Actuarial Mathematics for Modelling · Gross premiums and reserves

Reserves with Bonuses, Surrender Values and Surplus Analysis

Updated 11 October 2026 · Fact-checked

With-profits reserves value guaranteed benefits plus the bonuses the basis assumes, less future premiums. A surrender value is the cash paid on early exit. A paid-up value is the reduced sum assured that cash would buy instead. Analysis of surplus splits actual profit into mortality, interest and expense sources by comparing expected and actual experience.

Understand Reserves with Bonuses, Surrender Values and Profit Analysis

A with-profits policy pays a guaranteed sum assured plus bonuses. Bonuses are added from time to time out of surplus. Because the final benefit is bigger than the sum assured, a reserve that ignores bonuses is too low. You must decide what future bonuses to assume on the valuation basis.

Two common bonus types are simple reversionary bonus and compound reversionary bonus. Simple bonus is a percentage of the original sum assured each year. Compound bonus is a percentage of the sum assured plus bonuses already added. Once declared, a reversionary bonus is guaranteed. Future bonuses are not guaranteed, so a valuation basis may allow for them in full, in part, or not at all. The question will state which.

A policyholder may stop paying premiums. The insurer can then pay a surrender value in cash, or convert the policy to a paid-up policy. A paid-up policy has no more premiums and a reduced sum assured. The surrender value is usually based on the reserve or asset share, less a deduction for expenses and for the cost of early exit. The paid-up sum assured is found by equating the value of the paid-up benefit to the amount available.

Analysis of surplus asks where profit came from in a year. You compare what the valuation basis expected with what actually happened. The main sources are mortality, interest and expenses. The total of the sources equals the total surplus only if you use a consistent method and order. You also need the policy values at the start and end of the year.

Key rules to remember

Simple reversionary bonus benefit: Benefit after t years = S × (1 + b × t); S is the sum assured and b is the annual bonus rate on the original sum assured. This applies to bonuses already declared.
Compound reversionary bonus benefit: Benefit after t years = S × (1 + b)^t; b is applied to the sum assured plus bonuses already added.
Gross premium policy value with bonuses: tV = PV of future benefits (including assumed bonuses) + PV of future expenses − PV of future gross premiums; Use the stated valuation basis for mortality, interest, expenses and bonuses. Say which bonuses you include.
Net interest rate for compound bonus: i′ = (1 + i) ÷ (1 + b) − 1; Use only if the bonus attaches at the start of each year and the benefit is paid at a fixed point. Check the timing in the question.
Paid-up sum assured: Paid-up sum assured = amount available ÷ value of £1 of paid-up benefit at the current age and term (e.g. A(x+t:n−t)); The amount available is the surrender value or the policy value, as the question states. The factor must match the contract type and the remaining term. Divide by the factor; do not multiply.
Surrender value: Surrender value = reserve (or asset share) − deduction; The deduction may be a percentage or a fixed rupee amount. Use the rule given in the question.
Recursion for policy values (premium paid at start of year): (tV + P − e)(1 + i) = q × (S + bonus) + p × (t+1)V; Here q is the death probability in the year and p = 1 − q. If this holds on the basis, the expected profit is zero. Death benefit is taken at the end of the year.
Mortality profit: Mortality profit = (expected deaths − actual deaths) × (death benefit − (t+1)V); The term in brackets is the death strain at risk, also called the sum at risk. A positive answer means fewer deaths than expected.
Interest profit: Interest profit = (tV + P − e) × (actual i − basis i) × number of policies; Use the funds at the start of the year after premiums and expenses. Say which base you use.
Expense profit: Expense profit = (expected expense − actual expense) × (1 + i) × number of policies; Expenses at the start of the year are accumulated to year-end. Use the actual rate if you sequence interest first.

How to solve Reserves with Bonuses, Surrender Values and Profit Analysis questions

Use this order for any question on with-profits reserves, surrender or paid-up values, or surplus analysis. State your assumptions as you go.

1Read the contract type, term, premium frequency, bonus type and the date. Note what is guaranteed and what is a future assumption.
2Write down the valuation basis: mortality, interest, expenses and the bonus allowance. Check whether future bonuses are included.
3For a reserve, write the cashflows as a time line. Value future benefits and expenses, and subtract future premiums at the valuation rate.
4For a surrender or paid-up value, find the reserve or asset share first. Apply any deduction stated. Then, for paid-up, divide by the paid-up benefit factor for the remaining term.
5For surplus by source, list expected and actual figures for the year: deaths, interest rate, expenses and policy values. Fix an order and say it.
6Calculate each source with its formula. Keep the sign: positive for gain, negative for loss.
7Add the sources and check the total against the expected surplus where you can. A mismatch usually means an order or base error.
8State the answer in rupees with a short comment, such as which source drove the result.

Quickest way: Three-line check for paid-up and surplus questions

When to use it: Use in multiple-choice questions or when a written question gives most of the factors and asks only for the final figure.

For paid-up value: write amount available ÷ factor. Check the answer is smaller than the original sum assured when premiums have stopped early.
For mortality profit: write (expected − actual deaths) × (S − next-year policy value). Do not use S alone.
For interest profit: write (policy value + premium − expense) × rate difference × policies. If expense is paid at the start, subtract it before applying interest.
Sanity check the sign. Fewer deaths, higher interest and lower expenses give gains.

Common mistakes in Reserves with Bonuses, Surrender Values and Profit Analysis

Using the full sum assured instead of the sum at risk in mortality profit.
Students remember that a death costs the sum assured and forget that the reserve is released.
Fix: Use death benefit minus the policy value at the end of the year. Write the term as (S − (t+1)V) every time.
Multiplying instead of dividing by the paid-up factor.
The factor is a present value less than 1, so students confuse it with an accumulation.
Fix: Paid-up sum assured = amount available ÷ factor. Check the result is larger than the amount available, as a rupee of benefit costs less than a rupee.
Including future bonuses in the reserve when the basis excludes them, or leaving them out when the basis includes them.
The word bonus appears in the question and students add it automatically.
Fix: Underline the bonus instruction in the valuation basis. Only declared bonuses are guaranteed; value future bonuses only if told.
Applying the surrender deduction to the wrong figure.
Questions vary: a percentage of the reserve, a fixed amount, or a percentage of premiums paid.
Fix: Write the surrender rule in words first, then compute. Show the reserve, the deduction and the net value as three separate lines.
Applying interest to the wrong base in the interest profit.
Students use the policy value alone and miss premiums and start-of-year expenses.
Fix: Use policy value + premium − expense at the start of the year. State that base before you multiply.
Changing the order of sources mid-solution so the total does not add up.
Interest and expense effects overlap, so the split depends on the order.
Fix: Choose one order, state it, and use it for all sources. Then check the total against the actual profit.

Worked examples

Example 1

A with-profits endowment policy has a sum assured of ₹10,00,000. After 5 years the policyholder stops paying premiums. The policy value at that date is ₹1,60,000. The insurer's surrender rule is the policy value less a 10% deduction. The remaining term is 15 years. The value of ₹1 paid-up endowment benefit at the current age for 15 years is 0.72. Find the surrender value and the paid-up sum assured, assuming the paid-up value is bought with the surrender value and no future bonuses are included.

Show the solution

Deduction = 10% × ₹1,60,000 = ₹16,000.
Surrender value = ₹1,60,000 − ₹16,000 = ₹1,44,000.
Paid-up sum assured × 0.72 = ₹1,44,000.
Paid-up sum assured = ₹1,44,000 ÷ 0.72 = ₹2,00,000.
Check: ₹2,00,000 × 0.72 = ₹1,44,000. The paid-up amount is below the original ₹10,00,000, as expected.

Answer: Surrender value ₹1,44,000. Paid-up sum assured ₹2,00,000.

Example 2

An insurer has 1,000 identical with-profits policies in force at the start of a year. Each has sum assured ₹5,00,000. Per policy: policy value at start ₹60,000, annual premium ₹40,000, expense at start of year ₹2,000 on the basis, policy value at end of year ₹1,00,000. Basis: interest 5%, mortality rate 0.00725. Actual experience: 6 deaths, interest 6%, expense ₹2,500 per policy at start of year. Assume the basis values satisfy the recursion and death benefit is the sum assured. Find the profit from mortality, interest and expenses. Take interest on the expected base (policy value + premium − basis expense), and accumulate the expense difference at the actual rate.

Show the solution

Check the basis: (60,000 + 40,000 − 2,000) × 1.05 = 98,000 × 1.05 = ₹1,02,900. Expected outgo = 0.00725 × 5,00,000 + 0.99275 × 1,00,000 = 3,625 + 99,275 = ₹1,02,900. The basis gives zero expected profit.
Expected deaths = 1,000 × 0.00725 = 7.25.
Death strain at risk = 5,00,000 − 1,00,000 = ₹4,00,000.
Mortality profit = (7.25 − 6) × 4,00,000 = 1.25 × 4,00,000 = ₹5,00,000.
Interest profit = 1,000 × 98,000 × (0.06 − 0.05) = ₹9,80,000.
Expense profit = 1,000 × (2,000 − 2,500) × 1.06 = −₹5,30,000.
Total = 5,00,000 + 9,80,000 − 5,30,000 = ₹9,50,000.
Check directly: funds at year end = 1,000 × 97,500 × 1.06 = ₹10,33,50,000. Outgo = 6 × 5,00,000 + 994 × 1,00,000 = 30,00,000 + 9,94,00,000 = ₹10,24,00,000. Surplus = ₹9,50,000, which matches.

Answer: Mortality profit ₹5,00,000, interest profit ₹9,80,000, expense loss ₹5,30,000. Total surplus ₹9,50,000.

Exam tips

Write the valuation basis in a short list at the top of your answer. Examiners give marks for stating whether future bonuses are included.
Show the surrender or paid-up rule in words before calculating. Marks are often given for the method even if the arithmetic slips.
In surplus questions, name the order of sources. Add a one-line check that the sources add up to the actual surplus.
In the multiple-choice section, check the sign and size of the answer. A paid-up value above the original sum assured or a mortality gain with more deaths than expected points to an error.
For computer-based work, label each source column and keep the policy value, premium and expense in separate cells so you can trace the base.

Practice questions from Gross premiums and reserves

Reserves with Bonuses, Surrender Values and Profit Analysis: frequently asked questions

How do I calculate a paid-up policy value?

Find the amount available, usually the surrender value or policy value. Divide it by the value of £1 of paid-up benefit for the remaining term at the current age. The result is the paid-up sum assured. Include bonuses only if the question says to.

What is the difference between surrender value and paid-up value?

A surrender value is cash paid when the policyholder leaves. A paid-up value keeps the policy alive with no further premiums and a reduced benefit. Both come from the same pool of value, but they are paid out differently.

Why does the order of sources matter in analysis of surplus?

Interest and expense effects overlap, because expenses reduce the funds that earn interest. Different orders split the same total differently. Choose one order, state it, and use it for all sources.

Do I include future bonuses in the with-profits reserve?

Follow the valuation basis in the question. Declared bonuses are guaranteed and are included. Future bonuses are included only if the basis allows for them, in full or in part. State your assumption in the answer.