The variance-covariance method for a two-asset portfolio: volatility from weights and correlation, the confidence multiplier, the diversification benefit and the scaling mistakes.
Parametric VaR is the portfolio's standard deviation of value over the horizon times the normal quantile for the confidence level: VaR = z × σp × value. For an illustrative 50,000,000 portfolio of 30,000,000 in equities at 18 per cent volatility and 20,000,000 in bonds at 6 per cent, correlated at 0.20, portfolio volatility is 11.52 per cent a year and the one-day 99 per cent VaR is 844,289, or 1.689 per cent of value. The two positions alone would sum to 967,205.
Worked in full in Financial Risk Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The variance-covariance method assumes that daily returns are normally distributed with a mean of zero, and that the volatilities and correlation are stable over the horizon. All inputs are illustrative.
| Input | Equities | Bonds | Portfolio |
|---|---|---|---|
| Market value | 30,000,000 | 20,000,000 | 50,000,000 |
| Weight | 60% | 40% | 100% |
| Annual volatility | 18% | 6% | 11.5225% |
| Daily volatility (annual / √252) | 1.1339% | 0.3780% | 0.7258% |
| Correlation | 0.20 | ||
| Confidence, horizon | 99%, one day (z = 2.3263) | ||
σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ σ1 σ2
= 0.6² × 0.18² + 0.4² × 0.06² + 2 × 0.6 × 0.4 × 0.20 × 0.18 × 0.06
= 0.011664 + 0.000576 + 0.001037 = 0.013277, so σp = 11.5225% a year
Excel, weights in a column W and the covariance matrix in Cov: =SQRT(MMULT(TRANSPOSE(W),MMULT(Cov,W)))
Dividing by √252, which is 15.8745, gives a daily volatility of 0.7258 per cent. On 50,000,000 that is a one-day standard deviation of 362,925.
VaR = z × σp,daily × V = 2.3263 × 0.7258% × 50,000,000 = 844,289
Excel: =NORM.S.INV(0.99)*SigmaAnnual/SQRT(252)*Value
Read it as: on 99 days in 100 the portfolio should not lose more than 844,289 in a day. On the other day it loses more, and VaR says nothing about how much more. Over a year of 252 trading days, 2.5 days beyond VaR are expected.
Two simplifications sit inside that number. The expected daily return is set to zero, which is standard over one day because the mean is tiny next to the volatility; over longer horizons it starts to matter and some desks subtract it. And the positions are treated as linear: a 1 per cent move in the equity index is a 1 per cent move in the equity sleeve. A portfolio holding options or callable bonds needs their deltas, or a full revaluation, before the variance-covariance method can be used, because the normal distribution of the underlying no longer maps to a normal distribution of the position's value.
Each position on its own has a VaR of z × daily volatility × value: 791,349 for the equities and 175,855 for the bonds, a sum of 967,205. The portfolio VaR is 122,916 lower, a diversification benefit of 12.7 per cent. The same result comes from combining the standalone VaRs directly:
VaRp = √(VaR1² + VaR2² + 2ρ VaR1 VaR2) = √(791,349² + 175,855² + 2 × 0.20 × 791,349 × 175,855) = 844,289
That second form is how a desk aggregates VaR reported by different books, and it shows why VaRs cannot simply be added unless the correlation is 1.
| Correlation | VaR | % of value | Diversification benefit |
|---|---|---|---|
| −0.30 | 757,404 | 1.515% | 21.7% |
| 0.00 | 810,653 | 1.621% | 16.2% |
| 0.20 | 844,289 | 1.689% | 12.7% |
| 0.50 | 892,369 | 1.785% | 7.7% |
| 1.00 | 967,205 | 1.934% | 0.0% |
The correlation between equities and bonds is the input most likely to be wrong in a stress: if it moves from −0.30 to 0.50, VaR rises 17.8 per cent with no change in positions or volatilities. How much of a reported diversification benefit is left once correlations are re-estimated on stressed months is worked in how much diversification survives a crisis.
| Confidence | z | VaR |
|---|---|---|
| 95.0% | 1.6449 | 596,958 |
| 97.5% | 1.9600 | 711,319 |
| 99.0% | 2.3263 | 844,289 |
| 99.5% | 2.5758 | 934,832 |
Under normality, VaR at any confidence level is the same standard deviation times a different z, so a 95 per cent figure and a 99 per cent figure are the same information. That is no longer true once returns have fat tails, which is why the 99 per cent figure tends to be understated by the parametric method more than the 95 per cent one.
Build the portfolio variance from weights, volatilities and correlation, scale to the horizon, multiply by z and by value: 844,289 here, 12.7 per cent less than the sum of the parts. How that total splits back across the positions is in how to calculate component VaR. The book builds the same calculation one visible step at a time for a four-sleeve credit fund, with a switch between reported and crisis correlations, in the free workbook for this case; a blank version for your own positions is the VaR and expected shortfall model.
Parametric VaR assumes a distribution, usually normal, and needs only volatilities and correlations: here 2.3263 times a daily standard deviation of 362,925 gives 844,289. Historical VaR replays actual past returns on today's positions and reads off the loss at the chosen percentile, so it captures fat tails present in the sample but depends heavily on the window chosen.
Under the normal assumption, multiply by the ratio of the quantiles: 2.3263 divided by 1.6449, about 1.41. A one-day 95 per cent VaR of 596,958 becomes 844,289. If returns have fat tails the true 99 per cent figure is larger than this conversion suggests, so a scaled number should be labelled as such.
Because losses on different assets do not all happen on the same day unless correlation is 1. With a correlation of 0.20 the portfolio VaR is 844,289 against 967,205 for the two positions separately, a diversification benefit of 12.7 per cent. At a correlation of 1.00 the benefit is zero.
The parametric construction is worked one visible step at a time in the workbook for chapters 4, 6 and 7 of Financial Risk Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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