Guaranteed-Period Annuity Reserves

Why Separate the Guaranteed Payments?

A life annuity ordinarily pays only while the annuitant is alive. A guaranteed-period life annuity adds a promise: installments in a specified initial period are payable even if the annuitant dies during that period. Payments after the guarantee ends are still conditional on survival. A reserve must therefore account for two different kinds of future cash flow.

The key calculation is to value the remaining guaranteed installments as certain payments, then add the expected present value of the later installments that require survival. This tutorial uses annual payments and an annual effective interest rate unless stated otherwise. It is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: A life annuity with $g$ annual payments guaranteed pays each installment due in the first $g$ years whether or not the annuitant is alive. Payments due after that period are made only if the annuitant is alive on the payment date. A prospective reserve at a valuation date is the expected present value, at that date, of future benefits, conditional on the annuitant being alive then. This tutorial assumes no future premiums or expenses.

Set the Payment Dates and Valuation Date

Timing determines which payments are certain and which require survival. First consider an annuity-immediate: installments are due at the end of each year. At issue, an annuity with a $g$-year guarantee has guaranteed payments at times $1,\ldots,g$, followed by survival-contingent payments at times $g+1,g+2,\ldots$. At duration $k$, measured just after the payment at time $k$, the annuitant is alive at the valuation date and the next possible payment is one year later.

Write $R$ for the annual installment, $i$ for the effective annual interest rate, and $v=1/(1+i)$. Write ${}_tp_x$ for the probability that a life aged $x$ survives $t$ years. The life annuity-immediate factor $a_x$ is the expected present value of payments of 1 at the ends of future years while the life aged $x$ is alive on each payment date.

Reserve formula: At duration $k<g$, immediately after the payment at time $k$, let $r=g-k$ be the number of guaranteed installments remaining. The reserve for an annuity-immediate is $$ {}_kV_x = R\left( a_{\overline{r}|} + v^r\,{}_rp_x\,a_{x+r} \right). $$ Here $a_{\overline{r}|}=v+v^2+\cdots+v^r$ is the present-value factor for $r$ certain payments at the ends of the next $r$ years. The second term values the payments after the guarantee, conditional on surviving the remaining $r$ years.

At issue, set $k=0$ and $r=g$. Once the guarantee has ended, there are no guaranteed installments left: immediately after the final guaranteed payment, the reserve is $R a_{x+g}$, conditional on being alive then. The formula does not apply unchanged if the valuation date is immediately before a payment, if installments are due at the beginning rather than the end of each year, or if the benefit amount varies.

Why the Formula Works

The formula follows by splitting the future installments into disjoint groups. Each remaining guaranteed installment is payable whether or not death occurs, so its value is a certain-annuity value. A later installment is payable only if the life survives to its due date, so its expected present value includes the corresponding survival probability.

Value the guaranteed installments.
At duration $k$, the next $r$ payments are due one, two, through $r$ years later. Their total value is $R\sum_{j=1}^{r}v^j=R a_{\overline{r}|}$.

Condition on survival to the end of the guarantee.
The first payment after the guarantee is due $r+1$ years after the valuation date. For any payment after that point to occur, the life must survive the $r$ years to age $x+r$. That probability is ${}_rp_x$.

Value the contingent tail.
Conditional on survival to age $x+r$, the payments after the guarantee form a life annuity-immediate at that age. Discount its value $r$ years and multiply by the probability of reaching that age. The result is $R v^r{}_rp_xa_{x+r}$.

Equivalently, the tail can be written as a sum. The payment due $j$ years after the valuation date, for $j>r$, has expected present value $R v^j{}_jp_x$. Since ${}_jp_x={}_rp_x\,{}_{j-r}p_{x+r}$, summing these terms gives

R\sum_{j=r+1}^{\infty}v^j{}_jp_x = R v^r{}_rp_x \sum_{s=1}^{\infty}v^s{}_sp_{x+r} = R v^r{}_rp_x a_{x+r}.

This decomposition relies on the stated payment dates, survival-contingent payment rule, and interest basis. It does not turn the post-guarantee installments into certain payments: the survival probability remains part of their value.

Worked Calculations

Worked Example: One Guaranteed Payment and a Mortality Change

A hypothetical annuity-immediate pays $800 annually. At issue, one payment is guaranteed. The annual effective interest rate is $i=20\%$. The probability of surviving the first year is $0.75$; conditional survival probability in every later year is $0.60$. Find the reserve at issue. The stated survival probabilities apply to each successive year, and interest is effective annually.

Method. Value the first payment as certain, then value the contingent tail beginning at year 2. This avoids applying a survival probability to the guaranteed installment.

The discount factor is $v=1/1.20=5/6$. The guaranteed payment at time 1 contributes

800v=800\left(\frac{5}{6}\right)=\$666.666\ldots

A payment at time 2 requires survival through both years, so its survival probability is $0.75(0.60)=0.45$. Its discounted probability factor is

v^2(0.45)=\left(\frac{5}{6}\right)^2(0.45)=0.3125.

For each later year, the discounted probability factor is $v(0.60)=(5/6)(0.60)=0.5$ times the preceding factor. Thus the tail is a geometric series with first factor $0.3125$ and ratio $0.5$. Its value is

800\left(\frac{0.3125}{1-0.5}\right)=\$500.

The reserve is $800(5/6)+500=\$1,166.666\ldots$, or $1,166.67 to the nearest cent. The check is that the $800 guarantee contributes $666.67, not $800, because its payment is one year away; adding the $500 contingent tail gives the stated reserve.

Worked Example: Reserve During a Four-Year Guarantee

A hypothetical annuity-immediate pays $1,000 annually, with four payments guaranteed from issue. Calculate the reserve immediately after the second payment, conditional on the annuitant being alive. The annual effective interest rate is $i=25\%$, and the conditional probability of surviving each future year is $0.50$. Assume this probability remains constant after the valuation date.

Method. There are $r=4-2=2$ guaranteed installments remaining. Value those as certain payments and then add the tail after the guarantee. This avoids treating payments after year 4 as guaranteed.

Here $v=1/1.25=0.8$. The two remaining guaranteed payments, due one and two years after the valuation date, have factor

a_{\overline{2}|}=v+v^2=0.8+0.64=1.44.

The life must survive two years to reach the end of the guarantee, with probability ${}_2p=0.50^2=0.25$. At that point, the annuity-immediate factor for the remaining lifetime, under the constant survival assumption, is

a_{x+2} = \sum_{j=1}^{\infty}(v\,0.50)^j = \frac{0.4}{1-0.4} = \frac{2}{3}.

The tail factor, valued at the current date, is therefore

v^2\,{}_2p\,a_{x+2} = 0.64(0.25)\left(\frac{2}{3}\right) = \frac{8}{75}.

The reserve is

\$1{,}000\left(1.44+\frac{8}{75}\right) = \$1{,}546.666\ldots

Thus the reserve is $1,546.67 to the nearest cent. As a check, the first post-guarantee payment is three years after the valuation date: its survival probability is $0.50^3=0.125$, and its discounted probability factor is $0.8^3(0.125)=0.064$. Summing that factor and its later terms with ratio $0.4$ gives $0.064/(1-0.4)=8/75$, as above.

Worked Example: Annuity-Due Payments at Issue

A hypothetical annuity-due pays $500 annually, with three payments guaranteed from issue. The first payment is due immediately at time 0; later payments are due at times 1, 2, and so on. Find the reserve immediately before the first payment. Assume $i=10\%$ and a conditional annual survival probability of $0.90$ in every year. Interest and survival assumptions remain constant.

Method. The guaranteed payments are due at times 0, 1, and 2. The contingent tail starts at time 3 and is an annuity-due conditional on survival to that time. This makes the valuation date relative to the first payment explicit.

Here $v=10/11$. The guaranteed-payment factor is

1+v+v^2 = 1+\frac{10}{11}+\frac{100}{121} = \frac{351}{121}.

The combined one-year survival-and-discount factor is $v(0.90)=(10/11)(9/10)=9/11$. Conditional on survival to time 3, the annuity-due factor is the geometric series

1+\frac{9}{11}+\left(\frac{9}{11}\right)^2+\cdots = \frac{1}{1-9/11} = \frac{11}{2}.

The factor for surviving and discounting to time 3 is $(9/11)^3=729/1331$. Therefore, the contingent tail factor at issue is

\left(\frac{9}{11}\right)^3\left(\frac{11}{2}\right) = \frac{729}{2662}.

The total reserve is

\$500\left(\frac{351}{121}+\frac{729}{2662}\right) = \$500\left(\frac{8445}{2662}\right) = \$1{,}586.957\ldots

The reserve is $1,586.96 to the nearest cent. The first payment contributes the full $500 because it is due immediately; the next two guaranteed payments are discounted. The tail is included only after multiplying by the probability and discount factor for reaching time 3.

Common Errors and Practical Checks

The most reliable check is to write the future payment dates relative to the valuation date. Label each installment “certain” or “survival-contingent” before calculating. For an annuity-immediate valued just after a payment, the first future payment is one year away; for an annuity-due valued before its first payment, an installment is due immediately.

Wrong: At issue, treat every payment during the guarantee as immediately payable, so the guaranteed value is $R g$. Correction: Discount each guaranteed installment to the valuation date. For an annuity-immediate, use $R a_{\overline{g}|}$.
Wrong: Multiply guaranteed installments by the probability of survival. Correction: A guaranteed installment is payable even if death occurs before its due date, so no survival probability applies to it.
Wrong: Start the contingent tail at the end of the guarantee but omit the probability of reaching that date. Correction: Include ${}_rp_x$, where $r$ is the number of years remaining in the guarantee.
Wrong: Use an annuity-due factor for payments made at year-end, or an annuity-immediate factor for an installment due immediately. Correction: Identify the due dates first and use an annuity factor with matching timing.

This separation matters because the guarantee changes the payment condition, not the interest rate or the survival model. The guaranteed part is valued without mortality; mortality affects only the later installments. If the guarantee has expired, the reserve is simply the value of the remaining life annuity conditional on survival at the valuation date.

Check Your Understanding

For each problem, identify which future payments are guaranteed before calculating. Unless stated otherwise, use annual effective interest, annual installments, and no expenses or premiums.

An annuity-immediate pays $500 annually. Two payments are guaranteed from issue. The interest rate is $25\%$, and future conditional survival probabilities are $0.50$ each year. Find the reserve at issue. (Answer: $773.33)
An annuity-immediate pays $900 annually and has a three-year guarantee. Find the reserve just after the first payment, with $i=10\%$ and constant annual survival probability $0.80$. (Answer: $2,831.40)
An annuity-due pays $250 annually, with three payments guaranteed from issue. Find the reserve immediately before the first payment if $i=4\%$ and annual survival probability is $0.90$ in every year. (Answer: $1,925.10)
An annuity-immediate pays $700 annually and has a four-year guarantee. Find the reserve just after the second payment if $i=25\%$ and each future annual survival probability is $0.50$. (Answer: $1,082.67)
An annuity-immediate pays $1,200 annually, with two payments guaranteed from issue. The interest rate is $10\%$; first-year survival probability is $0.90$, and conditional survival probability in every later year is $0.80$. Find the reserve at issue. (Answer: $3,986.78)

What You'll Learn