Actuarial Mathematics for Modelling · Projecting expected future cashflows and profit testing

Profit Vector, Profit Signature and Discounted Profit

Updated 11 October 2026 · Fact-checked

The profit vector gives expected profit at each year end per policy in force at the start of that year. Multiply each entry after time 0 by the probability the policy is in force at the start of that year to get the profit signature. Discount the signature at the risk discount rate and sum to get the NPV.

Understand Profit Vector, Profit Signature and Discounted Profit

A profit test projects the cashflows of one policy year by year and asks: how much profit does the insurer expect, and when? Each year you take the premium, subtract expenses, add interest, then take away expected claims and the reserve that must be set up at year end. What is left is profit for that year.

The profit vector collects these yearly profits, written Pr_0, Pr_1, Pr_2, and so on. Each entry is per policy in force at the start of that year. It assumes the policy has survived to that point. It does not yet allow for the chance that the policy has already gone.

The profit signature fixes this. It gives the profit per policy issued. You multiply each profit vector entry by the probability that the policy is still in force at the start of that year. Time 0 needs no adjustment, because every policy is in force at issue. Since later years are weighted by survival, the signature is the right basis for adding cashflows across years.

The discounted profit, or net present value (NPV), is the present value of the signature at the risk discount rate. This rate reflects the return the insurer needs for the risk it takes. A positive NPV means the contract meets the required return on this basis.

Remember the order: profit vector first, then signature (survival weighting), then discounting. Mixing up the order is the main source of lost marks.

Key rules to remember

Profit vector entry for year t (t ≥ 1): Pr_t = (t-1V + P_t − e_t)(1 + i_t) − q_{x+t-1} × S_t − p_{x+t-1} × tV; Per policy in force at the start of year t. Premium and expense at start of year, claims and reserve at end of year. S_t is the death benefit. Add similar terms for surrender or other decrements if the question gives them.
Profit vector at time 0: Pr_0 = premium at time 0 − initial expenses − commission − reserve set up at time 0 (if any); Usually negative (new business strain). Only include cashflows that happen at time 0. If the premium and expenses fall at the start of year 1, they are handled in Pr_1, and Pr_0 is 0 or only pre-issue costs. Take care not to count the same premium or expense twice.
Profit signature: Π_0 = Pr_0 ; Π_t = (t-1)p_x × Pr_t for t ≥ 1; (t-1)p_x is the probability that a life aged x at issue is in force at the start of year t. Use the right survival term, not t p_x.
Survival probability to start of year t: (t-1)p_x = p_x × p_{x+1} × … × p_{x+t-2}; Each p = 1 − q. For t = 1 this is 1.
Net present value of profit: NPV = Σ Π_t × v^t, with v = 1 ÷ (1 + r); r is the risk discount rate, summed from t = 0 to the end of the term. This is different from the interest rate i used in the reserves and in the profit vector.
Profit margin: Profit margin = NPV ÷ PV of premiums; The PV of premiums uses the same survival probabilities and risk discount rate, so the premium at time t is weighted by (t-1)p_x × v^(t-1) if paid at the start of year t.

How to solve Profit Vector, Profit Signature and Discounted Profit questions

Use this method for any question on the profit vector, profit signature or NPV.

1List the data for each year: premium, expenses, interest rate, death benefit (and other benefits), reserves at start and end of the year, and mortality rates.
2Compute each profit vector entry Pr_t with the formula. Work year by year and show the pieces: start-of-year cashflows, interest, claims, end-of-year reserve.
3Find Pr_0 separately. It is usually the initial expense minus any premium paid at time 0, with any reserve at time 0 allowed for.
4Compute the survival probabilities (t-1)p_x for t = 1, 2, …, using p = 1 − q at each age.
5Build the profit signature: Π_0 = Pr_0 and Π_t = (t-1)p_x × Pr_t.
6Discount each Π_t at the risk discount rate using v^t, then add them to get the NPV.
7If asked, divide the NPV by the PV of premiums for the profit margin, and state the result with units.

Quickest way: Table method for the signature and NPV

When to use it: Use when the profit vector is given or after you have built it, and you need the signature and NPV fast, for example in a multiple-choice question.

Set up columns: t, Pr_t, (t-1)p_x, Π_t, v^t, Π_t × v^t.
Build the survival column by repeated multiplication: each new row is the previous row times the previous year's p.
Multiply Pr_t by the survival column, then by v^t, and sum the last column.
Check: the signature should be smaller than the vector in absolute terms for t ≥ 2, and Π_1 = Pr_1. If not, you used the wrong survival term.

Common mistakes in Profit Vector, Profit Signature and Discounted Profit

Using t p_x instead of (t-1)p_x when forming the signature.
Profit emerges at the end of year t, so it feels natural to use survival to time t.
Fix: The vector entry is already per policy in force at the start of year t, and it contains the year's own mortality. So weight by survival to the start of year t only, (t-1)p_x.
Discounting the profit vector instead of the signature.
Students skip the survival step because the vector looks like a complete set of cashflows.
Fix: Always convert to the signature before discounting. Discounting the vector ignores policies that have gone, so it gives a wrong NPV. It is typically too high when the later profits are positive.
Using the risk discount rate inside the profit vector, or the earned interest rate when discounting.
Both rates appear in the question, and both are called an interest rate.
Fix: Interest earned on the funds (i) goes in the profit vector. The risk discount rate (r) is used only for discounting the signature.
Mixing up the reserve terms: using the end-of-year reserve for all policies, or the wrong year's reserve.
The reserve at the start of the year and at the end of the year look alike.
Fix: Add the reserve at the start of the year (t-1V) with the premium. Take away p × tV at the end of the year, because only survivors carry a reserve forward. The death benefit S_t is paid for the q share instead.
Counting the time 0 premium or expense twice, or leaving out Pr_0 in the NPV.
Students merge year 1 with time 0 and lose track of which cashflows have been used.
Fix: Decide the timing first. Cashflows at issue go in Pr_0. Premiums and expenses at the start of year 1 can be in Pr_1 with the interest factor. Always include Π_0 in the NPV, as it is not discounted.
Rounding survival probabilities and discount factors too early.
Tables are done by hand and rounded to two or three decimals.
Fix: Keep at least five significant figures through the working, and round only the final answer.

Worked examples

Example 1

A three-year policy has profit vector Pr_0 = −₹150, Pr_1 = ₹50, Pr_2 = ₹80, Pr_3 = ₹120, all per policy in force at the start of each year. The mortality rates are q_x = 0.01, q_{x+1} = 0.015 and q_{x+2} = 0.02. (a) Find the profit signature. (b) Find the NPV at a risk discount rate of 10% per year.

Show the solution

Survival to the start of year 1: 1. To the start of year 2: p_x = 0.99. To the start of year 3: p_x × p_{x+1} = 0.99 × 0.985 = 0.97515.
Signature: Π_0 = −150. Π_1 = 1 × 50 = 50. Π_2 = 0.99 × 80 = 79.20. Π_3 = 0.97515 × 120 = 117.018.
Discount at 10%: v = 1/1.1. Π_1 v = 50 ÷ 1.1 = 45.4545. Π_2 v² = 79.20 ÷ 1.21 = 65.4545. Π_3 v³ = 117.018 ÷ 1.331 = 87.9174.
Sum: −150 + 45.4545 + 65.4545 + 87.9174 = 48.8264.

Answer: (a) Profit signature: (−₹150, ₹50, ₹79.20, ₹117.018). (b) NPV ≈ ₹48.83 per policy issued.

Example 2

For the first two years of a long-term contract, annual premium is ₹1,000 paid at the start of each year. Expenses are ₹150 at the start of year 1 and ₹50 at the start of year 2. Interest earned is 6% per year. The death benefit is ₹50,000 at the end of the year of death. Mortality: q_x = 0.002 and q_{x+1} = 0.003. Reserves per policy in force: 0V = 0, 1V = ₹300, 2V = ₹620. Assume no other decrements, and that Pr_0 = 0. Find Pr_1, Pr_2, the first two signature entries and their NPV at a risk discount rate of 8%.

Show the solution

Year 1: (0 + 1,000 − 150) × 1.06 = 850 × 1.06 = 901.
Year 1 claims: 0.002 × 50,000 = 100. Year 1 reserve for survivors: 0.998 × 300 = 299.4. Pr_1 = 901 − 100 − 299.4 = 501.60.
Year 2: (300 + 1,000 − 50) × 1.06 = 1,250 × 1.06 = 1,325.
Year 2 claims: 0.003 × 50,000 = 150. Year 2 reserve for survivors: 0.997 × 620 = 618.14. Pr_2 = 1,325 − 150 − 618.14 = 556.86.
Survival to the start of year 2: p_x = 0.998. Π_1 = 501.60. Π_2 = 0.998 × 556.86 = 555.746.
Discount at 8%: 501.60 ÷ 1.08 = 464.444. 555.746 ÷ 1.1664 = 476.460.
NPV of these two years = 464.444 + 476.460 = 940.904.

Answer: Pr_1 = ₹501.60 and Pr_2 = ₹556.86. The signature entries are Π_1 = ₹501.60 and Π_2 ≈ ₹555.75. The NPV of these two years is about ₹940.90 per policy issued.

Exam tips

Write out the table of t, Pr_t, survival, Π_t and v^t even in a multiple-choice question. Most wrong answers come from skipped survival terms.
In written questions, name each rate: i for earned interest, r for the risk discount rate. Examiners give marks for showing the right rate in the right place.
Look for the timing of premiums and expenses (start of year) and claims (end of year). Check this before you write any numbers.
If the question gives a profit vector, go straight to the signature. If it asks you to explain why the signature differs from the vector, say the vector is per policy in force and the signature is per policy issued.
State your assumptions, such as no other decrements or reserves per policy in force, and give the final NPV in rupees to two decimal places.

Practice questions from Projecting expected future cashflows and profit testing

Profit Vector, Profit Signature and Discounted Profit: frequently asked questions

What is the difference between the profit vector and the profit signature?

The profit vector gives profit at each year end per policy in force at the start of that year. The profit signature gives profit per policy issued, so it is the vector multiplied by the probability of being in force at the start of the year. The signature is the one you discount to get the NPV.

Why is Π_1 equal to Pr_1?

Every policy issued is in force at the start of year 1, so the survival probability is 1. The weighting only starts to matter from year 2, where it is p_x.

How do I calculate the net present value of a profit signature?

Multiply each signature entry Π_t by v^t, where v = 1 ÷ (1 + r) and r is the risk discount rate. Then add the results from t = 0 onwards. Π_0 is not discounted.

Which interest rate do I use in the profit vector and which in the NPV?

Use the rate earned on the insurer's funds in the profit vector, as stated in the basis. Use the risk discount rate in the NPV. These are often different, and mixing them up gives the wrong answer.

Can the profit vector be negative after time 0?

Yes. A large reserve increase or a high claim in a year can make the profit negative, so the signature has negative entries. This often happens with reserves that rise quickly, and it is why the NPV and the timing of profit both matter.