Strategies q( t ) = exp Investing Over the Lifecycle, Part II How life insurance and annuities manage the uncertainty of the time of death. 1 - exp () t b exp ( ) a-m b In this model, I make use of two pricing formulas for life insurance. First, I calculate the fair price of t a-m aq(permanent life insurance policy that pays p( tt )) == exp q ( t - 1)1 -- exp q ( t ) b exp b whenever the insured dies. Per dollar of payment, the price is analogous to the fair price of an annuity, butq(t) with exp a b- m instead of q( t ) =∞exp 1 - death exp t probabilities A₀ t = p( ) =∑ qprobabilities: ( t - 1) -t q( t ) b survival t = 0 (1 + r) () ( ) () ( ) ∞ p(t) L₀ == ∑∞ q(t) t p( t ) = q ( t (1 t = 0 A₀ = ∑ -+1)r) -t q ( t ) t = 0 (1 + r) E X H I BI TS 1 & 2 show q( t ) with these parameters of a 30-year-old. QUANT U Paul D. Kaplan In the last issue of Quant U, I presented a bare-bones lifecycle model of investing that focuses on spending and savings. In this issue, I present a model that shows how to deal with the uncertainties of death using annuities and life insurance. Modeling Mortality While the old saying about the certainty of death and taxes is true, what is not certain is when death will occur. Fortunately, there are very good models of the probability distribution of when death occurs. In the February/March 2015 issue of Quant U, I presented the Gompertz mortality model. According to the Gompertz model, the probability of a person surviving for at least t more years is a function that only has three additional parameters: 1 The person's current age ( a). 2 The mode of the distribution of the age of death ( m). 3 The dispersion of the age of death around the mode ( b). The b parameter is similar to the standard deviation parameter of a normal distribution. Given these parameters, the probability of surviving for at least t more years is as follows: q( t ) = exp 1 - exp () t b exp ( ) a-m b In this issue of Quant U, I set m to 86 and b p( t ) = q( t - 1) - q ( t ) to 10.48. 48 A₀ q(t) December/January 2018 = Morningstar ∑ t t = 0 (1 + r) ∞ The probability of dyingt in t years is given by q( t ) = exp 1 - exp b exp a b- m the difference in the survival probabilities for t - 1 and t years: () ( ) Annuities and Life Insurance ∞ q(t)companies provide two basic Life A₀ =insurance ∑ + r) t to help investors manage the t = 0 (1 insurance products uncertainty of the time of death: annuities ∞ p(t) L₀ ==life and insurance. ∑ t For purposes of this discussion, t = 0 (1 + r) by annuities, I mean single-premium immediatepayout annuities. Here is some detail on each of these1products: - q(t + 1|t) L1₀ = 1+r Annuities An annuity provides a guaranteed stream of + 1) income for theq(t annuitant, thus providing insurance q(t + 1|t) = against runningq(t) out of money before death. t exp a b- m q( t ) = exp 1 - exp In this model, I assumeb that the capital market provides a single constant rate of return r. L q(t)y H₀ == ∑the actuarially Hence, fair price of an annuity per t t = 0 (1 + r) dollar of annual payments per year in year p( t ) = q( t - 1) - q ( t ) 0 is: ( ) H₀ + F₀ = A₀c + L₀B ∞ q(t) A₀ = ∑ t t = 0 (1 + r) H₀ + F₀ - L₀B c = ∞ A₀ p(t) (In practice, insurance companies charge more L₀ == ∑ t t = 0 (1 + r) than fair price to earn profits.) 1+r (Ht-1 - yt-1 ) Ht = q(t|t - 1) Life Insurance 1 - q(t + 1|t) L1₀ =insurance is for investors who want to be Life 1+r sure to leave we shall see, life 1 + ar bequest. As (FAmost APt-1 -before AIt-1 )an investor FAt = t-1 +sense insurance makes the q(t|t - 1) has accumulated wealth to leave the q(t +enough 1) q(t + 1|t) = - t q(t) desired(1bequest. + ) - 1+ y + AI - c - AP - LP ) FRt = (1 + r) (FRt-1 t-1 t-1 t-1 t-1 L q(t)y H₀ == ∑ t exp 1 - exp t = 0 (1 + r) () t b exp ∞ p(t) L₀ == ∑ 1 - q(t +t 1|t) + 1) L1₀ +=t1|t) = 0 (1 + q(t q(t =1 +r)r q(t) n= f=n Where q1 (-t +q(t1 | t+) 1|t) means the probability of surviving -t L1₀ = (1 ) +q(t + 1)the investor survived into year q(t)y year+t1|t) +L 1,+ given that 1 r q(t = -1 H₀ == ∑ t q(t) = 0 (1 + t. It is tgiven by:r) p( t ) = q( t - 1) - q ( t ) () Second, ∞ in each year t, I calculate the fair - p(t) q(t + 1|t) L₀ q(t) ∑∞1life price insurance for year t + 1 alone. This is L1₀====of A₀ ++r)rtt t∑ = 0 (1 1 = 0 (1 + r) given tby: ( ) a-m b () + 1)bt exp 1 q(t - exp L q(t + F₀ 1|t) =q(t)y H₀ + = A₀c + L₀B H₀ == ∑ q(t) t t = 0 (1 + r) exp ( ) P= P ( y; a-m b Hence,H₀L1 +- qF₀( t-+L₀B 1 | t ) is the probability that the c = q(t)y H₀ +==F₀∑ =dies A₀in +year H₀ A₀c L₀Bt + 1, given that he investor t t = 0 (1 + r) survived to year t. This is the probability that the insurance +F₀ r- L₀B will need to make a H₀ +1company - (H cHt =+= F₀ in payout year 1. t-1 yt-1 ) H₀ = q(t|tA₀c A₀- 1)+t +L₀B D ( y; P ( y; The Model and Its Solution H₀ +11+F₀+r-r L₀B cFA Unlike presented of)this series, - AIIt-1 -+ (FA AP)int-1Part yt-1 Ht =t == the model t-1t-1 A₀ I(H q(t|t I do notq(t|t solve--a1)1) utility maximization problem. Instead, I assume a constant level of consumption 11+the (so long as I solve for +r r investor (H+t-1y -is+ ) which yalive), Ht = t-1AI + r)-(FR AP FR ) t-1 - LPt-1 ) FA t = (1 t-1(FA t-1 +AP t-1 t-1 - cAI-t-1 q(t|t 1) using the intertemporal budget constraint. q(t|t- -t 1) (1 + ) -1 As in Part 1I, +torform the intertemporal budget - cAI-t-1 ) - (FAt-1 ++AP FA + r) AIt-1 AP FRtt == (1q(t|t constraint, I first value t-1 + yto t-1calculate t-1 - the t-1 ofLPt-1 ) -(FR 1)need t a-m expcapital 1 -inexp human yearb 0. exp Let: b () ( ) + ywhich - c - APt-1 - LPt-1 ) r) (FR LFRt== (1 the+last year the t-1 in t-1 + AI t-1 investor earns income y = the amount of labor income the investor earns in years 0, 1, 2,... L n = qTa person can only earn income when Because alive, the income level needs to be multiplied by f = n- qT survival probabilities in the calculation of human n - n 0, we have: -j capital. For human capital iny year c y f + 1+ q P= q 1+ q ) ( ( ∑ j =1 n P ( y; c, T ) = c q f- j ) - qT ∑ (1 + ) + (1 + ) y q j =1 y q c q P( n= f=n = qc P= Pc ( y ) P ( y; Dc ( y ) D ( y; Dm (y; P ( y;(y; D m ∆P P P( c q DV01 = qc

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