Q5 Consider the rate of return of stocks ABC and XYZ.


a. Calculate the arithmetic average return on these stocks over the sample period.
b. Which stock has greater dispersion around the mean return?
c. Calculate the geometric average returns of each stock. What do you conclude?
d. If you were equally likely to earn a return of 20%, 12%, 14%, 3%, or 1% in each year (these are the five annual returns for stock ABC), what would be your expected rate of return?
e. What if the five possible outcomes were those of stock XYZ?
f. Given your answers to parts (d) and (e), which measure of average return, arithmetic or geometric, appears more useful for predicting future performance? (LO 18-1)

.        Arithmetic average: ̅rABC = 10%; ̅rXYZ = 10%

b.        Dispersion: σABC = 7.07%; σXYZ = 13.91%
Stock XYZ has greater dispersion.
(Note: We used 5 degrees of freedom in calculating standard deviations.)

c.        Geometric average:
rABC = (1.20 × 1.12 × 1.14 × 1.03 × 1.01)1/5 – 1 = 0.0977 = 9.77%
rXYZ = (1.30 × 1.12 × 1.18 × 1.00 × 0.90)1/5 – 1 = 0.0911 = 9.11%
Despite the fact that the two stocks have the same arithmetic average, the geometric average for XYZ is less than the geometric average for ABC. The reason for this result is the fact that the greater variance of XYZ drives the geometric average further below the arithmetic average.

d.        Your expected rate of return would be the arithmetic average, or 10%.

e.        Even though the dispersion is greater, your expected rate of return would still be the arithmetic average, or 10%.

f.        In terms of “forward-looking” statistics, the arithmetic average is the better estimate of expected rate of return. If the data reflect the probabilities of future returns, 10 percent is the expected rate of return for both stocks.

CASE STUDY:
CFA problem no. 2        Carl Karl, a portfolio manager for the Alpine Trust Company, has been responsible since 2020 for the City of Alpine’s Employee Retirement Plan, a municipal pension fund. Alpine is a growing community, and city services and employee payrolls have expanded in each of the past 10 years. Contributions to the plan in fiscal 2025 exceeded benefit payments by a three-to-one ratio.
The plan board of trustees directed Karl five years ago to invest for total return over the long term. However, as trustees of this highly visible public fund, they cautioned him that volatile or erratic results could cause them embarrassment. They also noted a state statute that mandated that not more than 25% of the plan’s assets (at cost) be invested in common stocks.
At the annual meeting of the trustees in November 2025, Karl presented the following portfolio and performance report to the board:

Karl was proud of his performance and was chagrined when a trustee made the following critical observations:
a.        “Our one-year results were terrible, and it’s what you’ve done for us lately that counts most.”
a. Extended time frame is essential to draw a conclusion. Though the one-year return was terrible, but this small duration of one year cannot be used as a base to draw inferences. Moreover, the mandate was to give priority to long-term investments.

b.        “Our total fund performance was clearly inferior compared to the large sample of other pension funds for the last five years. What else could this reflect except poor management judgment?”
b. Large proportion of pension funds has caused a decreased returns. The pension funds held a much larger share in equities whose returns significantly exceeded bond returns. Moreover, the Alpine fund manager was also instructed to hold down risk, investing at most 25% of fund assets in common stocks.

c.        “Our common stock performance was especially poor for the five-year period.”
c. Alphine’s stock was positive. On the contrary, over the five years, Alpine’s alpha was positive:

α = .133 – [ .075 + 0.9 x ( .138 – .075)] = .13%

d.        “Why bother to compare your returns to the return from Treasury bills and the actuarial assumption rate? What your competition could have earned for us or how we would have fared if invested in a passive index (which doesn’t charge a fee) are the only relevant measures of performance.”
Heavy weightage in bonds was determined upon by the Board and not the fund management.
d. Over the last five years, especially the last year, bond performance has been poor because Alpine was encouraged to hold this asset class.

Still, however, the Alpine fund fared much better than the index. Moreover, despite the underperformance of the bond index, it outperformed both for the five years. Its performance was also superior on a risk-adjusted basis. The major source of disappointment was the heavy asset allocation weighting towards bonds, which was the Board’s choice, not the fund manager’s.

e.        “Who cares about time-weighted return? If it can’t pay pensions, what good is it?” Appraise the merits of each of these statements and give counterarguments that Mr. Karl can use. (LO 18-1)
e. Manager’s ability is revealed in the time-weighted return. e. A trustee may care little about the time-weighted return, but this would reflect more about manager’s performance.