Discounted Payback Period

Study guide: Exam ALTAM

The Discounted Payback Period (DPP) of a policy is the first year by whose end the shareholders have recovered the money they put in at issue together with their required return. In profit testing it is the smallest $m$ for which the partial net present value of the profit signature, discounted at the risk discount rate, is zero or positive. It measures how long the capital committed to a policy stays at risk.

$\text{NPV}(m) = \sum_{t=0}^{m} \Pi_t\,(1+r)^{-t}$

$\text{DPP} = \min\left\{ m : \text{NPV}(m) \geq 0 \right\}$

Symbols: Following Dickson, Hardy & Waters, $\Pi_t$ is the expected profit at time $t$ per policy issued and $r$ is the risk discount rate. $\text{NPV}(m)$ is the partial NPV of the first $m$ years, and $\text{NPV}(n)$ is the policy's full NPV.
When there is none: If the full NPV is negative, the partial NPV never reaches zero and the policy has no discounted payback period. The shareholders never recover their outlay with the required return.
What moves it: A higher risk discount rate lengthens the period, because later profits count for less. Larger initial expenses and larger reserves lengthen it too, because reserves hold back profit in the early years and release it later.
What it does not show: Two policies with the same NPV can pay back at different times, and a policy that pays back quickly can still have a small NPV. The DPP is a measure of timing and risk, so it is reported with the NPV, the internal rate of return and the profit margin.

Payback Period of a 5-Year Term Insurance A 5-year term insurance on $(60)$ has the profit signature $(-250,\ 90.59,\ 90.14,\ 92.66,\ 88.95,\ 89.83)$ , worked out in Profit Testing. Find the discounted payback period at a risk discount rate of 10%.

Answer Discount each entry at 10% and accumulate the partial NPVs until the total turns non-negative.

$t$ $\Pi_t$ $\Pi_t(1.1)^{-t}$ $\text{NPV}(t)$

0 $-250.00$ $-250.00$ $-250.00$

1 90.59 82.35 $-167.65$

2 90.14 74.50 $-93.15$

3 92.66 69.62 $-23.53$

4 88.95 60.75 37.22

5 89.83 55.78 93.00

The partial NPV is still negative at time 3 and first becomes positive at time 4, so the discounted payback period is 4 years. The shareholders wait until the fourth year of a five-year policy to recover their $250 with a 10% return.

$t$	$\Pi_t$	$\Pi_t(1.1)^{-t}$	$\text{NPV}(t)$
0	$-250.00$	$-250.00$	$-250.00$
1	90.59	82.35	$-167.65$
2	90.14	74.50	$-93.15$
3	92.66	69.62	$-23.53$
4	88.95	60.75	37.22
5	89.83	55.78	93.00

Reserves Delay the Payback If the insurer holds no reserves for the same policy, the profit signature is $(-250,\ 174.30,\ 134.44,\ 90.05,\ 40.63,\ -14.46)$ . Find the discounted payback period at 10% and compare it with the policy that holds reserves.
Answer The present values of the entries for $t = 1$ to $5$ are 158.45, 111.11, 67.66, 27.75 and $-8.98$ .
 $\begin{align*} \text{NPV}(1) &= -250 + 158.45 \\ &= -91.55 \\ \text{NPV}(2) &= -91.55 + 111.11 \\ &= 19.56 \end{align*}$ 
The policy pays back in 2 years instead of 4. Its partial NPV rises to 114.97 at time 4 and falls to 105.99 at time 5, because of the loss in the last year. Without reserves, the profit emerges earlier and the NPV at 10% is higher than the 93.00 with reserves, but the loss in year 5 is left unfunded, which is why the insurer must hold at least the zeroized reserve.

How it appears on the exam

ALTAM asks candidates to calculate and interpret the discounted payback period with the NPV, IRR and profit margin of a profit test, for traditional, universal life and equity-linked policies.

Building the table: The usual route is a running total of the discounted signature, read off at the first year it is non-negative.

Comparisons: Questions may ask how the period changes under a stress-test assumption, a different risk discount rate or a different reserve basis.

Exam traps

Discounted, not undiscounted: The partial sums use the risk discount rate. Adding the undiscounted signature gives the plain payback period, which is shorter and not what the question asks for.

Signature, not vector: Each term is a profit per policy issued. Accumulating the profit vector ignores the policies that have ended and makes the payback look earlier.

Time 0 counts: The initial expenses at time 0 are the outlay being recovered. A payback period of $m$ means that the total first turns non-negative at the end of year $m$ .

Later losses: If the signature turns negative after the payback year, the partial NPV falls again. The definition still takes the first year it is non-negative, so a short period does not mean the later years are safe.