# 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: $$X=\{x_1,x_2,...,x_m\}$$

range(replicate(20000, sort(rnorm(500, sd = 0.01))[10]))