Ch. 12: Some Lessons from Capital Market History

Calculating Returns

• Dollar Returns
• $D_{t+1} + \Delta P_{t+1} = D_{t+1} + P_{t+1} - P_t$
• How do we compare investments that differ based on initial investment?
• Percentage Returns
• Scale dollar returns by initial investment
• $(D_{t+1} + \Delta P_{t+1})/P_t = (D_{t+1} + P_{t+1})/P_t - 1$ = dividend yield + capital gains yield

Statistics Review

• Average: $\bar{X} = \frac{1}{T} \sum^T_{t=1} X_t$
• Variance: $\sigma^2_X = \frac{1}{T-1} \sum^T_{t=1} (X_t-\bar{X})^2$
• Std. Deviation: $\sigma_X = \sqrt{\sigma^2_X} = \sqrt{\frac{1}{T-1} \sum^T_{t=1} (X_t-\bar{X})^2}$
• Z-Score
• $Z = \frac{X-\bar{X}}{\sigma_X}$
• How many standard deviations away from the mean is our observation?
• Probabilty that $|Z| \le n$ for Normal Distribution
• $n = 1$: 68%
• $n = 2$: 95%
• $n = 3$: 99%
• Covariance: $\sigma_{XY} = \frac{1}{T-1} \sum^T_{t=1} (X_t-\bar{X})(Y_t-\bar{Y})$
• Correlation: $Corr(X,Y) = \frac{\sigma_{XY}}{\sigma_X \sigma_Y}$
• Linear Regression
• Interested in $Y_t = \alpha + \beta X_t$
• Pick $\alpha$ and $\beta$ such that we minimize $\sum^T_{t=1} (Y_t-\alpha-\beta X_t)^2$
• $\alpha = \bar{Y} - \beta \bar{X}$ and $\beta = \frac{\sigma_{XY}}{\sigma^2_X}$
• Example
• Consider two stocks. Stock X has returns of 15%, 9%, 6%, and 12%. Stock Y has returns of 18%, 7%, 8%, and 14%.
• Calculate the averages, standard deviations, correlation, and beta.
• $\bar{X} = 10.5%$, $\bar{Y} = 11.75%$, $\sigma_X = 3.87%$, $\sigma_Y = 5.19%$, $Corr(X,Y) = 0.92$, and $\beta = 1.23$.

Historical Returns across Asset Classes

• See Figure 12.4
• See Figure 12.10
• Investors demand higher return for holding riskier assets.
• Volatility (Std. Deviation) is a measure of this risk.
• Useful to look at the return on a risky asset minus the return on a risk-free asset
• Use return on T-bills as risk-free rate of return

Arithmetic vs. Geometric Returns

• Example
• Consider the following case. You buy a stock for $\$$100. The first year it falls in value to \$$50. Then at the end of the second year, you sell the stock for$\100. No dividends were paid.
• What is the average return?
• Arithmetic Average Return
• What was you return in an average year over a particular period?
• See the above formula.
• Geometric Average Return
• What was your average compound return per year over a particular period?
• $PV = FV(1+R)^t$
• $[(1+R_1) \times (1+R_2) \times ... \times (1+R_T)]^{1/T} - 1$
• Geometric Average $\le$ Arithmetic Average because of compounding. Geometric Average is roughly equal to the Arithmetic Average minus one half the variance.
• Example
• Suppose a stock has the following returns over three years: 5%, -3%, and 12%. What are the arithmetic and geometric averages of this return series?
• Arithmetic Average = 4.67%; Geometric Average = 4.49%
• Which should we use?
• If you know the true arithmetic average, then use it.
• But we typically have only an estimate of the arithmetic average.
• Rule of thumb
• <20 years: arithmetic
• >40 years: geometric
• 20-40 years: split the difference between the two

Capital Market Efficiency

• Can an investor earn abnormal or excess returns?
• Are stocks fairly priced?
• How quickly and accurately do markets incorporate new information about securities?
• In an efficient capital market, current market prices fully reflect available information.
• See Figure 12.14
• Efficient Market Hypothesis
• Strong Form: prices reflect all information (public and private)
• Semi-strong Form: prices reflect all public information
• Weak Form: prices reflect all past market information (such as price and volume traded)
• Keep in mind
• Efficient markets do not mean that you can't make money.
• Efficient markets do mean that, on average, you will earn a return that is appropriate for the risk undertaken and there is not a bias in prices that can be exploited to earn excess returns.