Scaling Parametric VaR for a Hedged Long-Equity, Short-Futures Portfolio

# Portfolio VaR of a hedge portfolio (long index, short future): What total exposure to take to calculate VaR?


# Portfolio VaR of a hedge portfolio (long index, short future): What total exposure to take to calculate VaR?












Imagine a portfolio is made of 20m USD invested in equities and -18m USD of MSCI World futures (sell 18m USD short). The annualised volatility of the 20m USD in equities (Equ.) is 15% and the annualised volatility of the -18m USD in futures (F.) is 18%. Correlation between the equities and the future is 85%.

The aim is to calculate the portfolio parametric VaR using the portfolio volatility, which I calculate as:

```
Weight Equ.^2 * Vol Equ.^2 + Weight F.^2 * Vol future^2 + 2 * Weight Eq. * Weight F. * Vol Eq. * Vol F. * Corr(Eq,F):

(20m/20m)^2 * 0.15^2 + (-18m/20m) * .18^2 + 2 * (20/20) * (-18/20) * .15 * .18 * .85
```

I find an annualised volatility of 0.74%.

The parametric Value-at-Risk for long only portfolio can be calculated as:

```
Value-at-Risk = portfolio volatility * Z * Exposure.
```

I have the portfolio volatility, the Z (for 95% confidence interval we use 1.645), but I don't know what to use for the exposure? Do you take the portfolio value of 20m? the net of 20m and -18m? or 20m+18m? Or something else?

## Answer by Pontus Hultkrantz (score 4)

https://quant.stackexchange.com/a/53027

Firstly, your portfolio volatility of 0.74% is the variance, as the vol will be 8.6% relative your equity position. This is the Case 2 below. I will try to give you a derivation that you hopefully can find an intuition for.

Your portfolio consists of two assets

- A basket/collection of equity with a market price $S$ USD per unit of equity. Assume you hold $n_s$ such units with a total dollar market value $V_s = n_s \cdot S = +\$20\text{m}$. Define the yearly return by $R_s := \frac{S(t+1\text{Y})}{S(t)}-1$, having stdev $\sigma_s$.

- A basket of futures with price $F$ USD per unit of futures. Assume you hold $n_f$ such units with a total dollar market value of $V_f = n_f \cdot F = -\$18\text{m}$. Define the yearly return by $R_f := \frac{F(t+1\text{Y})}{F(t)}-1$, having stdev $\sigma_s$.

### Case 1: net the values $\$20\text{m} - \$18\text{m} = \$2\text{m}$

The portfolio market value (exposure) $V_p$ at time (year) $t$ is $$V_p \equiv V_p(t) = V_s(t) + V_f(t) = n_s \cdot S(t) + n_f \cdot F(t) = \$20\text{m} - \$18\text{m} = \$2\text{m},$$ and stochastic at year $t+1$, $$\begin{align} V_p(t+1) &= n_s \cdot S(t+1) + n_f \cdot F(t+1) \\ &= n_s \cdot S(t)(1+R_s) + n_f \cdot F(t)(1+ R_f) \\ &= V_s(t)(1+R_s) + V_f(t)(1+R_f). \\ &= \$20\text{m} \cdot (1+R_s) -\$18\text{m} \cdot (1+R_f)\end{align}$$ The 1Y stochastic change (PnL) in value of the portfolio is given by $$ \begin{align} \Delta V_p(t+1) :&= V_p(t+1)-V_p(t) \\ & = V_s(t)R_s + V_f(t)R_f \\ & = \$20\text{m} \cdot R_s -\$18\text{m} \cdot R_f, \end{align}$$ and the 1Y portfolio return $$ \begin{align} R_p = \frac{\Delta V_p(t+1)}{V_p(t)} \end{align} = \frac{V_s}{V_s+V_f}R_s + \frac{V_f}{V_s+V_f}R_f = w_1R_s + w_2R_f, $$ where $w_1 := V_s/V_p = \$20\text{m}/\$2\text{m} = 10$ and $w_1 + w_2 = 1$.

The stdev (monetary volatility) of the PnL is $$ \begin{align} \mathbb D[\Delta V_p(t+1)] &= \sqrt{(V_s\sigma_s)^2 + 2V_s V_f\sigma_s \sigma_f \rho + (V_f\sigma_f)^2 } \\ &= \sqrt{(20 \cdot 0.15)^2 + 2 \cdot 20\cdot (-18)\cdot 0.15\cdot 0.18\cdot 0.85 + (-18\cdot 0.18)^2} \\ &= \$1.7244\text{m} .\end{align}$$

The stdev (volatility) of the Portfolio return is $$ \begin{align} \mathbb D[R_p] &= \sqrt{(w_s\sigma_s)^2 + 2w_s w_f\sigma_s \sigma_f \rho + (w_f\sigma_f)^2 } \\ &= \sqrt{(10 \cdot 0.15)^2 + 2 \cdot 10\cdot (-9)\cdot 0.15\cdot 0.18\cdot 0.85 + (-9\cdot 0.18)^2} \\ &= 86.22 \% .\end{align}$$

Finally, $$ \begin{align} \text{Value-at-Risk}_{5\%} &= \text{portfolio volatility} \cdot \text{Z} \cdot \text{Exposure} \\ &= 86.22\% \cdot \text{Z} \cdot \$2\text{m} \\ &= \$1.7244m \cdot \text{Z} \\ & = \$2.8\text{m}. \end{align}$$

Notice that in percentage terms, the portfolio volatility is large, but that is because it is hedged such that the netted exposure is only $\$2m$. the unhedges equity position has a VaR of around $5m. Notice that if you use portfolio returns, and if you market exposure is zero, you will end up dividing by zero. You can then use the "alternative" approach of calculating the stdev as a monetary unit.

### Case 2: equity position only $\$20\text{m}$

This is the way you did it.

With $w_1 := V_s/V_s = 1$ and $w_2 = V_f/V_s = -18/20$ so that and $w_1 + w_2 \neq 1$ we get

the stdev (volatility) relative the Equity investment is $$ \begin{align} \mathbb D[R_p] &= \sqrt{(w_s\sigma_s)^2 + 2w_s w_f\sigma_s \sigma_f \rho + (w_f\sigma_f)^2 } \\ &= \sqrt{(20/20 \cdot 0.15)^2 + 2 \cdot 20/20\cdot (-18/20)\cdot 0.15\cdot 0.18\cdot 0.85 + (-18/20\cdot 0.18)^2} \\ &= 8.62 \% .\end{align}$$

Finally, $$ \begin{align} \text{Value-at-Risk}_{5\%} &= \text{portfolio volatility} \cdot \text{Z} \cdot \text{Exposure} \\ &= 8.62\% \cdot \text{Z} \cdot \$20\text{m} \\ & = \$2.8\text{m}. \end{align}$$

### Case 3: investment amount

We can also calculate returns relative to the cash investment. E.g. we have used $\$20\text{m}$ for the equity positions, and the margin requirement for the futures positions of let's say $20\%$ of $\$18\text{m}$. So the portfolio weights are calculated relative to $\$20\text{m} + 20\% \cdot \$18\text{m} = \$23.6\text{m}$.

### Conclusion

As long as you are consistent with how you calculate the portfolio weights and the exposure, you will end up with the same answer. So if you calculate the portfolio by normalising with $\$20\text{m}$, that is also your exposure in calculating VaR.