Financial Risk Manager Part 2

Financial Risk Manager Part 2

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In the context of the FRM Practice Exam Part I, consider a portfolio with a total investment of USD 1 million divided between emerging markets equities and US equities. Each equity component has a 1-day 95% Value at Risk (VaR) of USD 1.3 million and there is a correlation of 0.25 between their returns. The portfolio manager decides to sell USD 7 million of US equities and use the proceeds to purchase an additional USD 7 million of emerging markets equities. Concurrently, the Chief Risk Officer (CRO) recommends changing the risk measure from a 1-day 95% VaR to a 10-day 99% VaR. Assuming that the returns are normally distributed and the volatility of each equity component remains unchanged, determine the increase in the overall portfolio VaR resulting from both the portfolio rebalancing and the change in the risk measurement.

Explanation:

The correct answer is C, USD 7.034 million. The explanation involves a few steps to calculate the Value at Risk (VaR) for the portfolio both before and after the rebalancing, as well as the change in risk measure.

  1. Initial VaR Calculation: The initial portfolio VaR is calculated using the formula: VaRp=VaRu2+VaRe2+2⋅ρue⋅VaRu⋅VaRe\text{VaR}_p = \sqrt{\text{VaR}_u^2 + \text{VaR}_e^2 + 2 \cdot \rho_{ue} \cdot \text{VaR}_u \cdot \text{VaR}_e} where ρue\rho_{ue} is the correlation coefficient between US and emerging market equities. Plugging in the values: VaRp=1.32+1.32+2⋅0.25⋅1.3⋅1.3=$2.055 million\text{VaR}_p = \sqrt{1.3^2 + 1.3^2 + 2 \cdot 0.25 \cdot 1.3 \cdot 1.3} = \$2.055 \text{ million}

  2. Volatility Calculation: The volatilities (σ\sigma) of the individual assets are calculated using their respective VaR values:

    • For US equities: σ(u)=1.31.645⋅37=0.0214\sigma(u) = \frac{1.3}{1.645 \cdot 37} = 0.0214
    • For emerging markets equities: σ(e)=1.31.645⋅48=0.0165\sigma(e) = \frac{1.3}{1.645 \cdot 48} = 0.0165

  3. Rebalanced Position VaR: After rebalancing, the new positions are USD 30 million for US equities and USD 55 million for emerging markets equities. The new VaRs for each are:

    • \text{VaR}(u) = 1.645 \cdot 0.0214 \cdot 30 = \1.0561 \text{ million} $
    • \text{VaR}(e) = 1.645 \cdot 0.0165 \cdot 55 = \1.4928 \text{ million} $

  4. New Portfolio VaR: The new portfolio VaR, still using the 1-day 95% measure, is: VaRp,new=1.05612+1.49282+2⋅0.25⋅1.0561⋅1.4928\text{VaR}_{p,\text{new}} = \sqrt{1.0561^2 + 1.4928^2 + 2 \cdot 0.25 \cdot 1.0561 \cdot 1.4928}

  5. Change in Risk Measure: The risk measure changes from a 1-day 95% VaR to a 10-day 99% VaR. The 99% VaR for a 10-day horizon is approximately four times the 95% VaR for a 1-day horizon, assuming normal distribution and independence of returns over the period.

  6. Final VaR Increase: The increase in VaR due to the combined effect of rebalancing and changing the risk measure is calculated by multiplying the new 1-day VaR by four (for the 10-day horizon) and then comparing it to the initial 1-day VaR.

The detailed calculations are not provided in the snippet, but the final answer indicates that the increase in VaR due to both the rebalancing and the change in risk measure is USD 7.034 million.

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