Economic Scenario Generator

Study guide: Exam CP-312

An Economic Scenario Generator (ESG) is a model that simulates many joint future paths of economic and financial variables, such as interest rates across the yield curve, equity returns, credit spreads and defaults, inflation and exchange rates, so that an insurer can project its assets and liabilities under each path. A real-world ESG aims to reproduce how markets actually behave and is used for risk measurement, capital and asset-liability management. A risk-neutral ESG reproduces market prices and is used to value options and guarantees by risk-neutral valuation.

$dr_t = a\left(b - r_t\right)dt + \sigma\,dW_t$

$r_{t+\Delta} = r_t + a\left(b - r_t\right)\Delta + \sigma\sqrt{\Delta}\,Z$

$S_{t+\Delta} = S_t \exp\!\left(\left(\mu - \tfrac{1}{2}\sigma_S^2\right)\Delta + \sigma_S\sqrt{\Delta}\,Z\right)$

$\frac{1}{N}\sum_{j=1}^{N} \frac{S^{(j)}_T}{B^{(j)}_T} \approx S_0$

Symbols: The first two blocks are the Vasicek short rate model, a stochastic differential equation with speed of mean reversion $a$ , long-run level $b$ and volatility $\sigma$ , and its Euler-Maruyama step with $Z \sim N(0,1)$ . The third is a lognormal equity model with drift $\mu$ and volatility $\sigma_S$ . In the last, $B_T$ is the money market account and $N$ the number of scenarios.
Real-world versus risk-neutral: A real-world ESG uses drifts that include risk premiums, such as an equity return above the risk-free rate, and is judged on whether its distributions and stylized facts look like history. A risk-neutral ESG sets every asset's expected return to the risk-free rate, so that discounted prices are martingales, and is judged on whether it reproduces market prices. Real-world scenarios must not be used to price, and risk-neutral scenarios do not give realistic probabilities of loss.
Components: An ESG links an interest rate model, which may be one-factor like Vasicek or multi-factor, with equity, credit and inflation models through correlations or a cascade structure in which later variables depend on earlier ones. The equity model may add stochastic volatility or regime switching to capture fat tails and volatility clustering.
Calibration: Parameters are chosen to match targets: historical means, volatilities, correlations and tail percentiles for a real-world model, or the current yield curve and option prices for a risk-neutral one, as in model calibration. The starting point always comes from current market data, so that the scenarios begin from today's curve.
Validation: A real-world ESG is checked against stylized facts, such as mean reversion of rates, the frequency of inverted curves, fat-tailed equity returns and the occurrence of negative rates. A risk-neutral ESG is checked with the martingale test in the last block and by repricing the instruments it was calibrated to.
Uses: Stochastic reserves and capital under principle-based reserving (VM-20 and VM-21), asset adequacy and cash flow testing, economic capital, hedging programs and strategic asset allocation all run on ESG output.
Regulatory generators: In the U.S., prescribed scenarios for statutory reserves and capital came from the American Academy of Actuaries' interest rate and equity generators, and the NAIC has since field-tested a replacement, the Generator of Economic Scenarios (GOES). CP-312 reads a case study of the Academy interest rate generator and a review of the NAIC's second field test of GOES.

One Step of a Vasicek Short Rate A real-world ESG models the short rate with the Vasicek model, $r_0 = 3\%$ , $a = 0.2$ , $b = 4\%$ and $\sigma = 1\%$ . Generate the one-year rate in three scenarios with standard normal draws $Z = -1.2$ , $0.3$ and $1.5$ , using one Euler step with $\Delta = 1$ . Then find the model's long-run standard deviation and the long-run probability of a negative rate.
Answer The drift pulls the rate toward 4%, so before the random shock every scenario moves to $0.03 + 0.2(0.04 - 0.03) = 0.032$ .
 $\begin{align*} r_1 &= 0.032 + 0.01Z \\ r_1(Z = -1.2) &= 0.020 \\ r_1(Z = 0.3) &= 0.035 \\ r_1(Z = 1.5) &= 0.047 \end{align*}$ 
Over a long horizon, the Vasicek rate is normal with mean $b$ and variance $\sigma^2/(2a)$ .
 $\begin{align*} \text{SD}_\infty &= \frac{0.01}{\sqrt{2(0.2)}} \\ &= 0.0158 \\ P(r < 0) &= \Phi\!\left(\frac{0 - 0.04}{0.0158}\right) \\ &= \Phi(-2.53) \\ &= 0.0057 \end{align*}$ 
The scenarios show the stylized fact of mean reversion, and the model allows negative rates in about 0.6% of long-run outcomes. Whether that is acceptable is a validation question for the ESG's user. The exact one-year mean is $0.04 + (0.03 - 0.04)e^{-0.2} = 3.18\%$ , so a one-year Euler step is only an approximation, and production ESGs use monthly steps or the exact transition.

Martingale Test of Equity Scenarios An ESG produces four equally likely one-year equity scenarios with $S_0 = 100$ , $\sigma_S = 20\%$ and draws $Z = -1.15$ , $-0.32$ , $0.32$ and $1.15$ . The continuously compounded risk-free rate is 3%. Apply the martingale test to the risk-neutral scenarios, with $\mu = 3\%$ , and to real-world scenarios with $\mu = 7\%$ .
Answer Under the risk-neutral drift, $S_1 = 100\exp(0.03 - 0.02 + 0.2Z)$ , which gives 80.25, 94.74, 107.68 and 127.12.
 $\begin{align*} \bar S_1 &= \frac{80.25 + 94.74 + 107.68 + 127.12}{4} \\ &= 102.45 \\ e^{-0.03}\,\bar S_1 &= 99.42 \end{align*}$ 
The real-world scenarios are each $e^{0.04}$ times larger, so their discounted mean is 103.48. The risk-neutral set misses 100 only because four draws cannot reproduce the lognormal's full spread, and the error shrinks as the number of scenarios grows. The real-world set fails by about 3.5%, which is the equity risk premium built into its drift. Discounting those scenarios at the risk-free rate values the index above its market price, so they would misprice an equity-linked guarantee and must not be used for market-consistent values.

How it appears on the exam

CP-312 model questions: Candidates describe, apply and assess ESGs using the SOA's Economic Scenario Generators: A Practical Guide and its three-part series, explaining the difference between real-world and risk-neutral scenarios and which applications need each.

Calibration and validation: Questions ask how a generator is calibrated, which stylized facts or market prices its output should reproduce, and what tests would detect a problem, such as a failed martingale test or too few inverted yield curves.

Regulatory scenarios: Some questions ask about the Academy interest rate generator or the NAIC's GOES field test, and how a change of generator affects VM-20 and VM-21 reserves and capital.

Exam traps

Wrong measure for the job: Pricing a guarantee on real-world scenarios discounted at the risk-free rate misstates its value, and measuring a capital tail on risk-neutral scenarios uses probabilities that do not describe reality. Name the measure before using the scenarios.

A risk-neutral generator is not a forecast: Its drifts are set to reproduce prices, not to predict returns. A risk-neutral equity return of 3% is not a view that equities will earn 3%.

Calibrated parameters versus starting values: The long-run level and volatility come from calibration, but the starting yield curve comes from today's market. Starting every scenario at the long-run level hides the effect of current rates.

Discretization error: One Euler step per year in a mean-reverting model can misstate means and allow implausible values. Report the step size and check the output against the model's exact moments.