Exam One June 2024. Optimal Portfolio Allocation
An investment universe of the following risky assets with a dependence structure (correlation) applies to all questions below as relevant:
Question 1. Global Minimum Variance portfolio is obtained subject to the budget constraint:
• Derive the analytical solution for optimal allocations w* . Provide full derivation workings.
• Compute optimal allocations (Global MV portfolio) for the given investment universe.
Question 2. Consider the optimization for a target return m. There is no risk-free asset.
• Compute correlation levels by stressing the matrix × 1, ×1.3, ×1.8, subject to the upper limit 0.99 for each cross-asset correlation. Diagonal elements stay equal to one.
• Compute w* and portfolio risk σΠ = √w′Σw for m = 7% for three levels of correlation given.Hints: it is possible to compute this kind of optimal allocation via analytical formula. Negative and non- robust allocations (into ± 100s%) are possible, particularly for high correlation. Please do not reconfirm your numerical results via support.
Question 3. “Evaluating the PL more frequently make it appear more risky than it actually is.” Make the following computations to demonstrate this statement.
• Write down the formula for Sharpe Ratio and note that σ is scaled with time.
• Compute Daily, Monthly, and Quarterly Sharpe Ratio, for Annualised SR of 0 .53. Hint: this is an abstract computation, not related to Questions 1 and 2.
• Convert each Sharpe Ratio into Loss Probability (daily, monthly, quarterly, annual), using
Pr(PL < 0) = Pr(x < −SR).
where x is a standard Normal random variable.
Exam One June 2024. Understanding Value-at-Risk
Assume you are an analyst concerned with how risky NASDAQ-100 became over SP 500. Perform. the backtesting of Analytical VaR (99%/10day) on the data provided in .csv files.
Question 4. The quick guide is given below, but please refer to the tutorial and CQF material.
VaR10D,t = Factor × σt 代 写Optimal Portfolio Allocation 2024Python
× √10
• Compute the rolling standard deviation σt from 21 daily returns. Timescale of σt remains ‘daily’ regardless of how many returns are in the sample.
• To make a projection over 10 days, we use the additivity of variance σ10D = √σt(2) × 10.
• A breach occurs when the forward realised 10-day return is below the VaRt quantity.
r10D,t+10 < VaR10D,t given both numbers are negative.
VaR is fixed at time t and compared to the return from t to t+10, computed ln(St+10/St ). Alternatively, you can compare to ln(St+11/St+1) but state this assumption in your report upfront.
Prepare and present the following deliverables in your report: (a) The count and percentage of VaR breaches.
(b) Provide a plot which identifies the breaches with crosses or other marks.
(c) Provide a list of breaches with columns [Date, ClosingPrice, LogReturn, VaR 10D, Ret 10D]. Hint: you need to have True/False breach indicator column in Python, and filter with ‘== True’ .
(d) In your own words describe, was NASDAQ-100 more risky than SP 500 during COVID pandemic news 2020-Feb to 2020-Mar? What about the subsequent market correction period in 2021-2022?
Question 5. Implement the backtest of VaR10D,t but now with the input of EWMA σt(2)+1 from the
filtering formula below, instead of rolling std dev. Tutor will not reconfirm the formula/computation.
σt(2)+1 | t = λσt(2)| t−1 + (1 − λ) rt(2)
with λ = 0.72 value set to minimise out of sample forecasting error.
Hint: use the variance for the entire dataset to initialise the computation.
(a-c) Provide the same deliverables (a), (b) and (c) as in the previous Question.
(d) Briefly discuss the impact of λ on smoothness of EWMA-predicted volatility (3-4 lines).
Hint: you can discuss λ theoretically without recomputing EWMA-based backtest but, if you recompute for an extra illustration it is sufficient to do so for one market index only.
WX：codehelp
