FIN 402 Week 2 Measuring Risk and Return Example

How Much Did Rosa Earn, and How Much Could She Lose? Holding Period Returns, Real Returns, Expected Return, Standard Deviation and the Sharpe Ratio for a Stock Fund and a Bond Fund

[Student Name]

University of Phoenix

FIN/402: Investment Fundamentals and Portfolio Management

Week 2 Assignment

[Instructor Name]

[Date]

The investor, scenarios and probabilities are composites written for a model paper; methods, historical patterns and research findings come from the sources listed.

A year after investing her inheritance, Rosa Delgado, the composite respiratory therapist from Week 1, wants to know how she has done and how much risk she has taken. She bought shares of a broad stock index ETF at $480, received $6.50 per share in dividends and saw the price close the year at $512. She also holds a short-term bond fund. Inflation over the year was 3.2 percent, and Treasury bills yielded about 4.3 percent. A return means little until it is set beside the risk taken to earn it and the inflation that eroded it. This paper measures both for her two funds.

Holding Period Return

The holding period return adds the price change and any income, then divides by the starting price. For the ETF, the gain is $32 of price change plus $6.50 of dividends, or $38.50, divided by $480, a return of about 8.0 percent. That is the return Rosa earned before taxes over exactly one year.

Real Return

Inflation of 3.2 percent reduced what that return could buy. The exact real return divides one plus the nominal return by one plus inflation: 1.0802 divided by 1.032, minus one, about 4.7 percent. The shortcut of subtracting inflation gives 4.8 percent, close here but increasingly wrong at higher rates (Bodie et al., 2021). Rosa's purchasing power grew by about 4.7 percent.

Expected Return From Scenarios

Past returns describe one year; planning requires expectations. A simple model assigns probabilities to three economic scenarios for next year. For the stock fund, a boom with a 25 percent probability brings a 28 percent return, a normal year with a 50 percent probability brings 10 percent and a recession with a 25 percent probability brings a loss of 14 percent. Weighting each outcome by its chance and adding the results, 28 percent at one quarter, 10 percent at one half and negative 14 percent at one quarter, gives 7.0 plus 5.0 minus 3.5, an expected return of 8.5 percent.

For the bond fund, the same scenarios bring 2 percent, 4.5 percent and 8 percent; bonds tend to gain in recessions as rates fall. Its expected return is 0.5 plus 2.25 plus 2.0, or 4.75 percent.

Standard Deviation

Variance weights each squared deviation from the expected return by its probability. For the stock fund, the deviations are 19.5, 1.5 and negative 22.5 points; their squares, 380.25, 2.25 and 506.25, weighted by 0.25, 0.50 and 0.25, sum to about 222.8. The standard deviation, the square root, is about 14.9 percentage points. For the bond fund, the squared deviations of 7.56, 0.06 and 10.56 weighted the same way sum to about 4.56, a standard deviation of about 2.1 points. The stock fund offers almost twice the expected return with seven times the variability.

Reward per Unit of Risk

The Sharpe ratio divides the expected excess return over the risk-free rate by the standard deviation. For the stock fund, 8.5 minus 4.3, or 4.2 points, divided by 14.9 gives about 0.28. For the bond fund, 0.45 points divided by 2.1 gives about 0.21. By this measure, the stock fund rewards its risk slightly better, though neither ratio is high, because Treasury bills paid an unusually high safe return.

What History Shows

Over long periods in the United States, large company stocks have returned roughly 10 percent a year on average with a standard deviation near 20 percent, long-term government bonds about half that return with less variability and Treasury bills less still, with the gaps shifting from decade to decade (Bodie et al., 2021). The pattern supports the idea that higher average returns come with higher risk. Fama and French (2002) cautioned, however, that the large equity premium of the second half of the twentieth century was well above what dividend and earnings growth would predict, suggesting that rising valuations, which may not repeat, supplied part of it.

Returns Beyond One Country

Most long return histories come from the United States, which may flatter them. Dimson et al. (2002) assembled returns for 16 countries from 1900 to 2000 and found that stocks beat bonds and bills in every one of them, but that the United States was among the better performers over the century. Some markets suffered losses from war, hyperinflation or confiscation that U.S. investors never faced. A forecast built only on U.S. history therefore leans toward the optimistic end of the range. For Rosa, that argues for planning with an expected stock return somewhat below the long U.S. average and for holding international stocks, a step Week 3 considers.

Arithmetic and Geometric Averages

History is summarized in two ways. The arithmetic average of annual returns estimates a typical single year; the geometric average measures the compound growth an investor actually received over many years. A fund that gains 50 percent one year and loses 50 percent the next has an arithmetic average of zero but leaves the investor with 75 percent of her money, a geometric average of about negative 13.4 percent a year. For a long horizon like Rosa's, the geometric average is the more honest planning figure.

What the Numbers Mean for Rosa

The scenario model suggests that in a recession year, Rosa's stock fund could lose 14 percent, and history shows that worse years occur. If she held only stocks, a $70,000 position could fall by nearly $10,000 in a single year, and by more in a severe one. Whether she can tolerate that without selling at the bottom is a question for Week 3, where the two funds' combined risk will be lower than either fund's risk weighted alone, because bonds have tended to hold up when stocks fall.

Limits of the Measures

Standard deviation treats gains and losses symmetrically, while investors feel losses more. Scenario probabilities are judgments, not facts, and real returns can fall outside all three scenarios. The measures are useful for comparison, not for prediction.

Conclusion

Rosa's first year earned an 8.0 percent nominal and 4.7 percent real return. Looking forward, the stock fund offers a higher expected return with far more variability, and on a risk-adjusted basis the two funds are closer than raw returns suggest. These measures supply the inputs for building her portfolio.

References

Bodie, Z., Kane, A., & Marcus, A. J. (2021). Investments (12th ed.). McGraw Hill.

Dimson, E., Marsh, P., & Staunton, M. (2002). Triumph of the optimists: 101 years of global investment returns. Princeton University Press.

Fama, E. F., & French, K. R. (2002). The equity premium. The Journal of Finance, 57(2), 637-659. https://doi.org/10.1111/1540-6261.00437