ACTS 431 FORMULA SUMMARY Lesson 1: Probability Review 1. VarX)= E[X 2 ]- E[X] 2 2. V arax + by ) = a 2 V arx) + 2abCovX, Y ) + b 2 V ary ) 3. V ar X) = V arx) n 4. E X [X] = E Y [E X [X Y ]] Double expectation 5. V ar X [X] = E Y [V ar X [X Y ]] + V ar Y E X [X Y ]) Conditional variance Distribution, Density and Moments for Common Random Variables Continuous Distributions Discrete Distributions Name fx) F x) E[X] VarX) Exponential e x θ /θ 1 e x θ θ θ 2 1 x Uniform b a, x [a, b] a+b θ, x [a, b] 2 b a) 2 12 Gamma x α 1 e x θ /Γα)θ α x ft)dt αθ αθ2 Name fx) E[X] VarX) Poisson e λ λ x /x! λ λ Binomial ) p x 1 p) m x mp mp1 p) m x
2 Lesson 2: Survival Distributions: Probability Functions, Life Tables 1. Actuarial Probability Functions tp x - probability that x) survives t years tq x - probability that x) dies within t years t uq x - probability that x) survives t years and then dies in the next u years. t+up x = t p x u p x+t t uq x = t p x u q x+t = = t p x t+u p x = 2. Life Table Functions = t+u q x t q x 3. Mathematical Probability Functions tp x = S T x) t) tq x = F T x) t) d x = l x l x+1 nd x = l x l x+n q x = 1 p x = d x l x tp x = l x+t l x t uq x = l x+t l x+t+u l x t uq x = P r t < T x) t + u) = F T x) t + u) F T x) t) = = P r x + t < X x + t + u X > x) = F Xx + t + u) F X x + t) s X x) S x+u t) = S xt + u) S x u) S x t) = S x + t) S x) F x t) = F x + t) F x) 1 F x)
3 Lesson 3: Survival Distributions: Force of Mortality µ x = f x) S x) = d dx ln S x) µ x+t = f xt) S x t) = d dx ln S xt) t ) S x t) = t p x = exp µ x+s ds µ x t) = d tp x /dt = d ln t p x tp x dt f T x) t) = f x t) = t p x µ x t) t+u t uq x = t t tq x = sp x µ x s)ds sp x µ x s)ds = exp t If µ xt) = µ x t) + k for all t, then s p x = s p x e kt If µ x t) = ˆµ x t) + µ x t) for all t, then s p x = s ˆp xs p x If µ xt) = kµ x t) for all t, then s p x = s p x ) k ) µ x s)ds = exp x+t x ) µ s ds
4 Constant Force of Mortality Lesson 4: Survival Distributions: Mortality Laws µ x t) = µ tp x = e µt Uniform distribution De Moivre s law) 1 µ x t) = ω x t, t ω x tp x = ω x t ω x, t ω x tq x = t ω x, t ω x t uq x = u ω x, t + u ω x Beta distribution Generalized De Moivre s law) α µ x t) = ω x t ) ω x t α tp x =, t ω x ω x Gompertz s law: µ x = Bc x, c > 1 tp x = exp Bcx c t ) 1) ln c Makeham s law: Weibull Distribution µ x = A + Bc x, c > 1 tp x = exp At Bcx c t ) 1) ln c µ x = kx n S x) = e kxn+1 /n+1)
5 Complete Future Lifetime e x = e x = Lesson 5: Survival Distributions: Moments t t p x µ x+t dt tp x dt e x = 1 for constant force of mortality µ e x = ω x for uniform de Moivre) distribution 2 e x = ω x for generalized uniform de Moivre) distribution α + 1 [ E T x)) 2] = 2 t t p x dt V ar T x)) = 1 for constant force of mortality µ 2 ω x)2 V ar T x)) = for uniform de Moivre) distribution 12 ω x) 2 V ar T x)) = α + 1) 2 for generalized uniform de Moivre) distribution α + 2) n-year Temporary Complete Future Lifetime e x:n = e x:n = n n Curtate Future Lifetime e x = k k q x e x = t t p x µ x+t dt + n n p x tp x dt e x:n = n p x n) + n q x n/2) for uniform de Moivre) distribution e x:1 = p x +.5q x for uniform de Moivre) distribution e x:n = 1 e µn for constant force of mortality µ [ E T x) n) 2] n = 2 t t p x dt k=1 kp x k=1 e x = e x.5 for uniform de Moivre) distribution [ E Kx)) 2] = 2k 1) k p x k=1 V ar Kx)) = V ar T x)) 1 for uniform de Moivre) distribution 12
6 n-year Temporary Curtate Future Lifetime n 1 e x:n = k k q x + n n p x e x:n = k=1 n kp x k=1 e x:n = e x:n.5 n q x for uniform de Moivre) distribution [ E Kx) n) 2] n = 2k 1) k p x k=1
7 Lesson 6: Survival Distributions: Percentiles, Recursions, and Life Table Concepts Recursive formulas Life Table Concepts e x = e x:n + n p x e x+n e x:n = e x:m + m p x e x+m:n m, m < n e x = e x:n + n p x e x+n = e x:n 1 + n p x 1 + e x+n ) e x = p x + p x e x+1 = p x 1 + e x+1 ) e x:n = e x:m + m p x e x+m:n m = = e x:m 1 + m p x 1 + e x+m:n m ) m < n e x:n = p x + p x e x+1:n 1 = p x 1 + e x+1:n 1 ) T x = nl x = Y x = n e x = T x l x e x:n = n L x l x l x+t dt, l x+t dt, T x+t dt total future lifetime of a cohort of l x individuals total future lifetime of a cohort of l x individuals over the next n years Central death rate and related concepts nm x = n d x nl x q x m x = for uniform de Moivre) distribution 1.5q x nm x = µ x for constant force of mortality ax) = L x l x+1 d x ax) = 1 2 the fraction of the year lived by those dying during the year for uniform de Moivre) distribution
8 Lesson 7: Survival Distributions: Fractional Ages Function Uniform Distribution of Deaths Constant Force of Mortality Hyperbolic Assumption l x+s l x sd x l x p s x l x+1 /p x + sq x ) sq x sq x 1 p s x sq x /1 1 s)q x ) sp x 1 sq x p s x p x /1 1 s)q x ) sq x+t sq x /1 tq x ), s + t 1 1 p s x sq x /1 1 s t)q x ) µ x+s q x /1 sq x ) ln p x q x /1 1 s)q x ) sp x µ x+s q x p s xln p x ) p x q x /1 1 s)q x ) 2 m x q x /1.5q x ) ln p x qx/p 2 x ln p x ) L x l x.5d x d x / ln p x l x+1 ln p x /q x e x e x +.5 e x:n e x:1 e x:n +.5 n q x p x +.5q x
9 Lesson 8: Survival Distributions: Select Mortality When mortality depends on the initial age as well as duration, it is known as select mortality, since the person is selected at that age. Suppose q x is a non-select or aggregate mortality and q [x]+t, t =,, n 1 is select mortality with selection period n. Then for all t n, q [x]+t = q x+t.
1 Lesson 9: Insurance: Payable at Moment of Death - Moments - Part 1 Ā x = Actuarial notation for standard types of insurance e δt tp x µ x t) dt Name Present value random variable Symbol for actuarial present value Whole life insurance v T Ā x Term life insurance Deferred life insurance Deferred term insurance Pure endowment T n T > n v T T n v T T > n T n v T n < T n + m T > n T n v n T > n Ā 1 x:n n Āx n Ā1 x:m = n m Ā x A 1 x:n or n E x Endowment insurance v T v n T n T > n Ā x:n
11 Lesson 1: Insurance: Payable at Moment of Death - Moments - Part 2 Actuarial present value under constant force and uniform de Moivre) mortality for insurances payable at the moment of death Type of insurance APV under constant force APV under uniform de Moivre) Whole life µ µ+δ ā ω x ω x n-year term µ µ+δ 1 e nµ+δ) ) ā n ω x n-year deferred life µ µ+δ e nµ+δ) e δnā ω x+n) ω x n-year pure endowment e nµ+δ) e δn ω x+n)) ω x j-th moment Multiply δ by j in each of the above formulae Gamma Integrands u t n e δt dt = n! δ n+1 te δt dt = 1 δ 2 1 1 + δu)e δu) = āu uv u Variance If Z 3 = Z 1 + Z 2 and Z 1, Z 2 are mutually exclusive, then V arz 3 ) = V arz 1 ) + V arz 2 ) 2E[Z 1 ]E[Z 2 ] δ
12 Lesson 11: Insurance: Annual and m-thly: Moments A x = k q x v k+1 = kp x q x+k v k+1 Actuarial present value under constant force and uniform de Moivre) mortality for insurances payable at the end of the year of death Type of insurance APV under constant force APV under uniform de Moivre) Whole life n-year term q q+i q q+i 1 vp)n ) a ω x ω x a n ω x n-year deferred life q q+i vp)n vn a ω x+n) ω x n-year pure endowment vp) n v n ω x+n)) ω x
13 Lesson 12: Insurance: Probabilities and Percentiles Summary of Probability and Percentile Concepts To calculate P rz z) for continuous Z, draw a graph of Z as a function of T. Identify parts of the graph that are below the horizontal line Z = z, and the corresponding t s. Then calculate the probability of T being in the range of those t s. Note that ) P rz < z ) = P rv t < z ln z ) = P r t > = t p x, where t ln z ln z = = δ δ ln1 + i) The following table shows the relationship between the type of insurance coverage and corresponding probability: Type of insurance P rz < z ) Whole life n-year term n-year deferred life { { np x t p x t p x z v n z > v n nq x + t p x z < v n 1 z v n n-year endowment n-year deferred m-year term { z < v n t p x z v n nq x + n+m p x z < v m+n nq x + t p x v m+n z < v n 1 z v n In the case of constant force of mortality µ and interest δ, P rz z) = z µ/δ For discrete Z, identify T and then identify K + 1 corresponding to those T. In other words, P rz > z ) = P r T < t ) = k q x, where k = t - the greatest integer smaller than t P rz < z ) = P r T > t ) = k+1 p x, where k = t - the greatest integer smaller or equal to t To calculate percentiles of continuous Z, draw a graph of Z as a function of T. Identify where the lower parts of the graph are, and how they vary as a function of T. For example, for whole life, higher T lead to lower Z. For n-year deferred whole life, both T < n and higher T lead to lower Z. Write an equation for the probability Z less than z in terms of mortality probabilities expressed in terms of t. Set it equal to the desired percentile, and solve for t or for e kt for any k. Then solve for z which is often v t )
14 Recursive Formulas Lesson 13: Insurance: Recursive Formulas, Varying Insurances Increasing and Decreasing Insurance t n e δt dt = n δ n+1 u A x = vq x + vp x A x+1 A x = vq x + v 2 p x q x+1 + v 2 2p x A x+2 A x:n = vq x + vp x A x+1:n 1 A 1 x:n = vq x + vp x A 1 x+1:n 1 n A x = vp xn 1 A x+1 Ax:n 1 1 = vp x A x+1:n 1 te δt dt = 1 δ 2 1 1 + δu) e δu) = ān δ uv u δ ĪĀ ) x = µ for constant force µ + δ) 2 E[Z 2 2µ ] = for Z a continuously increasing continuous insurance, constant force. µ + 2δ) 3 ĪĀ ) 1 x:n + DĀ ) 1 x:n = nā1 x:n I Ā ) 1 x:n + DĀ) 1 x:n = n + 1)Ā1 x:n IA) 1 x:n + DA)1 x:n = n + 1)A1 x:n Recursive Formulas for Increasing and Decreasing Insurance IA) 1 x:n = A1 x:n + vp x IA) 1 x+1:n 1 IA) 1 x:n = A1 x:n + vp x IAA) 1 x+1:n 1 DA) 1 x:n = na1 x:n + vp x DA) 1 x+1:n 1 DA) 1 x:n = A1 x:n + DA) 1 x:n 1
Lesson 14: Relationships between Insurance Payable at Moment of Death and Payable at End of Year Summary of formulas relating insurances payable at moment of death to insurances payable at the end of the year of death assuming uniform distribution of deaths 15 Ā x = i δ A x Ā 1 x:n = i δ A1 x:n n Āx = i δ n A x I Ā ) 1 x:n = i δ IA)1 x:n I D) 1 x:n = i δ ID)1 x:n Ā x:n = i δ A1 x:n + A 1 x:n A m) x = i i m) A x 2 Ā x = 2i + i2 2 A x 2δ ĪĀ ) 1 x:n = IĀ) 1 x:n Ā1 x:n 1 d 1 ) δ Summary of formulas relating insurances payable at moment of death to insurances payable at the end of the year of death using claims acceleration approach Ā x = 1 + i).5 A x Ā 1 x:n = 1 + i).5 A 1 x:n n Āx = 1 + i).5 n A x Ā x:n = 1 + i).5 A 1 x:n + A 1 x:n A m) x = 1 + i) m 1)/2m A x 2 Ā x = 1 + i) 2 A x
16 Lesson 15: Annuities: Continuous, Expectation Actuarial notation for standard types of annuity Name Payment per Present value Symbol for actuarial annum at time t random variable present value Whole life 1 t T ā T ā x annuity Temporary life annuity 1 t mint, n) t > mint, n) ā T ā n T n T > n ā x:n Deferred life annuity t n or t > T 1 n < t T T n ā T ā n T > n n ā x Deferred temporary life annuity t n or t > T 1 n < t n + m and t T T > n + m T n ā T ā n n < T n + m ā n+m ā n T > n + m n ā x:m Certain-and-life 1 t maxt, n) t > maxt, n) ā n ā T T n T > n ā x:n Relationships between insurances and annuities ā x = 1 Āx δ Ā x = 1 δā x ā x:n = 1 Āx:n δ Ā x:n = 1 δā x:n General formulas for expected value n ā x = Āx:n Āx δ ā x = ā x = ā x:n = n ā x = Formulas under constant force of mortality n n a nt p x µ x+t dt v t tp x dt v t tp x dt v t tp x dt
17 ā x = 1 µ + δ ā x:n = 1 e µ+δ)n µ + δ n ā x = e µ+δ)n µ + δ Relationships between annuities ā x = ā x:n + n E x ā x+n ā x:n = ā n + n ā x
18 Relationships between insurances and annuities Lesson 16: Annuities: Discrete, Expectation ä x = 1 A x d A x = 1 dä x ä x:n = 1 A x:n d A x:n = 1 dä x:n Relationships between annuities A 1 x:n = vä 1 x:n a 1 x:n 1 + i)a x + ia x = 1 ä x:n = ä n + n ä x ä x:n = ä x n E x ä x+n Other annuity equations Accumulated value mthly annuities n ä x = n E x ä x+n ä x = ä x:n + n ä x a x = ä x 1 ä x:n = a x:n + 1 n E x = a x:n 1 + 1 n 1 ä x:n = ä k k 1p x q x+k 1 + ä nn 1 p x k=1 n 1 ä x:n = v k kp x, a x:n = k= ä x = 1 + i q + i, n v k kp x k=1 if q x is constant s x:n = äx:n ne x s x:n = s x+1:n 1 + 1 1 s x:n = s x+1:n 1 + n 1E x+1 s x:n = s x:n + 1 1 ne x ä m) x = 1 m v k m k m k= p x
19 General formulas for second moments E[Ȳ 2 x ] = E[Ÿ 2 x ] = E[Ÿ 2 x:n] = Lesson 17: Annuities: Variance ā 2 t tp x µ x+t dt ä 2 k k 1 q x k=1 n n 1 ä 2 k k 1 q x + n p x än 2 = ä 2 k k 1 q x + n 1 p x än 2 k=1 Special formulas for variance of whole life annuities and temporary life annuities 2 Ā x Āx) 2 V arȳx) = δ 2 = 2ā x 2 ā x ) ā x ) 2 δ 2 Ā x:n Āx:n) 2 V arȳx:n) = δ 2 = 2ā x:n 2 ā x:n ) ā x:n ) 2 δ 2 A x A x ) 2 V arÿx) = V ary x ) = d 2 = 2ä x 2 ä x ) d 2 + 2 ä x ä x ) 2 2 A x:n A x:n ) 2 V arÿx:n) = V ary x:n 1 ) = d 2 k=1
2 Lesson 18: Annuities: Probabilities and Percentiles To calculate a probability for an annuity, calculate the t for which ā t has the desired property. Then calculate the probability t is in that range. To calculate a percentile of an annuity, calculate the percentile of T, then calculate ā T. Some adjustments may be needed for discrete annuities or non-whole-life annuities If forces of mortality and interest are constant, then the probability that the present value of payments on a continuous whole life annuity will be greater than its actuarial present value is ) µ µ/δ P rā T x) > ā x ) = µ + δ
21 Lesson 19: Annuities: Varying Annuities, Recursive Calculations Increasing/Decreasing Annuities Īā )x = 1, if µ is constant µ + δ) 2 Īā )x:n + ) Dā x:n = nā x:n Recursive Formulas Ia) x:n + Da) x:n = n + 1)a x:n ä x = vp x ä x+1 + 1 a x = vp x a x+1 + vp x ā x = vp x ā x+1 + ā x:1 ä x:n = vp x ä x+1:n 1 + 1 a x:n = vp x a x+1:n 1 + vp x ā x:n = vp x ā x+1:n 1 + ā x:1 n ä x = vp xn 1 ä x+1 n a x = vp xn 1 a x+1 n ā x = vp xn 1 ā x+1 ä x:n = 1 + vq x ä n 1 + vp x ä x+1:n 1 a x:n = v + vq x a n 1 + vp x a x+1:n 1 ā x:n = ā 1 + vq x ā n 1 + vp x ā x+1:n 1
22 Lesson 2: Annuities: m-thly Payments In general: a m) x = ä m) x 1 m ) m 1 + i = 1 + im) = 1 dm) m m ) i m) = m 1 + i) 1 m 1 ) d m) = m 1 1 + i) 1 m ) m For small interest rates: ä m) x ä x m 1 2m a m) x a x + m 1 2m Under the uniform distribution of death UDD) assumption: ä m) x = ä x m 1 2m a m) x = a x + m 1 ä m) x 2m = αm)ä x βm) ä m) x:n = αm)ä x:n βm)1 n E x ) = αm) n ä x βm) n E x ā x = α )ä x β ) n ä m) x a m) x:n = äm) x:n 1 m + 1 m n E x ä m) x = 1 Am) x d m) Similar conversion formulae for converting the modal insurances to annuities hold for other types of insurances and annuities only whole life version is shown). αm) = id i m) d m) βm) = i im) i m) d m) i ) = d ) = ln1 + i) = δ
23 Woolhouse s formula for approximating ä m) x : ä m) x ä x m 1 2m m2 1 12m 2 µ x + δ) ā x ä x 1 2 1 12 µ x + δ) ä m) x:n ä x:n m 1 2m 1 n E x ) m2 1 12m 2 µ x + δ n E x µ x+n + δ)) n ä m) x n ä x m 1 2m n E x m2 1 12m 2 n E x µ x+n + δ) e x = e x + 1 2 1 12 µ x When the exact value of µ x is not available, use the following approximation: µ x 1 2 ln p x 1 + ln p x )
24 Lesson 21: Premiums: Fully Continuous Expectation The equivalence principle: The actuarial present value of the benefit premiums is equal to the actuarial present value of the benefits. For instance: whole life insurance Āx = P Ā x ) āx n year endowment insurance Āx:n = P Ā x:n ) āx:n n year term insurance Ā 1 x:n = P Ā 1 x:n) āx:n n year deferred insurance n Āx = n P n Āx) āx:n n pay whole life insurance Āx = n P Āx ) āx:n n year deferred annuity n ā x = P n ā x ) ā x:n P Ā x ) = 1 δā x ā x = 1 ā x δ P ) Ā x δāx Ā x = = 1 Āx)/δ 1 Āx P ) 1 Ā x:n = δ ā x:n δāx:n P ) Ā x:n = 1 Āx:n For constant force of mortality, P ) ) Ā x and P Āx:n are equal to µ. Future loss formulas for whole life with face amount b and premium amount π: ) 1 v L = bv T πā T = bv T T π = v T b + π ) π δ δ δ E[ L] = bāx πā x = Āx b + π ) π δ δ Similar formulas are available for endowment insurance.
25 Lesson 22: Premiums: Net Premiums for Discrete Insurances Calculated from Life Tables Assume the following notation 1) P x is the premium for a fully discrete whole life insurance, or A x /ä x 2) Px:n 1 is the premium for a fully discrete n-year term insurance, or A 1 x:n/ä x:n 3) Px:n 1 is the premium for a fully discrete n-year pure endowment, or Ax:n/ä 1 x:n 4) P x:n is the premium for a fully discrete n-year endowment insurance, or A x:n /ä x:n
26 Lesson 23: Premiums: Net Premiums for Discrete Insurances Calculated from Formulas Whole life and endowment insurance benefit premiums P x = 1 ä x d P x = da x 1 A x P x:n = 1 d ä x:n da x:n P x:n = 1 A x:n For fully discrete whole life and term insurances: If q x is constant, then P x = vq x and P 1 x:n = vq x. Future loss at issue formulas for fully discrete whole life with face amount b: L = v Kx+1 b + P ) x P x d d E[L ] = Āx b + P x d Similar formulas are available for endowment insurances. ) P x d Refund of premium with interest To calculate the benefit premium when premiums are refunded with interest during the deferral period, equate the premiums and the benefits at the end of the deferral period. Past premiums are accumulated at interest only. Three premium principle formulae np x P 1 x:n = P 1 x:na x+n P x:n n P x = P 1 x:n1 A x+n )
27 Lesson 24: Premiums: Net Premiums Paid on an m-thly Basis If premiums are payable mthly, then calculating the annual benefit premium requires dividing by an mthly annuity. If you are working with a life table having annual information only, mthly annuities can be estimated either by assuming UDD between integral ages or by using Woolhouse s formula Lesson 2). The mthly premium is then a multiple of the annual premium. For example, for h-pay whole life payable at the end of the year of death, hp m) x = A x a m) x:h = h P x a x:h a m) x:h
28 Lesson 25: Premiums: Gross Premiums The gross future loss at issue L g is the random variable equal to the present value at issue of benefits plus expenses minus the present value at issue of gross premiums: L g = P V Ben) + P V Exp) P V P g ) To calculate the gross premium P g by the equivalence principle, equate P g times the annuity-due for the premium payment period with the sum of 1. An insurance for the face amount plus settlement expenses 2. P g times an annuity-due for the premium payment period of renewal percent of premium expense, plus the excess of the first year percentage over the renewal percentage 3. An annuity-due for the coverage period of the renewal per-policy and per-1 expenses, plus the excess of first year over renewal expenses
29 Lesson 26: Premiums: Variance of Loss at Issue, Continuous The following equations are for whole life and endowment insurance of 1. For whole life, drop n. V arl ) = 2 Ā x:n ) ) 2 Ā x:n 1 + P ) 2 δ V arl ) = 2 Ā x:n Ā x:n ) 2 1 Ā x:n ) 2, If equivalence principle premium is used V arl ) = µ, For whole life with equivalence principle and constant force of mortality only µ + 2δ For whole life and endowment insurance with face amount b: V arl ) = 2 Ā x ) ) 2 Ā x b + P ) 2 δ V arl ) = 2 Ā x:n ) ) 2 Ā x:n b + P ) 2 δ If the benefit is b instead of 1, and the premium P is stated per unit, multiply the variances by b 2. For two whole life or endowment insurances, one with b units and total premium P and the other with b units and total premium P, the relative variance of loss at issue of the first to second is b δ + P ) / bδ + P )) 2.
3 Lesson 27: Premiums: Variance of Loss at Issue, Discrete The following equations are for whole life and endowment insurance of 1. For whole life, drop n. V ar L) = 2 A x:n A x:n ) 2) 1 + P ) 2 d 2 A x:n A x:n ) 2 If equivalence principle premium is used: V ar L) = 1 A x:n ) 2 For whole life with equivalence principle and constant force of mortality only: V ar L) = q1 q) q + 2 i If the benefit is b instead of 1, and the premium P is stated per unit, multiply the variances by b 2. For two whole life or endowment insurances, one with b units and total premium P and the other with b units and total premium P, the relative variance of loss at issue of the first to second is b d + P ) / bd + P )) 2.
31 Lesson 28: Premiums: Probabilities and Percentiles of Loss at Issue For level benefit or decreasing benefit insurance, the loss at issue decreases with time for whole life, endowment, and term insurances. To calculate the probability that the loss at issue is less than something, calculate the probabiitiy that survival time is greater than something. For level benefit or decreasing benefit deferred insurance, the loss at issue decreases during the deferral period, then jumps at the end of the deferral period and declines thereafter. To calculate the probability that the loss at issue is greater then a positive number, calculate the probability that survival time is less than something minus the probability that survival time is less than the deferral period. To calculate the probability that the loss at issue is greater than a negative number, calculate the probability that survival time is less than something that is less than the deferral period, and add that to the probability that survival time is less than something that is greater then the deferral period minus the probability that survival time is less than the deferral period. For a deferred annuity with premiums payable during the deferral period, the loss at issue decreases until the end of the deferral period and increases thereafter. The 1pth percentile premium is the premium for which the loss at issue is positive with probability p. For fully continuous whole life, this is the loss that occurs if death occurs at the 1pth percentile of survival time.