What a Payment Guarantee Changes
A life annuity normally pays only when the annuitant is alive at a payment date. A payment guarantee changes the contract during a stated initial period: payments in that period are made even if the annuitant dies. After the guarantee period, payments continue only if the annuitant is alive at the relevant payment dates. The guarantee therefore affects which payments depend on survival; it does not make the entire annuity certain.
To value the contract, first set out the payment dates. This is especially important because a due annuity pays at the beginning of each year, while an immediate annuity pays at the end. For the formulas below, payments are level, annual, and made in arrears or in advance as stated. The annuitant is age \(x\) at time 0. Interest is a deterministic effective annual rate \(i\), with \(v=1/(1+i)\), and mortality is described by \({}_kp_x\), the probability that a life aged \(x\) survives at least \(k\) years. Each payment is 1 unless a different amount is specified.
Payment Dates and Present-Value Formulas
For an annuity-due, payments occur at times \(0,1,2,\ldots\). With an \(n\)-payment guarantee, the payments at times \(0\) through \(n-1\) are certain. A later payment at time \(k\) is made only if the annuitant has survived \(k\) years. Let \(G^{due}_{x,n}\) denote the expected present value of this unit annuity.
For an annuity-immediate, payments occur at times \(1,2,3,\ldots\). Its first \(n\) payments, at times \(1\) through \(n\), are certain. Subsequent payments at time \(k\) require survival to time \(k\). Let \(G^{imm}_{x,n}\) denote the expected present value of this unit annuity.
These expressions assume the contract makes no further payments after the listed guarantee payments if death occurs during the guarantee period; in particular, they do not include a separate death benefit. They also assume payments are made at the stated annual dates. A contract with monthly payments, a guarantee measured in years rather than payment counts, or a benefit paid at death requires payment dates and conditions to be specified before using a valuation formula.
Why the Formulas Work
A present value is the sum of each possible payment amount multiplied by its discount factor. For a payment at time \(k\) that is conditional on survival, its expected amount is \(1\cdot{}_kp_x\): it is paid with probability \({}_kp_x\) and otherwise is zero. Discounting that expected amount gives \(v^k{}_kp_x\). A guaranteed payment has expected amount 1, so its present value is simply \(v^k\).
For the due contract, apply this reasoning to each payment date. At \(k=0,\ldots,n-1\), payment is certain, giving \(\sum_{k=0}^{n-1}v^k\). At \(k\ge n\), payment requires survival to time \(k\), giving \(\sum_{k=n}^{\infty}v^k{}_kp_x\). Adding the two groups gives the due formula. For the immediate contract, the same reasoning applies to its different schedule: guaranteed times are \(1,\ldots,n\), and survival-contingent times start at \(n+1\).
This is an application of expected present value: by linearity of expectation, the value is the sum of the expected present values of the individual payments. No independence between different survival events is needed for that sum. The survival probabilities must, however, represent the mortality assumption used for the annuitant.
Worked Valuations
Worked Example: Due Annuity with Constant Mortality
A hypothetical contract pays 1 at the beginning of each year while the annuitant is alive, with the first three payments guaranteed. The effective annual interest rate is \(i=0.05\). Assume a constant annual probability of death \(q=0.10\), independently of the deterministic interest rate. Find the expected present value, rounded to four decimal places.
Method. There are certain payments at times 0, 1, and 2. Payments at times 3 and later depend on survival. Under this constant mortality assumption, the one-year survival probability is \(p=1-q=0.90\), and \({}_kp_x=0.90^k\).
The discount factor is \(v=1/1.05=20/21\). For the survival-contingent payments, each successive expected discounted payment is multiplied by \(vp=(20/21)(0.90)=6/7\). Thus the tail is a geometric series:
The guaranteed-payment part is \(1+20/21+400/441=1261/441=2.8594\) to four decimal places. Since the common ratio \(6/7\) is less than 1, the infinite tail is \((6/7)^3/[1-(6/7)]=216/49=4.4082\) to four decimal places. Therefore the total is \(1261/441+216/49=7.2676\), rounded to four decimal places.
Check. The guaranteed payments alone are worth 2.8594. The additional value of 4.4082 is positive, as expected, because later payments can be made if the annuitant survives. A common mistake is to use \(q^k\) for survival; survival through \(k\) years is \(p^k=0.90^k\), not \(0.10^k\).
Worked Example: Immediate Annuity with a Two-Payment Guarantee
A hypothetical annuity pays 1 at the end of each year. Its first two payments are guaranteed, and later payments are made only if the annuitant is alive at the payment date. The effective annual interest rate is \(i=0.04\). Assume the annual probability of death is 0.20 in every year, independently of deterministic interest. Calculate the expected present value, rounded to four decimal places.
Method. The guaranteed payments occur at times 1 and 2. The first payment that depends on survival is at time 3. Here \(p=0.80\), \(v=1/1.04=25/26\), and \(vp=(25/26)(0.80)=10/13\).
The guaranteed part is \(v+v^2=25/26+625/676=1275/676=1.8861\) to four decimal places. Since \({}_kp_x=0.80^k\), the post-guarantee part is
Adding the two parts gives \(1.8861+1.9724=3.8585\), rounded to four decimal places. The exact fractions give the same rounded result: \(1275/676+13000/6591=3.8585\) to four decimal places.
Check. The second payment is guaranteed even if the annuitant dies before time 2. Starting the survival-contingent sum at time 2 would omit that guarantee and would value a different contract.
Worked Example: Due Annuity Using Specified Survival Probabilities
A hypothetical annuity-due pays 1 annually and guarantees its first four payments. The effective annual interest rate is \(i=0.05\). A mortality model gives \({}_4p_x=0.80\), \({}_5p_x=0.60\), \({}_6p_x=0.30\), and \({}_7p_x=0\). Find the expected present value, rounded to four decimal places.
Method. The certain payments are at times 0, 1, 2, and 3. The payments at times 4, 5, and 6 are survival-contingent. There are no later payments under the stated model because survival to time 7 has probability zero. With \(v=20/21\), the discounted certain payments are \(1\), \(0.9523810\), \(0.9070295\), and \(0.8638376\). Their sum is \(3.7232480\).
The contingent values are calculated payment by payment:
- At time 4: \(v^4{}_4p_x=0.8227025(0.80)=0.6581620\).
- At time 5: \(v^5{}_5p_x=0.7835262(0.60)=0.4701157\).
- At time 6: \(v^6{}_6p_x=0.7462154(0.30)=0.2238646\).
The contingent part is \(0.6581620+0.4701157+0.2238646=1.3521423\). The total is \(3.7232480+1.3521423=5.0753903\), or 5.0754 rounded to four decimal places.
Check. The supplied survival probabilities decrease with time, as survival probabilities should. The first survival probability used is \({}_4p_x\), because time 4 is the first payment date after the four due payments at times 0 through 3.
Common Timing Errors
The guarantee period is a count of scheduled payments, so its final guaranteed payment depends on whether payments are due or immediate. An effective way to avoid an off-by-one error is to list the times before substituting probabilities.
| Contract | Guaranteed payment times | First survival-contingent time |
|---|---|---|
| Due, \(n\)-payment guarantee | 0 through \(n-1\) | \(n\) |
| Immediate, \(n\)-payment guarantee | 1 through \(n\) | \(n+1\) |
For example, a wrong calculation for a due annuity with three guaranteed payments might put the guaranteed terms at times 1, 2, and 3, then start the survival-contingent terms at time 4. That calculation has shifted the entire due schedule one year later. The correction is to use certain terms \(1,v,v^2\), then begin the contingent sum at \(v^3{}_3p_x\). Conversely, for an immediate annuity with three guaranteed payments, the certain terms are \(v,v^2,v^3\), and the contingent sum starts at \(v^4{}_4p_x\).
Other errors include applying the survival probability to guaranteed payments, using an annual death probability in place of a survival probability, and forgetting that discounting and survival are separate factors. A post-guarantee payment at time \(k\) contributes \(v^k{}_kp_x\): \(v^k\) accounts for interest, while \({}_kp_x\) accounts for the chance of being alive.
Check Your Understanding
For each hypothetical contract, identify the guaranteed and survival-contingent dates before calculating. Round monetary values to four decimal places where applicable.
- An annuity-immediate guarantees its first two payments. What are the guaranteed payment times, and at what time does the first survival-contingent payment occur? (Answer: Guaranteed at times 1 and 2; first survival-contingent payment at time 3.)
- A due annuity guarantees two payments. Let \(i=0.05\), and suppose the annual probability of death is 0.20 in every year. Find its value per unit payment. (Answer: \(1+20/21+256/105=4.3905\).)
- An immediate annuity guarantees two payments. Let \(i=0.04\), \({}_1p_x=0.90\), \({}_2p_x=0.75\), \({}_3p_x=0.50\), and \({}_4p_x=0.20\), with no payments after time 4. Find its value per unit payment. (Answer: \(v+v^2+v^3(0.50)+v^4(0.20)=2.5016\), where \(v=1/1.04\).)
- A due annuity guarantees three payments, with \(i=0\), \({}_3p_x=0.70\), \({}_4p_x=0.40\), \({}_5p_x=0.10\), and \({}_6p_x=0\). Find the value per unit payment. (Answer: \(3+0.70+0.40+0.10=4.2000\).)
- At \(i=0\), compare a two-payment-guaranteed due annuity and a two-payment-guaranteed immediate annuity. For both, let \({}_2p_x=0.60\), \({}_3p_x=0.30\), and \({}_4p_x=0\). Find each value and their difference. (Answer: Due value \(2+0.60+0.30=2.9000\); immediate value \(2+0.30=2.3000\); due value exceeds immediate value by \(0.6000\).)