large numbers, which states that if a large enough group of people are insured, then the total number of losses is likely to be close to the predicted amount. The more advanced stochastic model assigns probabilities to the occurrence of these losses, so that the total loss can be more accurately predicted, especially for a smaller group. If you would like to know more on how these models can be used in insurance, I would recommend reading [Bowers] and [Promislow]. Acknowledgements I would like to thank Prof. Frank Coolen for his guidance and advice throughout this project. I would also like to thank Nicola Talbot for her guide L A TEX for Complete Novices, which has been invaluable to me in creating this project and understanding how to use L A TEX. 37

41 Appendix A Notation Here I am going to cover the notation first introduced in the 2nd year mathematics course Actuarial Mathematics II with help from [Gerber]. A.1 Lifetime Models The future lifetime of a person aged x is denoted by (X). As the time of death of the person is random, the future lifetime of the person can be modeled by a probability distribution T (X). The cumulative distribution of T (X) is given by: G x (t) = Pr(T t) t 0 (A.1.1) G x (t) can be interpreted as the probability of a life aged x dying within t years. Further notation covered in the 2nd year course is given below: tq x = G x (t) Equation (A.1.2) denotes the probability that a life aged x dies within t years. tp x = 1 G x (t) Equation(A.1.3) the probability that a life aged x survives at least t years. (A.1.2) (A.1.3) s tq x = s+t q x s q x (A.1.4) Equation(A.1.4) denotes the probability that a life aged x will survive s years and then die in the next t years. tp x+s = s+t p x sp x (A.1.5) Equation(A.1.5) denotes the conditional probability that a person will survive t years, given that they have attained the age x + s. µ x+t = g(t) 1 G(t) = d ln[1 G(t)] (A.1.6) dt Equation(A.1.6) is the force of mortality of (x) at the age x + t. i

42 The curtate future lifetime of X is defined as the number of completed years lived by x, and has the symbol K(X). Pr(K = k) = Pr(k T < k + 1) = k p x q x+k E(K(x)) = e x = k Pr(K = k) = k k p x q x+k k=1 k=1 (A.1.7) (A.1.8) A.2 Life Insurance I will only give the net single premiums of the life insurance policies as the construction of these, from the present value, was shown in the 2nd year course. Whole life insurance: payment of 1 unit when death occurs. A x = v k+1 kp x q x+k k=0 The commutation function for use with the life tables is: A x = M x D x (A.2.1) (A.2.2) Ā x = 0 v t tp x µ x+t dt (A.2.3) (A.2.1) has the benefit payment at the end of year of death and (A.2.3) has its at the moment of death. Term life insurance: payment of 1 unit if death occurs within n years. n 1 Ax:n 1 = v k+1 kp x q x+k k=0 (A.2.4) A 1 x:n = M x M x+n D x (A.2.5) n year deferred whole life insurance: payment of 1 unit, if the insured survives n years, at the end of year of death. n A x = v k+1 kp x q x+k k=n n A x = M x+n D x (A.2.6) (A.2.7) ii

43 A.3 Endowments Pure endowment of n years: payment of 1 unit, if the insured survives n years, at the end of year n. A 1 x:n = v n np x (A.3.1) A 1 x:n = D x+n D x (A.3.2) n year endowment: payment of 1 unit if death occurs within first n years, else payment is at the end of year n. n 1 A x:n = v k+1 kp x q x+k + v n np x k=0 A x:n = M x M x+n + D x+n D x (A.3.3) (A.3.4) A.4 Life Annuities Remember that the present value of an annuity-due with n annual payments of 1 unit is given by: ä n = 1 + v + v v n 1 (A.4.1) Whole life annuity: annual payment of 1 unit as long as the beneficiary lives, paid at the start of the year. ä x = ä k+1 kp x q x+k k=0 (A.4.2) ä x = N x D x (A.4.3) n year temporary life annuity due: payment of 1 unit at the start of years 1,...,n if the beneficiary is alive. n 1 ä x:n = ä k+1 kp x q x+k k=0 (A.4.4) ä x:n = N x N x+n D x (A.4.5) n year deferred life annuity due: payment of 1 unit at the start of the year as long as the beneficiary lives, if they survived n years. n ä x = ä k+1 kp x q x+k k=n n ä x = N x+n D x (A.4.6) (A.4.7) iii

44 Appendix B Useful Equations B.1 Proofs of Equations used in Section 2.3 Remember that the present value of a perpetuity due with annual payments of 1 unit is given by: ä = 1 + v + v 2 + = v k = 1 1 v = 1 (B.1.1) d Then we can think of an annuity due as the difference between two perpetuities, 1 starting at time 0, the other at time n, then: k=0 ä n = ä (v n + v n+1 + v n+2 + ) = ä v n ä = 1 d vn 1 d = 1 vn d If we then let n = k + 1 and take the expected value of both sides we get: (B.1.2) E(ä k+1 ) = E( 1 vk+1 ) d ä x = E(1) E(vk+1 ) E(d) ä x = 1 A x d (B.1.3) Use the expected loss equation for a whole life insurance policy at policy issue for sum insured 1 (as given by (1.1.1)), we see that: 0 = A x P x ä x A x = P x ä x (B.1.4) If we then combine (B.1.3) and (B.1.4) with x = x + k we get: d = P x+k + d = d = 1 A x+k ä x+k 1 P x+k ä x+k 1 ä x+k (B.1.5) iv

45 If we also combine rearranged versions of (B.1.3) and (B.1.4) with x = x + k then: d = 1 A x+k ä x+k d = (1 A x+k)p x+k A x+k (B.1.6) B.2 Proofs of Equations used in Section 2.6 We use the lifetime equations in conjunction with the life table notation to give us the following identities: kp x = l x+k l x q x+k = 1 p x+k = 1 l x+k+1 l x+k = l x+k l x+k+1 l x+k = d x+k l x+k If we multiply these together and rearrange we get: kp x q x+k = l x+k l x d x+k l x+k = d x+k l x d x+k = l x k p x q x+k B.3 Proofs of Equations used in Section ä x+k ä x = 1 k V x This can be shown by using A x+k = d ä x+k 1 from (B.1.5) and d + P x = 1 ä x (B.1.5). again from 1 k V x = 1 A x+k + P x ä x+k = d ä x+k + P x ä x+k = ä x+k (d + P x ) = äx+k ä x (B.3.1) v

46 Appendix C Life Tables These life tables are edited versions of the ones from [Gerber, Appendix E]. Since they contain all the numbers I have used to complete my examples, they will enable the reader to try them for themselves. Table C.1: Illustrative Life Tables Basic Functions and Net Single Premiums at i = 5% x l x d x D x N x M x x 0 10,000, ,200 10,000, ,427, , ,795,800 13,126 9,329, ,427, , ,782,674 11,935 8,873, ,097, , ,770,739 10,943 8,440, ,224, , ,759,796 10,150 8,029, ,784, , ,749,646 9,555 7,639, ,754, , ,740,091 9,058 7,268, ,115, , ,731,033 8,661 6,915, ,847, , ,722,372 8,458 6,580, ,932, , ,713,914 8,257 6,261, ,351, , ,705,657 8,250 5,958, ,089, , ,697,407 8,243 5,669, ,131, , ,689,164 8,333 5,395, ,461, , ,680,831 8,422 5,133, ,066, , ,672,409 8,608 4,885, ,932, , ,663,801 8,794 4,648, ,047, , ,655,007 8,979 4,423, ,398, , ,646,028 9,164 4,208, ,975, , ,636,864 9,348 4,004, ,767, , ,627,516 9,628 3,809, ,762, , ,617,888 9,906 3,624, ,952, , ,607,982 10,184 3,448, ,327, , ,597,798 10,558 3,281, ,879, , ,587,240 10,929 3,121, ,598, , ,576,311 11,300 2,969, ,476, , vi

47 x l x d x D x N x M x x 25 9,565,011 11,669 2,824, ,507, , ,553,342 12,133 2,686, ,682, , ,541,209 12,690 2,555, ,996, , ,528,519 13,245 2,430, ,440, , ,515,274 13,892 2,311, ,009, , ,501,382 14,537 2,198, ,698, , ,486,845 15,274 2,090, ,499, , ,471,571 16,102 1,987, ,409, , ,455,469 16,925 1,889, ,421, , ,438,544 17,933 1,796, ,531, , ,420,611 18,935 1,707, ,734, , ,401,676 20,120 1,623, ,027, , ,381,556 21,390 1,542, ,403, , ,360,166 22,745 1,465, ,861, , ,337,421 24,277 1,392, ,395, , ,313,144 25,891 1,322, ,002, , ,287,253 27,676 1,256, ,679, , ,259,577 29,631 1,193, ,423, , ,229,946 31,751 1,132, ,230, , ,198,195 34,125 1,074, ,097, , ,164,070 36,656 1,019, ,022, , ,127,414 39, , ,002, , ,088,075 42, , ,035, , ,045,725 45, , ,118, , ,000,135 49, , ,248, , ,950,994 52, , ,424, , ,898,004 57, , ,643, , ,840,879 61, , ,904, , ,779,258 66, , ,205, , ,712,711 71, , ,544, , ,640,918 77, , ,919, , ,563,495 83, , ,328, , ,480,001 90, , ,771, , ,389,943 97, , ,245, , ,292, , , ,750, , ,188, , , ,284, , ,075, , , ,846, , ,954, , , ,434, , ,823, , , ,048, , ,684, , , ,686, , vii

48 x l x d x D x N x M x x 65 7,534, , , ,347, , ,373, , , ,031, , ,201, , , ,737, , ,018, , , ,463, , ,823, , , ,209, , ,616, , , ,973, , ,396, , , ,756, , ,164, , , ,555, , ,920, , , ,372, , ,664, , , ,204, , ,396, , , ,050, , ,117, , , , , ,828, , , , , ,530, , , , , ,225, ,810 89, , , ,914, ,330 78, , , ,600, ,514 69, , , ,284, ,041 60, , , ,970, ,770 51, , , ,660, ,506 44, , , ,358, ,168 37, , , ,066, ,802 31, , , ,787, ,539 25, , , ,524, ,675 20, , , ,281, ,592 16, , , ,058, ,815 13, , , , ,973 10, , , , ,749 7, , , , ,890 5, , , , ,096 4, , , ,988 84,006 2, , , ,982 74,894 1, , , ,088 66,067 1, , , ,021 49, ,732 23, viii

49 Bibliography [Gerber] [Bowers] [Scott] [Wiser] [Patrik] [Fisher] [Slud] Life Insurance Mathematics, Hans U. Gerber, with exercises contributed by Samuel H Cox, Sprinjer, (1997). Actuarial Mathematics, Newton L. Bowers Jr [et all], 2nd edition, Schaumburg III: Society of Actuaries, (1997). Life Contingencies, W.F. Scott, Document used to support the lecture syllabus at Herriot-Watt university, (1996). Loss Reserving, R.F. Wiser; J.E. Cockley and A. Gardner, Foundations of Casualty Actuarial Science (Fourth Edition), Casualty Actuarial Society, Chapter 5, pp , (2001). Reinsurance, G.S. Patrik, Foundations of Casualty Actuarial Science (Fourth Edition), Casualty Actuarial Society, Chapter 7, pp , (2001). Loss Reserve Testing: A Report Year Approach, W.H. Fisher and J.T. Lange, PCAS LX, pp , (1973). Actuarial Mathematics and Life-Table Statistics, Eric V. Slud, Chapter 6: Commutation Functions, Reserves & Select Mortality (PDF), (2001). [Promislow] Fundamentals of Life Insurance Mathematics,David S. Promislow, John Wiley & Sons Ltd, (2006). ix