Rates of Return, Risk Measurement, and Diversification: Financial Accounting Study Notes

Rates of Return

Definition and Calculation

The rate of return measures the gain or loss on an investment over a specified period, expressed as a percentage of the initial investment. Returns typically come from two sources: dividends (or interest) and capital gains.

• Percentage Return Formula: $\text{Percentage return} = \frac{\text{capital gain} + \text{dividend}}{\text{initial share price}}$

• Dividend Yield: $\text{Dividend yield} = \frac{\text{dividend}}{\text{initial share price}}$

• Capital Gain Yield: $\text{Capital gain yield} = \frac{\text{capital gain}}{\text{initial share price}}$

Example: If you buy a share for $158.78, receive a $2.09 dividend, and sell it for $222.42, your percentage return is:

• $\text{Percentage return} = \frac{63.64 + 2.09}{158.78} = 0.414$ or 41.4%

• $\text{Dividend yield} = \frac{2.09}{158.78} = 0.0132$ or 13.2%

• $\text{Capital gain yield} = \frac{222.42 - 158.78}{158.78} = 0.4008$ or 40.08%

Nominal vs. Real Rate of Return

The nominal rate of return does not account for inflation, while the real rate of return adjusts for inflation, providing a more accurate measure of purchasing power.

• Formula: $1 + \text{real rate of return} = \frac{1 + \text{nominal rate of return}}{1 + \text{inflation rate}}$

• Example: If nominal return is 41.4% and inflation is 1.4%, $1 + \text{Real ROR} = \frac{1 + 0.414}{1 + 0.014} = 1.394$ $\text{Real ROR} = 39.4\%$

Market Indexes

Overview

Market indexes are statistical measures that track the performance of a group of assets, representing the overall market or a segment of it.

• Dow Jones Industrial Average (The Dow): Tracks 30 large "blue chip" stocks.

• Standard & Poor's Composite Index (S&P 500): Tracks 500 large U.S. stocks.

Example: The value of a $1 investment in equities, bonds, and bills since 1900 shows equities outperforming other asset classes over the long term.

Historical Returns

Portfolio	Average Annual Rate of Return	Average Premium (Excess return versus Treasury bill)
Treasury bill	3.7%	-
Treasury bonds	5.4%	1.7%
Common stocks	11.5%	7.8%

Expected Return

Calculation

The expected market return is often estimated as the sum of the interest rate on Treasury bills and a normal risk premium.

Year	Expected market return	Interest rate on Treasury bills	Normal risk premium
1981	21.7%	14%	7.7%
2018	9.4%	1.7%	7.7%

Historical Expected Returns

	1900–1928	1929–1957	1958–1987	1988–2020
Stocks	12.0%	9.8%	11.8%	12.4%
Treasury bills	4.9%	1.0%	6.0%	3.0%

Country Risk Premiums

Country risk premium is the additional return investors demand for investing in a particular country, reflecting its risk level.

• Risk premiums vary by country, with higher premiums for countries perceived as riskier.

Measuring Risk

Variance and Standard Deviation

Variance and standard deviation are statistical measures used to quantify the volatility of returns.

• Variance: Average value of squared deviations from the mean. $\text{Variance} = \frac{\sum (R_i - \bar{R})^2}{n}$

• Standard Deviation: Square root of variance. $\text{Standard deviation} = \sqrt{\text{Variance}}$

Example: For a set of returns, calculate the deviation from the mean, square each deviation, average them for variance, and take the square root for standard deviation.

Risk and Diversification

Portfolio Rate of Return

The portfolio rate of return is the weighted average of the returns of the assets in the portfolio.

• Formula: $\text{Portfolio return} = w_1 r_1 + w_2 r_2 + \ldots + w_n r_n$ where $w_i$ is the fraction invested in asset $i$, and $r_i$ is the rate of return of asset $i$.

Example: If a portfolio has 25% in gold and 75% in auto stocks, the portfolio return is calculated by multiplying each asset's return by its weight and summing the results.

Diversification

Diversification is a risk management strategy that mixes a wide variety of investments within a portfolio. The goal is to reduce unsystematic (firm-specific) risk.

• Specific Risk (Diversifiable Risk): Risk factors affecting only a particular firm.

• Market Risk (Systematic Risk): Economy-wide sources of risk that affect the overall market.

As the number of stocks in a portfolio increases, specific risk decreases, but market risk remains.

Sensitivity Analysis

Sensitivity analysis examines how changes in portfolio weights affect the expected return and risk under different economic scenarios (normal, boom, depression).

• By adjusting the proportion of assets, investors can analyze the impact on portfolio performance.

Thinking About Risk

Key Concepts

Understanding risk is essential for making informed investment decisions. Some risks may appear significant but are diversifiable, while market risks are macroeconomic and cannot be eliminated through diversification. Risk can be measured using statistical tools such as variance and standard deviation.

• Diversifiable risks can be reduced by holding a variety of assets.

• Systematic risks affect all investments and cannot be diversified away.

• Measuring risk helps investors make better decisions about asset allocation and expected returns.