Construct the optimal portfolio comprising two risky assets while considering the client’s risk
You have been assigned to construct the optimal portfolio comprising two risky assets (Portfolios A & B) while considering the client’s risk tolerance. The attached spread sheet shows historical monthly returns of the two portfolios, the S&P 500 and 90-day Treasury Bills. Also shown are the annualized returns for each for the period specified.
1.The first risky asset (Portfolio A) is a US equity strategy that uses publically available valuation, technical and sentiment factors to assess which stocks are over-priced and which are under-priced. Fundamental factors indicate the magnitude and quality of a company’s earnings and the strength of its balance sheet. Examples of such factors include: cash flow growth, cash flow return on invested capital, price to cash flow, and accruals which assess earnings quality (low quality earnings indicate that management may be manipulating earnings by adjusting accruals). Companies with favorable fundamental factors tend to outperform those with less favorable factors.
2.Technical and sentiment factors seek to identify mis-pricings resulting from investor behavior. Examples include: momentum and price reversals where investors tend to over-react to good news by bidding up prices ABOVE fair value and bad news by bidding down prices BELOW fair value; short interest on a stock which can indicate the investor sentiment about the company’s prospects; share buybacks which can indicate a positive signal from management’s optimism regarding a firm’s future prospects; and earnings / revenue surprise. Firms with favorable technical and sentiment factors also tend to outperform. For example, firms whose earnings and revenue exceed analysts’ expectations tend to continue to outperform vs. those firms that experience earnings surprise due to cost cutting.
3.Starting with the market portfolio, the US equity strategy over-weights those stocks with more favorable fundamental, technical and sentiment factors and under-weights or avoids those stocks with less-favorable or un-favorable factors. The strategy seeks to out-perform the market portfolio as represented by the S&P 500. The monthly returns of the US equity strategy are shown in the attached spreadsheet (Portfolio A).
4.The second risky asset (Portfolio B) is a global macro hedge fund. This strategy seeks to benefit from mis-pricings within and across broad asset classes by taking long and short positions in equity markets, bond markets and currencies. For example, if the manager believes that US equities will out-perform Japanese equities, the portfolio will go long S&P 500 futures and short TOPIX futures (TOPIX is a Japanese equity index). This long/short trade is not impacted by the overall direction of global equities, but rather the relative movement between US and Japanese equities. Similarly for bonds, if the manager believes that interest rates in the United Kingdom (UK) will decline more so than interest rates in Australia, then the manager will buy UK gilt futures (gilt is the 10-year UK bond) and short Australian 10-year bond futures. Again, this trade is not impacted by the overall direction of global interest rates, but rather the relative movement between UK and Australian rates. Recall that bond prices rise as interest rates decline. The global macro hedge fund is mostly market neutral meaning that long positions equal short positions thereby dramatically reducing systematic exposure (low beta).
5.Portfolios A & B are much more volatile than the risk free rate. You will find that their correlation is small indicating that there is a diversification benefit to be had from holding both in a portfolio (I don’t show the correlation, but you will need to calculate this using the excel function “=correl(range 1, range2)”.
You will be meeting with a client that is looking for investment advice from you based on your two strategies A & B. In preparation for your upcoming meeting with the client, your boss asks that you respond to the questions below and be ready to discuss. Hint: You will need to determine the correlations and volatilities for each risk premium.
Analytical Assignment
- Plot in Excel the risky asset opportunity set for Portfolios A & B. Hint: create the following table in excel assuming weights of portfolio A & B in 10 percentage point increments. Then calculate expected return and standard deviation for each allocation to A & B.
- Find the optimal complete portfolio based on your client’s indifference curve. Hint: Plot an indifference curve on the same graph you just created using the utility function formula from Chapter 6. Use the range of expected return and standard deviations shown in the table below. Assume U = 9% and a risk aversion coefficient (A) of 10 to complete the table below.
- Use the capital asset pricing model (CAPM) to determine the beta and alpha of Portfolio A & Portfolio B. Show the CAPM relationship graphically for BOTH Portfolio A and Portfolio B (separate graphs). The market portfolio is represented by the S&P 500 and the risk free rate is represented by 90 day Treasury Bill. Determine the beta for portfolio A & B using: i) the slope function in Excel; and ii) the formula for beta – the co-variance between the asset and the market divided by the variance of the market. This is explained in the Modules 6& 7 notes and pages 296 & 297 in the text. Recall the covariance between two assets is the volatility of asset 1 times the volatility of asset 2 times the correlation between them.
- Your client will notice that the Sharpe ratio of the hedge fund (Portfolio B) is much higher than that of the equity strategy (Portfolio A) and will ask why the optimal risky portfolio wouldn’t be 100% of Portfolio B. How would you respond?
- Your client vehemently believes in the semi-strong form of market efficiency as it relates to security selection. Is the performance of Portfolio A sufficient justification to convince the client otherwise – that markets are inefficient or at least less efficient? Why or why not?
- Given your client’s belief regarding market efficiency as it pertains to security selection, what portfolio substitution(s) would you make in your optimal portfolio? No calculations are necessary to answer this.
- Your client is expected to ask why you are recommending the optimal complete portfolio instead of the optimal portfolio even though the latter has a higher expected return. How will you respond?
- After meeting with the client, she appears to prefer the risk/return tradeoff of the optimal portfolio rather than that of the optimal complete portfolio. What does that indicate about your initial assumptions regarding the indifference curve?
| Weight Port A | Weight Port B | Return | Standard Deviation | Sharpe Ratio |
| 0% | 100% | |||
| 10 | 90 | |||
| 20 | 80 | |||
| 30 | 70 | |||
| 40 | 60 | |||
| 50 | 50 | |||
| 60 | 40 | |||
| 70 | 30 | |||
| 80 | 20 | |||
| 90 | 10 | |||
| 100 | 0 |
Determine the optimal allocation of A & B and draw in the Capital Allocation Line (CAL). You can arrive at an approximate optimal allocation using a table similar to the one shown above OR you can obtain a more precise optimal allocation using the formula shown in Chapter 7 (equation 7.13). When drawing the CAL on the efficient frontier graph plotted in Excel, you can manually draw a line starting at the risk free rate to the tangent point.
| Expected Return | Standard Deviation |
| 5% | |
| 7.5 | |
| 10 | |
| 12.5 | |
| 15 | |
| 17.5 | |
| 20 | |
| 22.5 | |
| 25 |
Calculate the expected alpha for each portfolio A & B using the intercept function in Excel and the index model of CAPM formula (equation 9.9 on page 302 – note that the terms are in excess return form). Ignore the error term and you have all the information to solve for alpha based on the monthly returns. Compare the betas and y-intercepts using the two different methods.
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