Economic Modelling · Valuing benefit guarantees using simulation

Accuracy, Variance Reduction and Nested Simulation in Actuarial Models

Q: How many simulations do I need for a given accuracy?

Use n = (z × s ÷ ε)², where ε is the half-width you accept and z is 1.96 for 95% confidence. You need a pilot run to estimate s. Round the answer up.

Q: When do antithetic variates work well?

They work when the payoff moves in one direction as the random inputs rise, so that the pair of payoffs is negatively correlated. Many guarantee payoffs behave this way. If the payoff is not monotonic, the gain can be small.

Q: What is a control variate in guarantee valuation?

It is a similar quantity with a known exact value, such as a European option priced by Black-Scholes. You simulate it with the guarantee and use the error in the control to correct the guarantee estimate.

Q: Why is nested simulation a problem for actuaries?

It needs inner simulations inside every outer scenario, so the runtime is N × M. This becomes too slow for frequent reporting, so firms use proxy models, closed-form formulae or fewer inner runs, which add approximation error.

Updated 11 October 2026 · Fact-checked

Monte Carlo gives an estimate with standard error s ÷ √n, so accuracy improves slowly as simulations grow. Variance reduction, such as antithetic or control variates, cuts the standard error without extra runs. Nested simulation runs inner simulations inside outer scenarios, for example to value a guarantee at a future date, and is very costly.

Understand Accuracy, Variance Reduction and Nested Simulation

A simulation estimate of a guarantee cost is an average of n simulated discounted payoffs. Each run is random, so the average is random too. Its spread is measured by the standard error, which is the sample standard deviation s divided by √n. A 95% confidence interval is roughly the estimate ± 1.96 × standard error.

The square root matters. To halve the standard error you need four times as many simulations. To get ten times more accuracy you need one hundred times the runs. That is slow and expensive, so actuaries look for smarter ways to reduce s itself. These are variance reduction techniques.

Antithetic variates pair each simulation with a mirror image. If you use random normal draws Z, you also run the path with −Z. The two payoffs are negatively correlated for a monotonic payoff, so their average has lower variance than two independent runs. The pair average is treated as one observation when finding the standard error.

Control variates use a related quantity whose true value you know exactly. For a guarantee, this could be a European put with a Black-Scholes price. You simulate both the guarantee payoff X and the control Y. You then adjust: X* = X − b(Y − E[Y]), where E[Y] is the known true mean. If X and Y are highly correlated, the variance falls sharply. The best b is Cov(X, Y) ÷ Var(Y).

Nested simulation (also called stochastic on stochastic) arises when you need a value at a future time inside each scenario. Outer scenarios are real-world paths to the future date. Inner simulations are risk-neutral runs from that date to value the guarantee. If you have N outer and M inner runs, the cost is N × M. This is very heavy, so firms use approximations, closed-form formulae or proxy models such as curve fitting to cut the cost.

Key rules to remember

Sample mean estimate: x̄ = (1 ÷ n) Σ xᵢ; Estimate of the expected discounted payoff, taken as the guarantee cost.
Standard error: SE = s ÷ √n, where s² = Σ(xᵢ − x̄)² ÷ (n − 1); Use the sample standard deviation of the n independent observations.
Approximate 95% confidence interval: x̄ ± 1.96 × s ÷ √n; Valid by the central limit theorem when n is large.
Simulations needed for a target standard error: n = (z × s ÷ ε)², with z = 1.96 for 95% confidence and ε the allowed half-width; Needs an estimate of s from a pilot run.
Antithetic estimator: Pair average Wᵢ = (Xᵢ + X′ᵢ) ÷ 2; Var(W) = (σ² ÷ 2)(1 + ρ); ρ is the correlation between the two payoffs in a pair. Gain needs ρ < 0.
Control variate estimator: X* = X − b(Y − E[Y]); Var(X*) = Var(X) − 2b Cov(X,Y) + b² Var(Y); E[Y] must be known exactly.
Optimal control coefficient: b = Cov(X, Y) ÷ Var(Y); Var(X*) = Var(X)(1 − ρ²); ρ is the correlation between X and Y. Higher |ρ| gives a larger reduction.
Nested simulation cost: Total inner runs = N × M; N outer scenarios, M inner simulations per scenario.

How to solve Accuracy, Variance Reduction and Nested Simulation questions

Use this order for any question on accuracy, variance reduction or nested simulation.

1Identify what is being estimated: a mean payoff, a probability or a quantile. State the estimator.
2Write the standard error as s ÷ √n. Check that s is the sample standard deviation of independent observations. For antithetic pairs, use the pair averages.
3For a sample-size question, set 1.96 × s ÷ √n equal to the required half-width and solve for n. Round up.
4For a variance reduction question, name the technique and state why it works: negative correlation for antithetic, known mean of a correlated control for control variates.
5Do the calculation. For a control variate, find b, compute the adjusted values and then the adjusted mean and standard error.
6For nested simulation, identify the outer (real-world) and inner (risk-neutral) stages and count total runs as N × M.
7State the result with units and an interpretation, for example a confidence interval for the guarantee cost.
8Add one comment on limits: bias, model error, extra effort or runtime.

Quickest way: Standard error and sample size shortcut

When to use it: Use it when the exam gives you s or a pilot result and asks how accuracy changes with n.

Remember SE ∝ 1 ÷ √n. Multiply n by k and SE falls by a factor of √k.
To cut SE by a factor f, multiply n by f².
For a control variate, new variance = old variance × (1 − ρ²). Use that directly.
For antithetic pairs, compare (1 + ρ) ÷ 2 per run-pair with the independent case.
For nested runs, multiply N by M at once and say if it is feasible.

Common mistakes in Accuracy, Variance Reduction and Nested Simulation

Thinking that doubling the simulations halves the standard error.
Students assume accuracy scales linearly with n.
Fix: SE depends on 1 ÷ √n. Doubling n reduces SE by a factor of √2. You need four times n to halve it.
Using the antithetic standard error with 2n separate observations.
Each pair gives two payoffs, so students count them as independent.
Fix: The two members of a pair are dependent. Average each pair, then compute s and SE from the n pair averages.
Using a control variate whose true mean is not known.
Students pick any correlated quantity.
Fix: The control must have an exactly known expected value, such as a Black-Scholes price or the discounted forward price. Otherwise the adjustment is itself biased.
Believing that antithetic variates always reduce variance.
The technique is taught as a free improvement.
Fix: It helps when the payoff is monotonic in the random inputs so that ρ < 0. For non-monotonic payoffs, the gain may be small or negative.
Confusing the outer and inner stages of nested simulation, or using real-world rates in both.
Students merge the two purposes.
Fix: The outer stage projects real-world scenarios to the future date. The inner stage values the guarantee there on a risk-neutral basis, then discounts.
Treating a low standard error as proof that the answer is right.
SE measures only sampling error.
Fix: Say that model error, wrong parameters and discretisation bias are not captured by SE.

Worked examples

Example 1

A simulation of 10,000 independent runs gives a mean guarantee cost of ₹4.20 per ₹100 of premium and a sample standard deviation of ₹15. (a) Find the standard error and a 95% confidence interval. (b) How many runs are needed for a 95% half-width of ₹0.10?

Show the solution

Standard error = 15 ÷ √10,000 = 15 ÷ 100 = 0.15.
The 95% half-width = 1.96 × 0.15 = 0.294.
The interval is 4.20 ± 0.294, which is (3.906, 4.494).
For half-width 0.10: n = (1.96 × 15 ÷ 0.10)² = (294)² = 86,436.
Round up if needed. The value is already an integer, so n = 86,436.

Answer: (a) SE = ₹0.15, 95% CI ≈ (₹3.91, ₹4.49). (b) About 86,436 runs.

Example 2

A simulation estimates the cost of a guarantee X. The control variate Y is a European put with known price E[Y] = 6.0. From the simulation: Var(X) = 100, Var(Y) = 64, Cov(X, Y) = 72, mean of X = 10.5 and mean of Y = 6.8. Find the control variate estimate of E[X] and the variance reduction.

Show the solution

The optimal b = Cov(X, Y) ÷ Var(Y) = 72 ÷ 64 = 1.125.
The adjusted mean = x̄ − b(ȳ − E[Y]) = 10.5 − 1.125 × (6.8 − 6.0).
1.125 × 0.8 = 0.9, so the adjusted mean = 10.5 − 0.9 = 9.6.
Correlation ρ = 72 ÷ √(100 × 64) = 72 ÷ 80 = 0.9.
Var(X*) = Var(X)(1 − ρ²) = 100 × (1 − 0.81) = 19.
The variance falls from 100 to 19, a reduction of 81%.

Answer: The control variate estimate is 9.6. The per-run variance falls from 100 to 19, which is a reduction of 81%.

Exam tips

Always show s ÷ √n and then interpret it. Examiners award marks for the interpretation, not only the number.
In written answers, explain why a variance reduction method works, using correlation, before giving a number.
For nested simulation, say what each stage represents and give the N × M cost. Then suggest a remedy such as proxy models or closed-form inner values.
In the computer-based paper, set a seed, report the SE next to the estimate and use sample sd for s. Check that your control variate has a known mean.
For MCQs, test the root rule first: to halve SE, multiply n by 4.

Practice questions from Valuing benefit guarantees using simulation

Accuracy, Variance Reduction and Nested Simulation in other exams

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

FRM Part IVariance Reduction Techniques

Accuracy, Variance Reduction and Nested Simulation: frequently asked questions

How many simulations do I need for a given accuracy?

Use n = (z × s ÷ ε)², where ε is the half-width you accept and z is 1.96 for 95% confidence. You need a pilot run to estimate s. Round the answer up.

When do antithetic variates work well?

They work when the payoff moves in one direction as the random inputs rise, so that the pair of payoffs is negatively correlated. Many guarantee payoffs behave this way. If the payoff is not monotonic, the gain can be small.

What is a control variate in guarantee valuation?

It is a similar quantity with a known exact value, such as a European option priced by Black-Scholes. You simulate it with the guarantee and use the error in the control to correct the guarantee estimate.

Why is nested simulation a problem for actuaries?

It needs inner simulations inside every outer scenario, so the runtime is N × M. This becomes too slow for frequent reporting, so firms use proxy models, closed-form formulae or fewer inner runs, which add approximation error.