ModelRisk Optimizer example · Corporate finance

Investment Portfolio Chosen by the Optimizer

Which mix of equities, bonds and real estate earns most while the one-year 95% value at risk stays within the policy's limit?

Complexity: ★★★★★
Free Excel workbook (CF-09)
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From the model’s saved run

The Optimizer’s plan against the starting plan

The two allocations, and how each does in the same 10,000 simulated years.

Decision	Unit	Starting plan	Optimizer’s plan	Change
Global equities	of reserve	40.0%	43.4%	3.4%
Government bonds	of reserve	30.0%	19.0%	−11.0%
Corporate bonds	of reserve	20.0%	27.8%	7.8%
Listed real estate	of reserve	10.0%	9.8%	−0.2%

Measure	Unit	Starting plan	Optimizer’s plan	Change
Mean return	a year	5.37%	5.57%	0.2%	✔ better
Expected gain	US$m a year	26.9	27.8	1.0	✔ better
Value at risk, 95%	US$m	42.7	49.6	6.9
Value at risk, 99%	US$m	88.9	98.0	9.1
Expected shortfall, 97.5%	US$m	96.7	107.9	11.2
Chance of a large loss (Inputs sheet)	of years	0.7%	1.0%	0.2%
Limit on the 95% value at risk (Inputs sheet)	US$m	50.0	50.0

Expected gain = mean return × amount invested. A large loss is one above the amount on the Inputs sheet.

One-year loss — cumulative probability

S-curve (cumulative probability) of Starting plan, Optimizer’s plan, Limit on the 95% VaR from the saved simulation.

The business question

A company holds a US$500 million reserve: 40% in global equities, 30% in government bonds, 20% in corporate bonds and 10% in listed real estate. CF-06 measured its one-year 95% value at risk with fat-tailed returns that crash together, and found it well inside the policy's US$50 million limit. Is the unused risk worth taking, and in which assets?

Here the ModelRisk Optimizer chooses the four weights together.

Who it is for: Treasurers and investment committees of corporate reserves, pension and insurance portfolios; risk managers.
Main outputs: The Optimizer's allocation against a starting plan (CF-06's allocation) · for both plans: the mean return and expected gain, value at risk at 95% and 99%, expected shortfall and the chance of a large loss · the efficient frontier: the best mean return and allocation at five VaR limits · the distribution of the loss.

What the Optimizer decides

The Optimizer changes the blue cells as a budget group: the four weights keep their total of 100%. The starting plan is CF-06's allocation.

Decision	Unit	Starting plan	Optimizer’s plan	Lowest	Highest
Global equities	of reserve	40.0%	43.4%	0.0%	60.0%
Government bonds	of reserve	30.0%	19.0%	0.0%	60.0%
Corporate bonds	of reserve	20.0%	27.8%	0.0%	60.0%
Listed real estate	of reserve	10.0%	9.8%	0.0%	60.0%

Goal: Maximise the mean return
Limit: Value at risk, 95% at most 50 (the 95th percentile of the loss)
Limit: The whole reserve invested: weights add up to 100%, in every plan tried

The Optimizer’s run: Vertex Detector, 23 plans tried, each on 10,000 samples, in 984.9 seconds.

How the model works

Four sheets after this one, read from left to right. Each one feeds the next.

Inputs sheet
The amount invested and the policy's limit on the one-year 95% value at risk. The degrees of freedom of the fat-tailed returns and of the Student copula; each asset class's expected return and volatility; the six correlations.
Decisions sheet
The four cells the Optimizer changes — the weights of the asset classes, in a budget group that keeps their total at 100% — with the policy's bounds, the starting plan they are compared with, the goal (the highest mean return) and the limit (the 95% VaR within the policy's limit).
Portfolio sheet
Each iteration draws four correlated U values from a Student copula (VoseCopulaMultiT) and turns them into fat-tailed returns (VoseStudent3) with each class's mean and volatility. Both plans meet the same returns. The return and loss of each are recorded as outputs.
Results sheet
Both plans on the same 10,000 simulated years: mean return, expected gain, value at risk, expected shortfall and the chance of a large loss. The efficient frontier, from an Optimizer run at each of five VaR limits; the distribution of the loss.

ModelRisk techniques

ModelRisk Optimizer: four weights in a budget group
Value at risk as a percentile limit
Efficient frontier across five VaR limits
Fat-tailed returns VoseCopulaMultiT, VoseStudent3

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The results in detail

The efficient frontier

The best mean return at five limits on the 95% VaR: the Optimizer run again at each.

Best mean return at each limit on the 95% VaR

Chart: Best plan at each limit (the frontier), Starting plan, Optimizer’s plan.

Limit on the 95% value at risk	Mean return a year	Equities weight	Gov. bonds weight	Corp. bonds weight	Real estate weight
VaR at most US$40m	5.22%	41.0%	30.0%	23.8%	5.2%
VaR at most US$45m	5.41%	47.2%	32.3%	12.0%	8.5%
VaR at most US$50m	5.46%	43.4%	28.5%	12.0%	16.2%
VaR at most US$55m	5.53%	38.1%	28.7%	3.5%	29.7%
VaR at most US$60m	5.68%	40.3%	22.8%	4.2%	32.7%

Filled in when the workbook was built: one Optimizer run at each limit. Bold: the policy's limit.

How to read these results

The starting plan — CF-06's 40% equities, 30% government bonds, 20% corporate bonds and 10% real estate — keeps its 95% VaR well inside the US$50 million limit, at about US$42 million, for a mean return of about 5.4% a year.
The Optimizer moves about a third of the government bonds into equities and corporate bonds: its plan holds about 43% equities, 19% government bonds, 28% corporate bonds and 10% real estate. The VaR rises to about US$48 million, inside the limit, and the mean return by about 0.2 points, some US$1 million a year.
The frontier repeats the run at four other limits. Loosening the limit from US$40 million to US$60 million lifts the best mean return found from about 5.2% to about 5.7% a year. Each run stopped at its 15-minute time limit, so a longer run may find slightly better plans.
Corporate bonds and real estate are near-substitutes: quite different mixes earn almost the same, so their weights jump between frontier points and may differ from one run to the next.
The tail beyond the VaR grows too: the Optimizer’s plan's 99% VaR and expected shortfall are higher. A limit on the 95% VaR alone does not control them (CF-06). To do so, add a second limit, for example on the 99th percentile of the loss.

Run the Optimizer yourself. Open the workbook, open the Optimizer from the ModelRisk ribbon (the saved task opens with it) and press Start; then simulate to compare the plans.

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Inputs and assumptions

Every assumption in the model, with its uncertainty and where it comes from. The figures are illustrative.

Portfolio 2 inputs

How much is invested, and the risk policy's limit.

Assumption	Unit	Value
Amount invested	US$m	500
Limit on the one-year 95% value at riskRisk policy: 10% of the reserve. The Optimizer's limit.	US$m	50

Tails 3 inputs

What makes the returns fat-tailed and crash together.

Assumption	Unit	Value
Degrees of freedom of the returnsLower = fatter tails; 5 fits monthly equity data.		5
Degrees of freedom of the Student copulaLower = more joint extremes.		4
A large loss20% of the portfolio.	US$m	100

Asset classes 4 inputs

Expected return and volatility over one year. The weights are decisions: see the Decisions sheet.

Asset class	Expected return a year	Volatility a year
Global equitiesLong-run assumptions of the investment committee.	7.0%	16.0%
Government bonds	3.5%	6.0%
Corporate bonds	4.5%	8.0%
Listed real estate	6.0%	18.0%

Correlations 4 inputs

Between the one-year returns. Enter the yellow cells above the diagonal; the cells below it mirror them.

	Equities	Gov. bonds	Corp. bonds	Real estate
Global equities	1.00	−0.20	0.60	0.70	Equities and government bonds tend to move apart.
Government bonds	−0.20	1.00	0.40	0.00
Corporate bonds	0.60	0.40	1.00	0.50
Listed real estate	0.70	0.00	0.50	1.00

Units and conventions

Money: US$ million (US$m); a loss is positive, a gain negative.
Weight: The share of the reserve in an asset class; the four add up to 100%.
Value at risk: The loss exceeded in only 5% of years (VaR 95% = the 95th percentile of the loss).
Expected shortfall: The average loss in the worst 2.5% of years (ES 97.5%).
Horizon: One year; no rebalancing during the year.

Key assumptions and limitations

Returns: CF-06's fat-tailed model: Student returns (5 degrees of freedom) with the stated means, volatilities and correlations, joined by a Student copula so that the asset classes tend to crash together.
Policy: At most 60% of the reserve in any one asset class; no borrowing and no short positions (no weight below 0%).
Optimizer: Each plan the Optimizer tries is judged on the same 10,000 simulated years (common random numbers, seed 1) as the simulation on the Results sheet, so plans are compared fairly and the limit the Optimizer met holds on the Results sheet too.
Data: The portfolio and all figures are invented for this example.

How to use it

Enter your amount, VaR limit, return and volatility assumptions and correlations on the Inputs sheet, and your policy's bounds on each weight on the Decisions sheet.
On the ModelRisk ribbon, open the Optimizer: the task is saved in this workbook. Press Start. When it finishes, apply the best solution: the weights on the Decisions sheet change.
Run a simulation from the ModelRisk ribbon (10,000 samples with a fixed seed are set) to compare the new plan with the starting plan on the Results sheet.
For a frontier of your own, choose Start ▾ ▸ Start Efficient Frontier Analysis in the Optimizer and sweep the limit on the 95% VaR over the range you want.
To watch the search from the beginning, copy the starting plan onto the Optimizer’s plan first.

ModelRisk functions used

ModelRisk Optimizer: Chooses the four weights: the highest mean return, with the 95% VaR at most the policy's limit. The weights form a budget group, which keeps their total at 100%.
VoseEigenValues: The eigenvalues of the correlation matrix: all above 0 for a valid matrix.
VoseCopulaMultiT: Four correlated U values each iteration, with tail dependence: joint crashes.
VoseStudent3: Fat-tailed returns from the U values, with each asset class's mean and standard deviation.
VoseOutput: The return and the loss of both plans, recorded at every sample.
VoseSimMean, VoseSimPercentile: The mean return and the value at risk of both plans.
VoseSimCVARp, VoseSimProbability: The expected shortfall and the chance of a large loss of both plans.

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An illustrative example: the figures are invented for teaching and do not describe a real company, project or asset. Last updated 9 October 2026.