Portfolio Optimization using Risk Parity and Sharpe Ratio

Risk Parity

The second optimization model we will implement is risk parity. By definition, risk parity seeks to equalize the individual risk contribution of each constituent in a portfolio. Therefore, it is an purely risk-based portfolio, i.e., it does not require the asset expected returns as an input parameter. Instead, it only requires the asset covariance matrix, Q. For this project, the covariance matrix should be estimated from the PCA factor model in Section 2. This optimization model will yield a risk parity portfolio for us to compare against the other investment strategies. Important note: We have studied several alternative formulations of risk parity in class. However, due to its simplicity for numerical implementation, it is recommended that you implement the convex risk parity formulation where we had the portfolio variance term minus the sum of natural logarithms.

 

This formulation can be easily implemented if we use the ‘fmincon()’ function in MATLAB, which accepts the use of logarithms. Please do not forget to normalize your optimal solution in order to recover the actual asset weights. To reiterate, this is the recommended form of risk parity, but you are free to implement any of the alternative methods seen in class if that is easier for you. The third and final optimization model we will implement in this project is the Sharpe ratio maxi- mization model. In its regular form, the Sharpe ratio is not a concave function which we can easily maximize. However, as shown in class, we can reformulate the Sharpe ratio into a convex problem. Use the convex reformulation of the Sharpe ratio optimization model to build the corresponding portfolio. As with the previous optimization models, short sales are disallowed.

 

Note: The Sharpe ratio is based on the asset excess returns. However, please note that the MATLAB template already converts your asset returns into excess returns by subtracting the monthly risk- free rate of return from each observation. Therefore, your asset returns are already adjusted for use during Sharpe ratio optimization.

 

1 How good is the fit of the PCA factor model with p = 3?

 

2 How does the distribution of our factor returns look like? Does the assumption of factor normality hold? We can plot a histogram using the principal components extracted from the historical asset returns.

 

3 Do the values of skewness and kurtosis of our factor returns approximate those of a normal distribution?

 

4 After we optimize all four of our portfolios, what are the estimated values of VaR and CVaR?

 

5 Do the CVaR portfolios have a lower VaR/CVaR value when compared against the risk parity and maximum Sharpe ratio portfolios?

 

6 How do the asset risk contributions of each portfolio look like? By definition, risk parity should be the most risk-diverse. How do the other portfolios compare?

 

7 What is the ex-ante Sharpe ratio of each four portfolios?

 

8 We can also use an area plot to show the changes per period in the asset weights and/or asset risk contributions. This will show us how our optimal weights and risk contributions change every time we rebalanced a portfolio. This will allow us to evaluate if our portfolios were concentrated or well-diversified.

 

9 We can compute the ex-post performance measures discussed in class, such as the realized portfolio return and volatility, Sharpe ratio, and the observed VaR and CVaR.

 

10 We should plot the wealth evolution of the four portfolios to observe their change in value over time (i.e., plot the portfolio value over the out-of-sample period).

 

11 How do the two optimal CVaR portfolios compare to one another? Is there a pronounced difference driven by the two Monte Carlo simulation methods?

 

12 Another possible analysis is to compare the ex-ante and ex-post Sharpe ratios of our different portfolios during each investment period. Did the expected values match their realizations?