# Value-at-Risk for a portfolio model with Gearing

My models: Say I want to construct a portfolio so I maximize my expected return while keeping my risk (measured by Value-at-Risk) lower than my risk target.

$$\max \sum x_i \mu_i \\ VaR_{0.05} \leq \text{RiskTarget} \\ \sum x_i = W \text{, (wealth)}$$ Now suppose I can gear my investments up to factor $$k$$ so I add the $$\max \sum x_i \mu_i -rB \\ VaR \leq \text{Risk Target} \\ \sum x_i = W+B \\ B\leq Wk$$

My Value-at-Risk is based on 500 scenarios and my 5% VaR is the absolute value of 10th lowest observation in my potential PnL vector. Say there is $$m$$ different assets I can invest in. $$x_i$$ is the value of asset $$i$$ in my portfolio and not amount of shares.

My Findings: When I experiment with this model, I get more or less the same results even for low interest rates. That is because my Value-at-Risk is an absolute risk measure and if I buy $$k$$ times more then my VaR increases roughly with same factor, and my risktarget doesn't allow that.

My Question: Should I compute VaR differently now that borrowing money have entered the model?

In other words: Given my 500 scenarios ($$R^{250 \times m }$$ matrix) how should I compute the 500 possible outcomes for a given portfolio: $$X=\{x_1,x_2,...,x_m\}$$

## 1 Answer

You need to factor borrowing costs into the scenarios (and the currently low interest rates help, so you may want to check with higher rates as well). Since you compute VaR from the scenarios, this will push VaR to the left (in terms of returns, i.e. make it worse).

The key question is if your risk budget can withstand such an increase in VaR, which will go up as well by roughly your leverage factor.

In any case, I would be wary to base an investment on a VaR number computed from an order statistic with only 500 scenarios. (Btw, the 10th smallest return is rather the 2% VaR.) An example: Assume portfolio returns were normally distributed (i.e. the nice case), and you looked at daily returns/VaR. The standard deviation of those daily returns is 1%. If I sample 500 returns from such a distribution, the order statistics will vary quite a bit. In R, you could say

range(replicate(20000, sort(rnorm(500, sd = 0.01))[10]))
## [1] -0.02667598 -0.01523161

So I would expect substantially different positions when you repeat the optimisation several times with different scenarios from the same distribution. You may want to consider either increasing the numbers of scenarios, or using an alternative (parametric/semiparametric) method to compute VaR. The paper The Hidden Risks of Optimizing Bond Portfolios under VaR explains some of these problems.