Columbia University in the City of New York
Vanguard (1)
It is June of 2003, and Prof. Juran is reviewing his retirement portfolio, wondering whether he ought to reallocate his investments. Currently his portfolio consists of positions in eight Vanguard index mutual funds:
|Ticker Symbol |VEURX |VPACX |VFINX |VWESX |VIGRX |VEXPX |NAESX |VWIGX |
|Fund Type |Europe |Pacific |USA |Bonds |Growth |Explorer |Sm-Cap |Int’l |
|Proportion |13.5% |13.1% |12.6% |21.3% |9.4% |10.1% |9.9% |10.1% |
Note: Vanguard does not allow Juran to sell mutual funds short.
Juran wants to optimize his portfolio based on the following historical price results (all are in dollars per share):
| |VEURX |VPACX |VFINX |VWESX |VIGRX |VEXPX |NAESX |VWIGX |
|Nov-01 |20.30 |6.79 |103.08 |9.21 |25.75 |54.01 |18.06 |14.56 |
|Dec-01 |20.18 |6.90 |105.35 |8.84 |26.45 |56.64 |18.90 |14.79 |
|Jan-02 |20.12 |6.55 |107.08 |8.60 |26.79 |60.87 |20.03 |14.93 |
|Feb-02 |19.28 |6.09 |103.59 |8.79 |26.20 |58.71 |19.49 |14.37 |
|Mar-02 |19.41 |6.55 |104.62 |8.74 |26.27 |57.65 |19.44 |14.55 |
|Apr-02 |20.26 |6.79 |105.85 |8.49 |26.14 |61.13 |20.61 |15.08 |
|May-02 |20.17 |7.04 |100.31 |8.69 |24.43 |59.99 |20.81 |15.22 |
|Jun-02 |20.13 |7.37 |98.67 |8.72 |23.79 |57.08 |19.90 |15.06 |
|Jul-02 |19.44 |6.99 |91.33 |8.71 |21.83 |52.53 |18.99 |14.42 |
|Aug-02 |17.28 |6.52 |84.28 |8.70 |20.80 |45.78 |16.14 |12.97 |
|Sep-02 |17.27 |6.47 |84.83 |9.07 |20.93 |45.90 |16.10 |12.97 |
|Oct-02 |15.00 |6.10 |75.26 |9.26 |18.70 |42.83 |14.94 |11.42 |
|Nov-02 |16.44 |5.90 |81.87 |8.86 |20.42 |44.60 |15.42 |12.22 |
|Dec-02 |17.25 |6.11 |86.68 |8.93 |21.41 |48.60 |16.79 |12.98 |
|Jan-03 |16.21 |5.88 |81.15 |9.23 |19.95 |45.49 |15.66 |12.16 |
|Feb-03 |15.45 |5.71 |79.02 |9.20 |19.44 |44.20 |15.23 |11.75 |
|Mar-03 |14.94 |5.42 |77.82 |9.43 |19.38 |42.93 |14.77 |11.37 |
|Apr-03 |14.71 |5.55 |78.27 |9.28 |19.70 |43.51 |14.95 |11.24 |
|May-03 |16.70 |5.60 |84.73 |9.46 |21.03 |47.31 |16.35 |12.34 |
|Jun-03 |17.92 |5.88 |87.89 |9.93 |21.45 |51.53 |17.58 |13.07 |
a. Convert these historical results from dollars per share into percent returns, and estimate the expected return on Juran’s portfolio over the next month, assuming that these historical events are representative of future results. State clearly any important assumptions you are making in your model.
b. Find the lowest-risk portfolio consisting of these eight funds, using the scenario approach to optimization. State clearly any important assumptions you are making in your model. What proportion of Juran’s portfolio ought to be invested in each of these funds? What is the expected return of your recommended portfolio? What is the standard deviation of the return?
c. Juran is not only interested in risk; he also wants to ensure that his portfolio will be expected to yield at least 0.15% (0.0015 in decimals) over the next month. Adjust your model to reflect this goal, and re-solve the problem. What proportion of Juran’s portfolio ought to be invested in each of these funds? What is the expected return of your recommended portfolio? What is the standard deviation of the return?
d. Juran is concerned that you may be over-fitting your model to recent historical events, which may not accurately reflect the future. In the interests of diversification, he asks you to create a portfolio that invests at least 2.5% in each of the eight funds, and no more than 40% in any single fund. Also, he wants the portfolio’s expected return not to be below zero. Adjust your model to reflect these goals, and re-solve the problem. What proportion of Juran’s portfolio ought to be invested in each of these funds now? What is the expected return of your recommended portfolio? What is the standard deviation of the return?
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