Question

A large institutional investor is considering three Exchange Traded Funds:

Fund Expected Return Standard deviation Internet

ETF (Q) 11% 16%

Health Care ETF (H). 9% 12%

T-bill money-market ETF 4%

The two ETFs, Q and H are uncorrelated.

1. Find the proportions of each asset, and the expected return and standard deviation of the tangency portfolio.

2. What is the reward-to-variability (Sharpe) ratio of the best feasible capital allocation line?

3. Suppose this investor is highly risk averse with a risk aversion coefficient equals to 8. a. What is the composition of the optimal portfolio in terms of all available funds? b. What are the expected return and standard deviation of your optimal portfolio?

4. A less sophisticated investor would like to use only the Internet ETF and Health Care ETF, and require an expected return of 13%, what must be the investment proportions of her portfolio? Compare your result to the optimized portfolio in part 3.a. What do you conclude?

Answer 1

1) From the information given in question, we can find the following:

     Internet ETF (Q) -                     Expected Return = E_Q = 11%

                                                         Standard Deviation = _Q = 16%

                                                         Proportion in Portfolio = W_Q

    Health Care ETF (H) -              Expected Return = E_H = 9%

                                                         Standard Deviation = _H = 12%

                                                         Proportion in Portfolio = W_H = 1 - W_Q

    Since the two ETFs, Q & H are uncorrelated, so as per minimum variance formula,

                                    

                                               = (12%)^2 / ((16%)^2 + (12%)^2)

                                               = 36%

                                So, W_H = 1 - W_Q = 1 - 36% = 64%

    So, The Expected Return of the Portfolio = E_P = (E_Q * W_Q) + (E_H * W_H) = (11% * 36%) + (9% * 64%) = 9.72%

    So, The Standard Deviation of the Portfolio = σ_P = Sqrt ((_Q²*W_Q²) + (_H²*W_H²))

                                                                                           = Sqrt ((16%² * 36%²) + (12%² * 64%²))

                                                                                           = 9.60%

Hence for the Tangency Portfolio:

                           Proportion of ETF (Q) = W_Q = 36%

                           Proportion of ETF (H) = W_H = 64%

                           Expected Return = E_P =9.72%

                           Standard Deviation = σ_P =9.60%

2) Let us compare thy Expected Return and Standard Deviation of individual ETF with the Portfolio

       

       From the table it can be inferred that the Tangency Portfolio (P) has the highest Expected Return with Lowest Variability.

       So, the best possible capital allocation line goes through the Tangency Portfolio.

      Now,

      Here, T-bill money-market ETF with 4% Expected Return is the Risk Free Asset.

So, Reward-to-Variability (Sharpe) ratio = (9.72% - 4%) / 9.60% = 59.58%

3) As discussed in solution of "part 2" the Optimal Portfolio in terms of all available funds is the Tangency portfolio which is a blend of both ETF(Q) and ETF(H).

   Also there is a Risk Free Asset i.e. the T-bill money-market ETF with 4% Expected Return.

Now we have to compare the Tangency Portfolio with the Risk Free Asset.

This can be done using the "Utility Formula" which will measure the utility of the Tangency portfolio with respect to Risk Free Asset (T-bill money-market ETF)

The Utility Formula is: U_P = E_P – 0.5 x R_A x σ_P².

                                                                            where, U_P = Utility score of a portfolio compared to a risk-free asset.

                                                                                          E_P = Expected return of the portfolio.

                                                                                          R_A = Risk Aversion coefficient of the investor.

                                                                                         σ_P² = Square of volatility (Standard Deviation) of a portfolio.

Here, E_P = 9.72% ; R_A = 8 ; σ_P² = 9.60%

So, in this case, U_P = 9.72% – 0.5 x 8 x 9.60²

From the above Utility value it can be inferred that by subtracting the portfolio risk (for the risk free investor) from the expected result, there is a Risk-free return that is higher than the return from Risk Free Asset(T-bill money-market ETF).

So, it is more profitable to invest in Tangency Portfolio (Combination of ETF(Q) and ETF(H)) as compared to Risk Free Asset(T-bill money-market ETF).

(a) As derived in "problem no 1" the Optimum Portfolio has the following composition:

                           Proportion of ETF (Q) = W_Q = 36%

                           Proportion of ETF (H) = W_H = 64%

(b) the Optimum Portfolio has the following:

                           Expected Return = E_P =9.72%

                           Standard Deviation = σ_P =9.60%

4) A less sophisticated investor would like a Portfolio with only the Internet ETF and Health Care ETF. he expects a return of 13%.

   So, Expected Return of Portfolio = E_P = 13%,

          Expected Return of Internet ETF (Q) = E_Q = 11%,

          Expected Return of Health Care ETF (H) = E_H = 9%

          Proportion of Internet ETF (Q) in Portfolio = W_Q

          Proportion of Health Care ETF (H) in Portfolio = W_H = 1 - W_Q

Now, The Expected Return of the Portfolio = E_P = (E_Q * W_Q) + (E_H * W_H)

                                                                            13% = (9.72% * W_Q) + (4% * W_H)

                                                                             13% = (9.72% * W_Q) + (4% * (1-W_Q))

                                                                              13% = 5.72% * W_Q + 4%

                                                                               W_Q = 200%

So, it can be seen that the investor needs to invest 200% of his entire fund for the Portfolio in Internet ETF (Q). But this is a hypothetical situation which is not possible.

So, he has to invest in the Tangency Portfolio, which has the following parameters:

    for the Tangency Portfolio:

                           Proportion of ETF (Q) = W_Q = 36%

                           Proportion of ETF (H) = W_H = 64%

                           Expected Return = E_P =9.72%

                           Standard Deviation = σ_P =9.60%