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.