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 $\beta = 1.23$.
  • Use your calculator.

Historical Returns across Asset Classes

• See Figure 12.4
• See Figure 12.10
• Risk-Return Tradeoff
  • Investors demand higher return for holding riskier assets.
  • Volatility (Std. Deviation) is a measure of this risk.
• Risk Premium (Excess Return)
  • 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.