Annuity Values with Time-Varying Interest Rates

Why the Rate Must Match the Payment Time

A life annuity’s value depends on two distinct features: whether each payment is made, and how much that payment is worth at the valuation date. Survival probabilities describe the first feature. When interest rates vary by period, the second requires a separate discount factor for each payment time. A single power of one discount factor is generally not correct.

Assume throughout that interest rates are deterministic, so the interest accumulated to a given time is known at the valuation date. Mortality is described by the survival probabilities ${}_tp_x$ for a life aged $x$. The rates are not required to be independent of mortality, because the rates here are fixed rather than random. If interest rates were random, their dependence on survival or other variables would need to be modeled; the formulas below would not in general suffice.

Definition: Let $A(t)$ be the accumulated value at time $t$ of one unit invested at time 0. The discount factor to time 0 is $D(t)=1/A(t)$. Under a deterministic interest-rate schedule, a life-annuity payment of amount $b(t)$ due at time $t$ has expected present value $b(t)\,{}_tp_xD(t)$, provided the payment is made only if the life is alive at time $t$.

Discrete Payments: Accumulate Rates Up to Each Due Date

Suppose $i_k$ is the effective annual rate earned during the period from time $k-1$ to time $k$. The accumulation from time 0 to integer time $k$ is the product of the accumulation factors for all $k$ periods, not just the factor for the $k$th period. Therefore,

A(k)=\prod_{j=1}^{k}(1+i_j), \qquad D(k)=\frac{1}{\prod_{j=1}^{k}(1+i_j)}.

For a temporary annuity-due paying $b_k$ at times $k=0,1,\ldots,n-1$, while the life is alive at each payment time, its expected present value is

\sum_{k=0}^{n-1} b_k\,{}_kp_xD(k), \qquad D(0)=1.

For an annuity-immediate paying at times $k=1,\ldots,n$, the corresponding value is

\sum_{k=1}^{n} b_k\,{}_kp_xD(k).

The time index determines both factors in each term: ${}_kp_x$ is the probability the life survives to the payment time, and $D(k)$ discounts from that same time. These sums are the direct valuation formulas; if the interest rate is constant, they reduce to familiar annuity formulas. With changing rates, do not use a constant-rate annuity factor unless it has been shown to produce the same payment-date discount factors.

Worked Example: A Three-Payment Annuity-Due

A hypothetical policy pays $1,000 at times 0, 1, and 2, provided the annuitant is alive at each payment time. The effective annual interest rates are 5% in year 1 and 4% in year 2. The conditional probabilities of surviving year 1 and year 2, given survival to the start of the respective year, are 0.90 and 0.80. Find the expected present value.

Method. Value each payment at its own due date. The payment at time 0 is certain and is not discounted. The survival probabilities are ${}_0p_x=1$, ${}_1p_x=0.90$, and ${}_2p_x=(0.90)(0.80)=0.72$.

The cumulative discount factors are $D(0)=1$, $D(1)=1/1.05$, and $D(2)=1/[(1.05)(1.04)]=1/1.092$. Thus the expected present value is

1000(1)(1) +1000(0.90)\frac{1}{1.05} +1000(0.72)\frac{1}{1.092} = 1000\left(1+\frac{0.90}{1.05}+\frac{0.72}{1.092}\right) =\$2{,}516.48.

The final amount is rounded to the nearest cent. As a check, the three terms are approximately $1,000, $857.14, and $659.34, which sum to $2,516.48. The common mistake avoided here is discounting the time-2 payment by only the second-year rate; it must be discounted through both years.

Worked Example: An Annuity-Immediate with Changing Survival and Interest

A hypothetical annuity pays $1,200 at the end of each of years 1, 2, and 3 if the annuitant is alive. The conditional annual survival probabilities are 0.95, 0.90, and 0.85, respectively. The effective interest rates for those years are 2%, 4%, and 6%. Calculate the expected present value.

Method. First find the cumulative survival probabilities: ${}_1p_x=0.95$, ${}_2p_x=(0.95)(0.90)=0.855$, and ${}_3p_x=(0.95)(0.90)(0.85)=0.72675$. The cumulative discount factors are $D(1)=1/1.02$, $D(2)=1/[(1.02)(1.04)]=1/1.0608$, and $D(3)=1/[(1.02)(1.04)(1.06)]=1/1.124448$.

Apply survival and discounting at each payment time:

\begin{aligned} \operatorname{EPV} &=1200\left( \frac{0.95}{1.02} +\frac{0.855}{1.0608} +\frac{0.72675}{1.124448} \right)\\ &=1117.6470588\ldots+967.1945701\ldots+775.58055\ldots\\ &=\$2{,}860.42. \end{aligned}

The final value is rounded to the nearest cent. The calculation uses each conditional survival probability only to build cumulative survival; using 0.90 alone as the probability of surviving to time 2 would omit the need to survive year 1. It also uses the product of the first two accumulation factors to discount the time-2 payment.

Continuous Payments: Use the Accumulation Over the Entire Elapsed Time

For continuous payments, let $\delta(s)$ denote the deterministic force of interest at time $s$. The accumulation from 0 to $t$ and the corresponding discount factor are

A(t)=\exp\left(\int_0^t \delta(s)\,ds\right), \qquad D(t)=\exp\left(-\int_0^t \delta(s)\,ds\right).

If a continuous annuity pays at a rate $c(t)$ per year while the life is alive, its expected present value over a payment period from 0 to $n$ is

\int_0^n c(t)\,{}_tp_x \exp\left(-\int_0^t \delta(s)\,ds\right)\,dt.

This expression accounts for changes in the force of interest by integrating it from the valuation time all the way to each payment time. Annual effective rates alone do not specify the discount factor at every fractional time. To value continuous payments from annual rates, an additional assumption about interest accumulation within each year is needed—for example, a specified force of interest or another stated interpolation convention.

Why the expected-value formulas hold: For a discrete payment at time $k$, let its payment indicator be 1 if the life is alive at $k$, and 0 otherwise. Its present value is $b_kD(k)$ times that indicator. The indicator has expectation ${}_kp_x$, so the expected present value is $b_kD(k){}_kp_x$. Adding the expected values gives the discrete sum. For continuous payments, the present value is the integral of $c(t)D(t)$ times the indicator that the life is alive at $t$. Taking expectations inside this nonnegative integral gives the stated integral with ${}_tp_x$. Thus survival weights the payment at each time, while $D(t)$ discounts that same payment.

Worked Example: A Continuous Annuity with Two Interest Periods

A hypothetical annuity pays continuously at $1,000 per year for at most two years while the annuitant is alive. The force of mortality is constant at $\mu_x=0.04$, so ${}_tp_x=e^{-0.04t}$. The force of interest is 0.02 during the first year and 0.05 during the second year. Find the expected present value.

Method. On $0\leq t\leq1$, the accumulated force of interest is $0.02t$, so the product of survival and discount factors is $e^{-0.04t}e^{-0.02t}=e^{-0.06t}$. On $1\leq t\leq2$, the accumulated force of interest is $0.02+0.05(t-1)$. The combined exponent for survival and discounting is therefore $0.04t+0.02+0.05(t-1)=0.09t-0.03$.

Split the integral at the time the force of interest changes:

\begin{aligned} \operatorname{EPV} &=1000\left[ \int_0^1 e^{-0.06t}\,dt +\int_1^2 e^{-(0.09t-0.03)}\,dt \right]\\ &=1000\left[ \frac{1-e^{-0.06}}{0.06} +e^{-0.06}\frac{1-e^{-0.09}}{0.09} \right]\\ &=\$1{,}871.22. \end{aligned}

The final value is rounded to the nearest cent. The first integral is approximately $0.9705911067$, and the second is approximately $0.9006284110$, giving a total of approximately $1,871.22. The common mistake avoided is applying the second-year force of interest to the first year as well; the second-year discount accumulation begins only after time 1.

Common Traps and Practical Checks

Changing rates make indexing especially important. Before evaluating a sum or integral, identify the payment times, the survival probability at each time, and the accumulated discount factor from time 0 to that time. The following checks help catch errors:

Do not discount a time-0 annuity-due payment. Its discount factor is $D(0)=1$, and its survival probability is ${}_0p_x=1$.
Do not use only the latest annual rate. A payment at time $k$ is discounted using every period’s rate from 1 through $k$.
Do not confuse conditional and cumulative survival. Multiply conditional annual survival probabilities to obtain ${}_kp_x$.
Do not invent fractional-year discount factors. If the cash flow is continuous or occurs between integer times, the interest assumption must define accumulation within the year.
Check that each term uses the same time index. A time-$k$ payment should be paired with ${}_kp_x$ and $D(k)$, not with factors belonging to another date.

These valuations are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. Their central discipline is to track time consistently: survival determines whether a payment occurs, and the full interest accumulation to its due date determines its present value.

Check Your Understanding

For each problem, identify the payment times and use the cumulative discount factor appropriate to each time.

A temporary annuity-due pays $500 at times 0 and 1 if the annuitant is alive. The year-1 effective interest rate is 10%, and ${}_1p_x=0.80$. Find its expected present value. (Answer: $863.64)
An annuity-immediate pays $1,000 at times 1 and 2. The cumulative survival probabilities are ${}_1p_x=0.90$ and ${}_2p_x=0.72$; the effective interest rates are 5% in year 1 and 3% in year 2. Find its expected present value. (Answer: $1,522.88)
A continuous annuity pays $600 per year for one year while the annuitant is alive. The force of interest is 0.04 and the force of mortality is 0.06, both constant over the year. Find its expected present value. (Answer: $570.98)
A continuous annuity pays $1,000 per year for two years while the annuitant is alive. The force of mortality is constant at 0.02; the force of interest is 0.01 in the first year and 0.03 in the second year. Find its expected present value. (Answer: $1,931.73)

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