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

Risk Discount Rate and Cost of Capital in Profit Testing

Updated 11 October 2026 · Fact-checked

The risk discount rate (RDR) is the rate used to discount a profit test's profit signature. It reflects the return the insurer needs for the risk taken. Capital held beyond reserves has a cost: the gap between the rate earned on it and the RDR. You allow for this in the discounted profit.

Understand Risk Discount Rate and Cost of Capital

A profit test projects the profit the insurer expects to emerge each year from a policy. These profits arrive in the future and are uncertain. To judge whether the product is worth selling, you convert them to one present value. You do this by discounting at the risk discount rate.

The RDR is not the investment return on the assets. It is the return the insurer requires on the money it puts at risk. A riskier product, or one with more uncertain assumptions, calls for a higher RDR. The higher the RDR, the lower the net present value (NPV) of the profits. So a higher RDR makes a product look less attractive.

There are two common ways to choose an RDR. One is to build it up from a risk-free rate plus a margin for the risk of the product. The other is to use the return the shareholders require on their capital. Capital asset pricing ideas can help: a higher systematic risk (beta) justifies a higher required return. Any choice needs a stated reason, and the choice should be tested by sensitivity analysis.

Regulators require the insurer to hold solvency capital in addition to reserves. This capital is tied up for the life of the policy. It usually earns only the interest rate on the assets backing it, which is normally lower than the RDR. The shareholder therefore loses value each year. That loss is the cost of capital. A profit test that ignores it overstates profit.

To allow for it, you add the capital to the reserve strain in the cashflows. You set up the extra capital at the start of each year and release it, with interest, at the end of the year. The profit vector then falls in years when capital is set up and rises in years when it is released. The NPV at the RDR falls compared with the same test without capital.

Key rules to remember

Discounted profit (NPV) at the RDR: NPV = Σ (t = 1 to n) Π_t × v_r^t, where v_r = 1 ÷ (1 + r); Π_t is the profit signature element for year t (including the policy's chance of being in force) and r is the risk discount rate.
Profit vector with capital: Profit_t = (tV + C_(t−1) + P − e)(1 + i) − q × (S + extra cashflows) − p × (t+1V + C_t); All amounts are per policy in force at the start of the year. tV is the reserve and C is the required capital at the start (t−1 to t) and end of the year. P is the premium, e the expenses, S the sum assured and q the probability of claim in the year, with p = 1 − q. The end-of-year reserve and capital are weighted by the survival probability p. Multiply the result by the probability of being in force at the start of the year to get the profit signature. State the interest rate i on the capital.
Cost of capital for one year: Cost = C × (r − i); C is the capital held for the year, r is the RDR and i is the earned rate on the capital. It is positive when r is greater than i.
Present value of the cost of holding capital: PV(cost) = Σ (t = 1 to n) C_(t−1) × (r − i) × p × v_r^t; p is the probability that the policy is in force at the start of the year. This gives the NPV reduction from holding capital when each year's capital is C_(t−1) and it is released at the year-end.
Profit margin: Profit margin = NPV of profits ÷ PV of premiums, both at the RDR; Use the same RDR for both parts.

How to solve Risk Discount Rate and Cost of Capital questions

Use this method for any question on choosing the RDR or allowing for solvency capital in a profit test.

1Read the question and note the RDR, the interest rate earned on assets and the interest rate on capital. Note the capital basis, such as a percentage of the reserve or of the sum assured.
2Set up the year-by-year table with the in-force probabilities. Write down the premium, expenses, interest and claims for each year.
3Add the reserve and the required capital as the amount that must be held at the end of each year. The amount at the start of a year is the previous year's end amount.
4Work out the profit vector for each year without forgetting the interest on the opening amount.
5Convert the profit vector to the profit signature by multiplying by the probability that the policy is in force at the start of each year.
6Discount the profit signature at the RDR to get the NPV. Do not discount at the investment rate.
7If you are asked about the cost of capital, compare the NPV with and without the capital, or use Σ C × (r − i) × v_r^t.
8State the result in words. Say whether the product meets the target, and comment on how sensitive it is to the RDR.

Quickest way: Find the cost of capital without rebuilding the full test

When to use it: Use this when you already have the profit signature without capital and you are asked how much the capital reduces the NPV.

List the capital held at the start of each year and the probability the policy is in force.
Multiply each year's capital by (r − i) and by the in-force probability.
Discount each answer by v_r^t and add the amounts. This total is the reduction in NPV.
Subtract it from the original NPV and check that the sign is sensible: the NPV must fall when r is greater than i.
Be careful with the timing of the capital. Only use this shortcut when the capital is set up at the start of the year and released at the end.

Common mistakes in Risk Discount Rate and Cost of Capital

Discounting the profit signature at the investment return instead of the RDR.
Both are interest rates, and the investment return is used elsewhere in the same test to grow assets.
Fix: Use the investment rate for the interest earned in each year. Use the RDR only for the final discounting.
Ignoring the capital requirement and treating reserves as the only amount held.
Earlier questions on profit testing mention only reserves.
Fix: Read the question for any capital or solvency margin. Add it to the amount held at the end of each year.
Applying the cost of capital as C × r instead of C × (r − i).
Students forget that the capital earns interest while it is held.
Fix: Remember the capital earns i. Only the shortfall (r − i) is the loss to the shareholder.
Using the profit vector instead of the profit signature when discounting.
The two tables look similar.
Fix: Multiply the profit vector by the probability that the policy is in force at the start of the year. Then discount.
Saying a higher RDR makes a product more profitable.
A higher rate sounds like a higher return.
Fix: A higher RDR reduces the present value of future profits. Check the direction every time.

Worked examples

Example 1

A policy produces a profit signature of ₹0 in year 1, ₹4,000 in year 2 and ₹6,000 in year 3. Calculate the NPV at a risk discount rate of 10% a year, and then at 15% a year. Comment on the result.

Show the solution

At 10%: v = 1 ÷ 1.10 = 0.909091.
Year 2 factor: v² = 0.826446. Year 3 factor: v³ = 0.751315.
NPV = 0 + 4,000 × 0.826446 + 6,000 × 0.751315.
4,000 × 0.826446 = 3,305.78. 6,000 × 0.751315 = 4,507.89.
NPV at 10% = 3,305.78 + 4,507.89 = 7,813.67.
At 15%: v² = 1 ÷ 1.3225 = 0.756144. v³ = 1 ÷ 1.520875 = 0.657516.
4,000 × 0.756144 = 3,024.58. 6,000 × 0.657516 = 3,945.10.
NPV at 15% = 3,024.58 + 3,945.10 = 6,969.68.

Answer: NPV at 10% is about ₹7,814 and at 15% about ₹6,970. The higher RDR gives a lower NPV, so the product looks less attractive when the required return is higher.

Example 2

An insurer holds capital of ₹20,000 per policy at the start of each of the 3 years of a policy. The capital is released at the end of each year. It earns 4% a year. The risk discount rate is 10% a year. The policy is in force at the start of year 1, year 2 and year 3 with probabilities 1, 0.9 and 0.8. Find the present value of the cost of holding this capital.

Show the solution

The annual loss on each unit of capital is r − i = 0.10 − 0.04 = 0.06.
Year 1 cost: 20,000 × 0.06 × 1 = 1,200. Discount: 1,200 ÷ 1.10 = 1,090.91.
Year 2 cost: 20,000 × 0.06 × 0.9 = 1,080. Discount: 1,080 ÷ 1.21 = 892.56.
Year 3 cost: 20,000 × 0.06 × 0.8 = 960. Discount: 960 ÷ 1.331 = 721.26.
Total = 1,090.91 + 892.56 + 721.26 = 2,704.73.

Answer: The present value of the cost of holding the capital is about ₹2,705 per policy. This amount reduces the NPV of the profits.

Exam tips

Write the RDR and the earned rate side by side at the top of your answer so that you never swap them.
For written questions, always add a sentence on how the choice of RDR affects the result. Examiners reward this comment.
Show the capital as its own line in the cashflow table. This makes your working easy to follow and earns method marks.
In multiple-choice questions, check the direction first: a higher RDR means a lower NPV, and a higher capital means a lower NPV when r is greater than i.
In the computer-based paper, keep r and i in separate named cells or variables so a change in one does not alter the other.

Practice questions from Projecting expected future cashflows and profit testing

Risk Discount Rate and Cost of Capital in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Risk Discount Rate and Cost of Capital: frequently asked questions

What is the risk discount rate in a profit test?

It is the rate used to discount the profit signature to a present value. It reflects the return the insurer needs for the risk of the product. It is normally higher than the risk-free rate.

How do I choose a risk discount rate for life insurance?

You can start from a risk-free rate and add a margin for the risk of the product. You can also use the return that shareholders require on their capital. The choice should be justified and tested with sensitivity analysis.

Why does holding solvency capital reduce profit?

The capital is tied up and normally earns less than the rate the shareholder requires. The difference between the RDR and the earned rate is a loss each year. This loss lowers the NPV.

Is the risk discount rate the same as the interest rate used on assets?

No. The asset rate is the return you assume the insurer earns on its investments. The RDR is the return required for the risk taken. They are used at different steps of the profit test.