# Annuities. Lecture: Weeks Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 1 / 43

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1 Annuities Lecture: Weeks 9-11 Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 1 / 43

2 What are annuities? What are annuities? An annuity is a series of payments that could vary according to: timing of payment beginning of year (annuity-due) with fixed maturity n-year annuity-due time n-1 n time more frequently than once a year annuity-due payable m-thly 1 m 1 m 1 m 1 m 1 m 1 m 0 1 m 2 m m 1 2 m 1 m payable continuously continuous annuity 2 time end of year (annuity-immediate) n-year annuity-immediate time n-1 n time annuity-immediate payable m-thly 1 m m 1 1 m m 1 m 1 1 m 0 1 m 2 m m 1 2 m 2 time time varying benefits Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 2 / 43

3 What are annuities? annuities-certain Review of annuities-certain annuity-due payable annually ä n = n v k 1 = 1 vn d annuity-immediate a n = n k=1 v k = 1 vn i payable m times a year ä (m) n = 1 m mn 1 v k/m = 1 vn d (m) a (m) n = 1 m mn v k = 1 vn i (m) k=1 ā n = continuous annuity n 0 v t dt = 1 vn δ Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 3 / 43

4 Chapter summary Chapter summary Life annuities series of benefits paid contingent upon survival of a given life single life considered actuarial present values (APV) or expected present values (EPV) actuarial symbols and notation Types of annuities discrete - due or immediate payable more frequently than once a year continuous varying payments Current payment techniques APV formulas Chapter 5 of Dickson, et al. Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 4 / 43

5 Whole life annuity-due Whole life annuity-due Pays a benefit of a unit \$1 at the beginning of each year that the annuitant (x) survives. The present value random variable is Y = ä K+1 where K, in short for K x, is the curtate future lifetime of (x). The actuarial present value of a whole life annuity-due is ä x = E[Y ] = E [ ] ä K+1 = ä k+1 Pr[K = k] = ä k+1 k qx = ä k+1 k p x q x+k Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 5 / 43

6 Whole life annuity-due current payment technique Current payment technique By writing the PV random variable as Y = I(T > 0) + vi(t > 1) + v 2 I(T > 2) + = one can immediately deduce that [ ] ä x = E[Y ] = E v k I(T > k) = = v k E[I(T > k)] = A straightforward proof of v k p k x = k v k I(T > k), v k Pr[I(T > k)] Ex = A 1 x: k. ä k+1 k qx = v k k p x is in Exercise 5.1. Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 6 / 43

7 Whole life annuity-due - continued Current payment technique - continued The commonly used formula ä x = payment technique for evaluating life annuities. v k p k x is the so-called current Indeed, this formula gives us another intuitive interpretation of what life annuities are: they are nothing but sums of pure endowments (you get a benefit each time you survive). The primary difference lies in when you view the payments: one gives the series of payments made upon death, the other gives the payment made each time you survive. Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 7 / 43

8 Whole life annuity-due some useful formulas Some useful formulas By recalling that ä K+1 = 1 vk+1, we can use this to derive: d relationship to whole life insurance [ ] 1 v K+1 ä x = E = 1 d d (1 A x). Alternatively, we write: A x = 1 dä x. very important formula! the variance formula Var[Y ] = Var [ ] 1 ä K+1 = d 2 Var [ v K+1] = 1 [ 2 d 2 A x (A x ) 2]. Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 8 / 43

9 Whole life annuity-due illustrative example Illustrative example 1 Suppose you are interested in valuing a whole life annuity-due issued to (95). You are given: i = 5%, and the following extract from a life table: x l x Express the present value random variable for a whole life annuity-due to (95). 2 Calculate the expected value of this random variable. 3 Calculate the variance of this random variable. Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 9 / 43

10 Other types temporary life annuity-due Temporary life annuity-due Pays a benefit of a unit \$1 at the beginning of each year so long as the annuitant (x) survives, for up to a total of n years, or n payments. The present value random variable is {ä Y = K+1, K < n ä n, K n = ä. min(k+1,n) The APV of an n-year life annuity-due can be expressed as ä x: n = E[Y ] = = n 1 ä k+1 p k x q x+k + ä n p n x using the current payment technique n 1 v k k p x. Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 10 / 43

11 Other types some useful formulas Some useful formulas Notice that Z = v min(k+1,n) is the PV random variable associated with an n-year endowment insurance, with death benefit payable at EOY. Similar to the case of the whole life, we can use this to derive: relationship to whole life insurance [ ] 1 Z ä x: n = E = 1 ( ) 1 Ax: d d n. Alternatively, we write: A x: n = 1 dä x: n. very important formula! the variance formula Var[Y ] = 1 d 2 Var[Z] = 1 [ 2 d 2 A x: n ( ) ] 2 A x: n. Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 11 / 43

12 Other types deferred whole life annuity-due Deferred whole life annuity-due Pays a benefit of a unit \$1 at the beginning of each year while the annuitant (x) survives from x + n onward. The PV random variable can be expressed in a number of ways: Y = { 0, 0 K < n ä = n K+1 n vn ä K+1 n = ä K+1 ä n, K n. The APV of an n-year deferred whole life annuity can be expressed as n äx = E[Y ] = k=n v k p k x = E n x ä x+n = ä x ä x: n. Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 12 / 43

13 Other types variance formula Variance of a deferred whole life annuity-due To derive the variance is not straightforward. The best strategy is to work with and use Y = { 0, 0 K < n v n ä K+1 n, K n Var[Y ] = E[Y 2 ] (E[Y ]) 2 = ) 2 ( v 2n ä k+1 n k qx k=n ä n x Apply a change of variable of summation to say k = k n and then the variance of a whole life insurance issued to (x + n). The variance of Y finally can be expressed as Var[Y ] = 2 d v2n n p x (äx+n ä 2 x+n ) 2 ) ( n ä x n äx). Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 13 / 43

14 Other types illustrative example Illustrative example 2 Suppose you are interested in valuing a 2-year deferred whole life annuity-due issued to (95). You are given: i = 6% and the following extract from a life table: x l x Express the present value random variable for this annuity. 2 Calculate the expected value of this random variable. 3 Calculate the variance of this random variable. Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 14 / 43

15 Other types recursive relationships Recursive relationships The following relationships are easy to show: ä x = 1 + vp x ä x+1 = E x ä x+1 = 1 + vp x + v 2 2 p x ä x+2 = E x + 2 E x ä x+2 In general, because E s are multiplicative, we can generalized this recursions to n 1 ä x = kex = kex + kex k=n apply change of variable k = k n = ä x: n + nex k Ex+n = ä x: n + n E k =0 = ä x: n + n E x ä x+n = ä x: n + ä n x x k k =0 E x+n The last term shows that a whole life annuity is the sum of a term life annuity and a deferred life annuity. Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 15 / 43

16 Other types parameter sensitivity äx i = 1% i = 5% i = 10% i = 15% i = 20% äx c = c = c = c = c = x x Figure : Comparing APV of a whole life annuity-due for based on the Standard Ultimate Survival Model (Makeham with A = , B = , c = 1.124). Left figure: varying i. Right figure: varying c with i = 5% Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 16 / 43

17 Life annuity-immediate whole life Whole life annuity-immediate Procedures and principles for annuity-due can be adapted for annuity-immediate. Consider the whole life annuity-immediate, the PV random variable is clearly Y = a K so that APV is given by a x = E[Y ] = Relationship to life insurance: Y = 1 i ( 1 v K ) = 1 i leads to 1 = ia x + (1 + i)a x. a K p k x q x+k = k=1 v k p k x. [ 1 (1 + i)v K+1 ] Interpretation of this equation - to be discussed in class. Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 17 / 43

18 Life annuity-immediate other types Other types of life annuity-immediate For an n-year life annuity-immediate: Find expression for the present value random variable. Express formulas for its actuarial present value or expectation. Find expression for the variance of the present value random variable. For an n-year deferred whole life annuity-immediate: Find expression for the present value random variable. Give expressions for the actuarial present value. Details to be discussed in lecture. Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 18 / 43

19 Life annuities with m-thly payments Life annuities with m-thly payments In practice, life annuities are often payable more frequently than once a year, e.g. monthly (m = 12), quarterly (m = 4), or semi-annually (m = 2). Here, we define the random variable K x (m), or simply K (m), to be the complete future lifetime rounded down to the nearest 1/m-th of a year. For example, if the observed T = for a life (x) and m = 4, then the observed K (4) is Indeed, we can write K (m) = 1 m mt, where is greatest integer (or floor) function. Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 19 / 43

20 Life annuities with m-thly payments whole life annuity-due Whole life annuity-due payable m times a year Consider a whole life annuity-due with payments made m times a year. Its PV random variable can be expressed as Y = ä (m) K (m) +(1/m) = 1 vk(m)+(1/m) d (m). The APV of this annuity is Variance is E[Y ] = ä (m) x Var[Y ] = = 1 m v h/m h/mpx = 1 A(m) x d (m). h=0 [ ] Var v K(m) +(1/m) ( ) d (m) 2 = 2 (m) Ax ( ) A (m) 2 x ( ) d (m) 2. Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 20 / 43

21 Life annuities with m-thly payments some useful relationships Some useful relationships Here we list some important relationships regarding the life annuity-due with m-thly payments (Note - these are exact formulas): 1 = dä x + A x = d (m) ä (m) x + A (m) x ä (m) x = d d (m) äx 1 d (m) ( A (m) x A x ) = ä (m) 1 ä x ä (m) ( A (m) x A x ) ä (m) x = 1 A(m) x d (m) = ä (m) ä(m) A(m) x Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 21 / 43

22 Life annuities with m-thly payments other types Other types of life annuity-due payable m-thly n-year term PV random variable Y = ä (m) APV symbol E[Y ] = ä (m) x: n current payment technique other relationships = 1 m = ä (m) x min(k (m) +(1/m),n) mn 1 h=0 relation to life insurance = 1 d (m) [ n-year deferred PV random variable Y = v n ä (m) APV symbol E[Y ] = ä (m) n x current payment technique = 1 m v h/m n E x ä (m) x+n ] 1 A (m) x: n K (m) +(1/m) n h=mn other relationships = n E x ä (m) relation to life insurance = 1 [ d (m) E v h/m x+n = ä(m) x p h/m x I(K n) p h/m x n x A (m) n x ä (m) x: n ] Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 22 / 43

23 Life annuities with m-thly payments illustrative example Illustrative example 3 Professor Balducci is currently age 60 and will retire immediately. He purchased a whole life annuity-due contract which will pay him on a monthly basis the following benefits: \$12,000 each year for the next 10 years; \$24,000 each year for the following 5 years after that; and finally, \$48,000 each year thereafter. You are given: i = 3% and the following table: x 1000A (12) x p 5 x Calculate the APV of Professor Balducci s life annuity benefits. Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 23 / 43

24 Continuous whole life annuity (Continuous) whole life annuity A life annuity payable continuously at the rate of one unit per year. One can think of it as life annuity payable m-thly per year, with m. The PV random variable is Y = ā T (x). where T is the future lifetime of The APV of the annuity: ā x = E[Y ] = E [ ā T ] = = 0 ā t p t x µ x+t dt use integration by parts - see page 117 for proof 0 v t p t x dt = 0 texdt Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 24 / 43

25 Continuous whole life annuity - continued One can also write expressions for the cdf and pdf of Y in terms of the cdf and pdf of T. For example, Pr[Y y] = Pr [ 1 v T δy ] [ ] log(1 δy) = Pr T log v Recursive relation: ā x = ā x: 1 + vp x ā x+1 Variance expression: Var [ [ ] ] 1 v T ā T = Var Relationship to whole life insurance: Ā x = 1 δā x δ = 1 [ 2 Ā δ 2 x ( ) ] 2 Ā x Try writing explicit expressions for the APV and variance where we have constant force of mortality and constant force of interest. Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 25 / 43

26 Continuous temporary life annuity Temporary life annuity A (continuous) n-year temporary life annuity pays 1 per year continuously while (x) survives during the next n years. {ā The PV random variable is Y = T, 0 T < n = ā ā n, T n min(t,n) The APV of the annuity: ā x: n = E[Y ] = + n n 0 ā t t p x µ x+t dt ā n p t x µ x+t dt = Recursive formula: ā x: n = ā x: 1 + vp x ā x+1: n 1. n 0 v t p t x dt. To derive variance, one way to get explicit form is to note that Y = (1 Z) /δ where Z is the PV r.v. for an n-year endowment ins. [details in class.] Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 26 / 43

27 Continuous deferred whole life annuity Deferred whole life annuity Pays a benefit of a unit \$1 each year continuously while the annuitant (x) survives from x + n onward. The PV random variable is { 0, 0 T < n Y = v n ā T n, T n { 0, 0 T < n =. ā T ā n, T n The APV [expected value of Y ] of the annuity is n āx = n E x ā x = ā x ā x: n = The variance of Y is given by Var[Y ] = 2 δ v2n n p x (āx+n ā 2 x+n n ) ( v t p t x dt. ) 2 n āx Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 27 / 43

28 Continuous special mortality laws Special mortality laws Just as in the case of life insurance valuation, we can derive nice explicit forms for life annuity formulas in the case where mortality follows: constant force (or Exponential distribution); or De Moivre s law (or Uniform distribution). Try deriving some of these formulas. You can approach them in a couple of ways: Know the results for the life insurance case, and then use the relationships between annuities and insurances. You can always derive it from first principles, usually working with the current payment technique. In the continuous case, one can use numerical approximations to evaluate the integral: trapezium (trapezoidal) rule repeated Simpson s rule Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 28 / 43

29 Other forms varying benefits Life annuities with varying benefits Some of these are discussed in details in Section You may try to remember the special symbols used, especially if the variation is a fixed unit of \$1 (either increasing or decreasing). The most important thing to remember is to apply similar concept of discounting with life taught in the life insurance case (note: this works only for valuing actuarial present values): work with drawing the benefit payments as a function of time; and use then your intuition to derive the desired results. Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 29 / 43

30 Evaluating annuities Methods for evaluating annuity functions Section 5.11 Recursions: For example, in the case of a whole life annuity-due on (x), recall ä x = 1 + vp x ä x+1. Given a set of mortality assumptions, start with ä x = v k k p x and then use the recursion to evaluate values for subsequent ages. UDD: deaths are uniformly distribution between integral ages. Woolhouse s approximations Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 30 / 43

31 Evaluating annuities UDD Uniform Distribution of Deaths (UDD) Under the UDD assumption, we have derived in the previous chapter that the following holds: A (m) x = i i (m) A x Then use the relationship between annuities and insurance: ä (m) x = 1 A(m) x d (m) This leads us to the following result when UDD holds: where ä (m) x = α (m) ä x β (m), α (m) = s (m) ä (m) = i 1 1 i (m) d d (m) β (m) = s(m) 1 d (m) 1 = i i(m) i (m) d (m) Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 31 / 43

32 Evaluating annuities Woolhouse s formulas Woolhouse s approximate fomulas The Woolhouse s approximate formulas for evaluating annuities are based on the Euler-Maclaurin formula for numerical integration: 0 g(t)dt = h g(kh) h h2 g(0) g (0) h4 720 g (0) + for some positive constant h. This formula is then applied to g(t) = v t p t x which leads us to g (t) = v t p t x (δ µ x+t ). We can obtain the following Woolhouse s approximate formula: ä (m) x ä x m 1 2m m2 1 12m 2 (δ + µ x) Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 32 / 43

33 Evaluating annuities Woolhouse s formulas Approximating an n-year temporary life annuity-due with m-thly payments Apply the Woolhouse s approximate formula to ä (m) x: n = ä(m) x n E x ä (m) x+n This leads us to the following Woolhouse s approximate formulas: Use 2 terms (W2) Use 3 terms (W3) ä (m) x: n ä x: n m 1 2m (1 n E x) ä (m) x: n ä x: n m 1 2m (1 n E x) Use 3 terms (W3*) m2 1 [δ + µ 12m 2 x n E x (δ + µ x+n )] use approximation for force of mortality (modified) µ x 1 2 [log(p x 1) + log(p x )] Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 33 / 43

34 Woolhouse s formulas numerical illustrations Numerical illustrations We compare the various approximations: UDD, W2, W3, W3* based on the Standard Ultimate Survival Model with Makeham s law µ x = A + Bc x, where A = , B = and c = The results for comparing the values for: ä (12) x: 10 ä (2) x: 25 with i = 10% with i = 5% are summarized in the following slides. Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 34 / 43

35 Woolhouse s formulas numerical illustrations Values of ä (12) x: 10 with i = 10% x ä x ä (12) x 10Ex Exact UDD W2 W3 W3* Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 35 / 43

36 Woolhouse s formulas numerical illustrations Values of ä (2) x: 25 with i = 5% x ä x ä (2) x 25Ex Exact UDD W2 W3 W3* Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 36 / 43

37 Woolhouse s formulas visualizing the differences Figure : Visualizing the different approximations for ä (2) x: 25 Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 37 / 43 Exact minus UDD Exact minus W age x age x Exact minus W Exact minus W3* age x age x

38 Woolhouse s formulas illustrative example Illustrative example 4 You are given: i = 5% and the following table: Approximate ä (12) 50: 3 1 UDD assumptions x l x µ x based on the following methods: 2 Woolhouse s formula using the first two terms only 3 Woolhouse s formula using all three terms 4 Woolhouse s formula using all three terms but approximating the force of mortality Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 38 / 43

39 Additional practice problems Practice problem 1 You are given: l x = 115 x, for 0 x 115 δ = 4% Calculate ä 65: 20. Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 39 / 43

40 Additional practice problems Practice problem 2 You are given: µ x+t = 0.03, for t 0 δ = 5% Y is the present value random variable for a continuous whole life annuity of \$1 issued to (x). [ Calculate Pr Y E[Y ] ] Var[Y ]. Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 40 / 43

41 Additional practice problems Practice problem 3 - modified SOA MLC Spring 2012 For a whole life annuity-due of \$1,000 per year on (65), you are given: Mortality follows Gompertz law with µ x = Bc x, for x 0, where B = and c = 1.1. i = 4% Y is the present value random variable for this annuity. Calculate the probability that Y is less than \$11,500. Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 41 / 43

42 Additional practice problems Practice problem 4 - SOA MLC Spring 2014 For a group of 100 lives age x with independent future lifetimes, you are given: Each life is to be paid \$1 at the beginning of each year, if alive. A x = A x = 0.22 i = 0.05 Y is the present value random variable of the aggregate payments. Using the Normal approximation to Y, calculate the initial size of the fund needed in order to be 95% certain of being able to make the payments for these life annuities. Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 42 / 43

43 Other terminologies Other terminologies and notations used Expression temporary life annuity-due annuity-immediate Other terms/symbols used term annuity-due n-year term life annuity-due immediate annuity annuity immediate Lecture: Weeks 9-11 (STT 455) Annuities Fall Valdez 43 / 43

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