Discrete Annuities Under Constant Force of Mortality

Introduction

A discrete life annuity makes payments at specified times, and a payment is made only if the annuitant is alive at the relevant payment date. Valuation therefore combines two distinct reductions: interest discounting for waiting to receive a payment, and a survival probability for the possibility that the payment will not be made. Under a constant force of mortality, the survival probabilities have a particularly simple form, so many discrete annuity values become finite or infinite geometric series.

The key is to match each payment date to the probability of being alive then. An annuity-due has a payment immediately, before any future survival is required. An annuity-immediate has its first payment one year later, so that payment depends on surviving one year. The same distinction continues through every later payment.

Definition: Suppose the force of mortality is constant at $\mu$ per year at the annuitant's age and throughout the future lifetime being valued. Then the probability of surviving $t$ years is ${}_tp_x=e^{-\mu t}$, and the probability of dying within $t$ years is ${}_tq_x=1-e^{-\mu t}$. In particular, the one-year survival probability is $p=e^{-\mu}$. The effective annual interest rate is $i$, and $v=1/(1+i)$ is the annual discount factor.

Annual Survival Probabilities Under Constant Force

The force of mortality measures the instantaneous rate of mortality, conditional on survival to that time. If it is constant, $\mu_x=\mu$, then survival over successive intervals follows an exponential pattern. The probability of surviving to year $k$ is $e^{-\mu k}$, which is also $p^k$, where $p=e^{-\mu}$. This gives the survival probability at every annual payment date directly.

For example, if $\mu=\ln 2$ per year, then $p=e^{-\ln 2}=1/2$. The probabilities of being alive at years 1, 2, and 3 are $1/2$, $1/4$, and $1/8$, respectively. These are probabilities at the payment dates, not probabilities of dying in those years.

Derivation: By definition, the force of mortality satisfies $\mu=-\frac{d}{dt}\ln({}_tp_x)$. If $\mu$ is constant, integrate from 0 to $t$. Since ${}_0p_x=1$, this gives $\ln({}_tp_x)=-\mu t$, and therefore ${}_tp_x=e^{-\mu t}$. Equivalently, survival over $k$ one-year intervals is $p^k=(e^{-\mu})^k=e^{-\mu k}$.

For a payment of 1 at time $k$, the expected present value is the payment amount multiplied by both its discount factor and its probability of being made. Thus the expected present value of that payment is $v^k{}_kp_x$. Under constant force, this product is

v^k{}_kp_x=(ve^{-\mu})^k=(vp)^k.

Define $r=vp=e^{-\mu}/(1+i)$. Each successive annual payment therefore has an expected present value $r$ times that of the preceding payment. This repeated ratio is what makes the geometric-series formulas useful.

Temporary Discrete Annuities

A temporary annuity makes payments for at most $n$ years. For an annuity-due with annual payments of 1 at times $0,1,\ldots,n-1$, the payment at time 0 is certain for an annuitant who is alive at issue. The later payments require survival to their respective dates. For an annuity-immediate, payments of 1 are made at times $1,2,\ldots,n$, each conditional on survival to its payment date.

Key formulas: Under constant force $\mu$ and effective annual interest rate $i$, let $r=ve^{-\mu}$. The temporary annuity-due value is $\ddot{a}_{x:\overline{n}|}=\sum_{k=0}^{n-1}r^k$, and the temporary annuity-immediate value is $a_{x:\overline{n}|}=\sum_{k=1}^{n}r^k$. For $r\ne1$, these are $\ddot{a}_{x:\overline{n}|}=(1-r^n)/(1-r)$ and $a_{x:\overline{n}|}=r(1-r^n)/(1-r)$.

These formulas follow by adding the expected present values of the individual payments. The due series begins with $1$, because the time-0 payment is neither discounted nor contingent on future survival. The immediate series begins with $r$, since its first payment is one year away and is made only if the annuitant survives that year. If $r=1$, the finite sums still apply directly and each value equals $n$; the quotient formulas have a zero denominator and should not be used in that case.

Worked Example: Four-Year Temporary Annuity-Due

A hypothetical contract pays \$1,000 at the beginning of each year for at most four years, with the first payment immediately. Assume an effective annual interest rate of $i=4\%$ and a constant force of mortality $\mu=\ln 2$ per year. Find the expected present value.

Method. List the payment times for an annuity-due: $0,1,2,3$. Survival to time 0 is certain; the later payments require survival for one, two, or three years. The annual survival probability is $p=e^{-\ln 2}=0.5$, and $v=1/1.04=25/26$. Therefore $r=vp=(25/26)(1/2)=25/52$.

The annuity factor is

1+\frac{25}{52}+\left(\frac{25}{52}\right)^2+\left(\frac{25}{52}\right)^3 =1+0.4807692+0.2311391+0.1111245 =1.8230328.

Multiplying by the annual payment gives $1{,}000(1.8230328)=\$1{,}823.0328$, or \$1,823.03 to the nearest cent. As a check, the four terms correspond to expected present values of approximately \$1,000.00, \$480.77, \$231.14, and \$111.12; their sum is \$1,823.03. The common mistake avoided here is discounting or applying a one-year survival probability to the immediate payment.

Worked Example: Six-Year Temporary Annuity-Immediate

A hypothetical benefit pays \$750 at the end of each year for at most six years, provided the recipient is alive at the payment date. Assume $i=6\%$ effective annually and a constant force $\mu=\ln(1.25)$ per year. Find the expected present value.

Method. The one-year survival probability is $p=e^{-\ln(1.25)}=0.8$, and the discount factor is $v=1/1.06=50/53$. The first payment is at time 1, so the series starts with $r=vp=(50/53)(0.8)=40/53$, not with 1. The six-year annuity factor is

\sum_{k=1}^{6}\left(\frac{40}{53}\right)^k =\frac{40}{13}\left[1-\left(\frac{40}{53}\right)^6\right] =2.5083041.

The payment present value is $750(2.5083041)=\$1{,}881.2281$, or \$1,881.23 to the nearest cent. The last payment contributes only if the recipient survives six years, so its factor is $r^6$, not $r^5$. The sum includes six terms, for payment dates 1 through 6.

Whole-Life Discrete Annuities

A whole-life annuity continues to make annual payments for as long as the annuitant is alive. Under the same constant-force and constant-interest assumptions, the expected present values form an infinite geometric series with ratio $r=ve^{-\mu}$. If $0\le r<1$, the series converges. For ordinary nonnegative interest and positive force of mortality, this condition holds.

Key formulas: Under constant force $\mu$ and effective annual interest rate $i$, the whole-life annuity-due value is $\ddot{a}_x=\sum_{k=0}^{\infty}r^k=1/(1-r)$. The whole-life annuity-immediate value is $a_x=\sum_{k=1}^{\infty}r^k=r/(1-r)$, where $r=e^{-\mu}/(1+i)$ and $0\le r<1$. Thus $\ddot{a}_x=1+a_x$.

The whole-life formulas are the limits of the temporary formulas as $n$ increases. For $0\le r<1$, $r^n$ tends to 0, so $(1-r^n)/(1-r)$ tends to $1/(1-r)$. The immediate value is one ratio $r$ smaller because its payments start at time 1. This relationship also follows directly from the payment schedules: the annuity-due has one additional payment at time 0.

Worked Example: Whole-Life Annuity-Due

A hypothetical retirement benefit pays \$1,200 immediately and then at the beginning of each subsequent year while the recipient is alive. Assume an effective annual interest rate $i=3\%$ and a constant force of mortality $\mu=\ln(1.1)$ per year. Find the expected present value.

Method. The one-year survival probability is $p=e^{-\ln(1.1)}=1/1.1=10/11$, and $v=1/1.03=100/103$. Hence

r=vp=\frac{100}{103}\frac{10}{11}=\frac{1000}{1133}.

Because $r<1$, the whole-life geometric series converges. The annuity-due factor is

\ddot{a}_x=\frac{1}{1-r} =\frac{1}{1-\frac{1000}{1133}} =\frac{1133}{133} =8.5187970.

The expected present value is $1{,}200(8.5187970)=\$10{,}222.5564$, or \$10,222.56 to the nearest cent. As a check, the time-0 payment contributes \$1,200 with certainty, and the value of all later payments is $1{,}200(r/(1-r))$; together these equal $1{,}200/(1-r)$. The common mistake avoided is using the annuity-immediate factor for a contract that pays immediately.

Valuation Checks and Common Traps

A reliable calculation starts with the payment schedule rather than a memorized formula. Write down the payment dates, then attach the survival probability and discount factor for each date. Under constant force and a level annual payment, the combined factor for a payment at time $k$ is $r^k$, but the starting value of $k$ depends on the annuity type.

Starting an annuity-due series at $r$. Its immediate payment has factor $r^0=1$. Starting at $r$ incorrectly omits the payment at time 0.
Starting an annuity-immediate series at 1. Its first payment is at time 1 and requires both one year of discounting and one year of survival, so its factor is $r$.
Using $e^{-\mu}$ as the whole present-value ratio. This is the annual survival probability only. The ratio between successive expected present values is $ve^{-\mu}$, which includes interest discounting.
Using the wrong number of payment terms. A temporary annuity-due for $n$ years pays at times $0$ through $n-1$; an annuity-immediate pays at times $1$ through $n$. Each has $n$ payments if the annuitant survives to the relevant dates.
Applying the constant-force formulas to a different mortality pattern without justification. If force varies with time, then ${}_kp_x$ need not equal $p^k$, and the expected present values may not form a geometric series. Use the appropriate survival probabilities for each date instead.

These annuity values are useful whenever annual benefits or premiums depend on survival and the assumed mortality model has constant force over the period being valued. They also provide a quick internal check: for the same payment amount and duration, a due annuity includes its certain time-0 payment, while an immediate annuity does not. For a whole-life annuity, a larger $r$ means larger expected present values, since later payments are discounted and survival-weighted less heavily.

Check Your Understanding

For each problem, identify the payment dates before selecting a formula. Round dollar values to the nearest cent.

A hypothetical three-year annuity-due pays \$500 at times 0, 1, and 2. Assume $i=4\%$ effective annually and constant force $\mu=-\ln(0.9)$ per year. Find its expected present value. (Answer: \$1,307.14)
A hypothetical five-year annuity-immediate pays \$800 at the end of each year if the recipient is alive. Assume $i=5\%$ and constant force $\mu=-\ln(0.8)$ per year. Find its expected present value. (Answer: \$1,902.73)
A hypothetical whole-life annuity-due pays \$1,000 annually. Assume $i=4\%$ and a constant one-year survival probability of $0.95$. Find its expected present value. (Answer: \$11,555.56)
A hypothetical four-year annuity-due pays \$300 at times 0, 1, 2, and 3. Assume $i=6\%$ and constant force $\mu=\ln 2$ per year. Calculate $r$, then find the expected present value. (Answer: $r=25/53$; \$539.74)

What You'll Learn