FIN 340 Module 4 Risk and Return Assignment Example

Risk, Return and Diversification for Two Clients' Portfolios

[Student Name]

Southern New Hampshire University

FIN 340: Fundamentals of Investments

Module Four Assignment

[Instructor Name]

[Date]

The organization, setting and figures below are a composite written as a model document. No real employer, client, colleague or patient is described.

Risk, Return and Diversification for Two Clients' Portfolios

Introduction

Project One concluded that the composite client Tessa Lindqvist should hold mainly stocks for retirement and that Gordon and Renee Abernathy need a balanced portfolio. This paper measures what those choices mean in terms of return and risk. It computes the expected return and standard deviation of five stock and bond mixes, shows how diversification lowers risk, tests what happens when stocks and bonds move more closely together and explains why concentrated holdings add risk without adding expected return.

Inputs

The estimates below are rounded long-run figures, not forecasts. Stocks are broad U.S. equities, bonds are investment-grade U.S. bonds of intermediate maturity and the risk-free asset is Treasury bills.

Assumed annual return and risk

Standard deviation measures how widely annual returns scatter around their average. For stocks, a standard deviation of 17 percent means that in about two years out of three, returns would fall between -7.5 and 26.5 percent, and in a bad year, about one in forty, they could lose roughly 24 percent or more.

Portfolio Return

A portfolio's expected return is the weighted average of its assets' returns. For the 60/40 mix, with sixty cents of each dollar in stocks:

E(Rp) = 0.60 x 9.5% + 0.40 x 4.5% = 5.7% + 1.8% = 7.5%.

Portfolio Risk

Portfolio risk is not a weighted average, because the assets do not move together perfectly. Markowitz (1952) showed that portfolio variance depends on each asset's variance and on how the assets move together:

Variance = (wS x sS) squared + (wB x sB) squared + 2 x wS x wB x correlation x sS x sB.

For the 60/40 mix with a correlation of 0.1:

(0.60 x 0.17) squared = 0.010404; (0.40 x 0.065) squared = 0.000676; 2 x 0.60 x 0.40 x 0.1 x 0.17 x 0.065 = 0.000530.

The total is 0.011610, and its square root is 0.1078, a standard deviation of 10.8 percent. The weighted average of the two standard deviations would be 12.8 percent, so diversification removed about 2 points of risk at no cost in expected return.

Five Mixes Compared

The Sharpe ratio divides each mix's expected return above the Treasury bill rate by its standard deviation, giving reward per unit of risk.

Expected return, risk and Sharpe ratio (correlation 0.1)

Adding even a modest share of the other asset lowers risk more than it lowers return, so the mixes have higher Sharpe ratios than either asset alone. The 60/40 and 40/60 mixes give the most reward per unit of risk under these assumptions.

A Stress Test

The benefit of combining stocks and bonds depends on their correlation. For most of the two decades before 2022, the correlation between U.S. stock and bond returns was near zero or negative. In 2022, when inflation and rising rates hit both, the S&P 500 fell about 18 percent and broad bond indexes fell about 13 percent. Repeating the calculation with a correlation of 0.5 raises the 60/40 portfolio's standard deviation from 10.8 to 11.7 percent and the 40/60 mix's from 8.2 to 9.4 percent, lowering both Sharpe ratios to about 0.34 and 0.32. Diversification still helps, but less, which is a reason to hold some short-term Treasury bills or cash for near-term needs rather than relying on bonds alone.

Systematic and Unsystematic Risk

Diversification across many stocks removes risk specific to individual companies, such as a lost contract or a failed product, leaving only market-wide or systematic risk. Statman (1987) estimated that a portfolio needs at least 30 to 40 randomly chosen stocks before the remaining specific risk becomes small relative to the cost of adding more. Sharpe (1964) showed that in equilibrium, investors are paid only for systematic risk, measured by beta, because anyone can shed company-level risk simply by owning more names. Gordon Abernathy's three engineering and construction stocks make up about $120,000 of the couple's savings. They carry a large dose of specific risk, and because they depend on the same industry as Gordon's paycheck, a downturn in construction could reduce his income and his savings at the same time. That risk earns no extra expected return.

Pricing the Risk That Remains

Sharpe's model turns this idea into a required return: start from the Treasury bill yield and add beta times the premium investors expect from owning the whole market. With Treasury bills at 3.5 percent and a market premium of 6 points, a stock with a beta of 1.0 should be expected to earn 9.5 percent, the same as the market. Engineering and construction firms tend to be cyclical, with betas often between 1.2 and 1.4. At a beta of 1.3, the required return is 3.5 + 1.3 x 6, or 11.3 percent. That higher figure rewards only the stocks' sensitivity to the overall market. The extra swings that come from one firm winning or losing a highway contract do not raise the required return at all, so a portfolio of three such stocks carries far more total volatility than a fund with the same beta, with no added reward. For Gordon, that is the case for trimming the position gradually, spreading the sales over several tax years to manage the capital gains.

Application to the Clients

For Tessa's retirement savings, a 90/10 mix offers an expected 9 percent with risk only slightly below an all-stock portfolio, which suits her long horizon and the goal-based split from Project One. A bad year, two standard deviations below the mean, would be a loss of about 22 percent, a fall she can afford in money she will not use for decades. For the Abernathys, the 60/40 mix offers the best reward per unit of risk and a bad-year loss of about 14 percent, about half the loss of an all-stock portfolio. Moving gradually toward 40/60 as retirement approaches would cut the bad-year loss further, to about 10 percent.

Conclusion

Combining assets that do not move in lockstep reduces risk without reducing return, which is the only free benefit in investing. The benefit shrinks when correlations rise, and it is wasted when a portfolio holds a few concentrated stocks whose specific risk the market does not reward.

References

Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7(1), 77-91. https://doi.org/10.1111/j.1540-6261.1952.tb01525.x

Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. The Journal of Finance, 19(3), 425-442. https://doi.org/10.1111/j.1540-6261.1964.tb02865.x

Statman, M. (1987). How many stocks make a diversified portfolio? Journal of Financial and Quantitative Analysis, 22(3), 353-363. https://doi.org/10.2307/2330969