The following table shows a sample of historical total returns of shares in two different companies A and B.
| Stock Returns | ||
| Total effective annual returns | ||
| Year | ##r_A## | ##r_B## |
| 2007 | 0.2 | 0.4 |
| 2008 | 0.04 | -0.2 |
| 2009 | -0.1 | -0.3 |
| 2010 | 0.18 | 0.5 |
What is the historical sample covariance (##\hat{\sigma}_{A,B}##) and correlation (##\rho_{A,B}##) of stock A and B's total effective annual returns?
To find the covariance there are 2 steps. First we need to find the historical average returns (##\bar{r}_A## and ##\bar{r}_B##), then use the covariance formula to find the historical sample covariance (##\hat{\sigma}_{A,B}##).
###\begin{aligned} \bar{r} &= \frac{r_{0 \rightarrow 1} + r_{1 \rightarrow 2} + r_{2 \rightarrow 3} + ... +r_{T-1 \rightarrow T}}{T} \\ \bar{r}_A &= \frac{0.2 + 0.04 + -0.1 + 0.18}{4} = 0.08 \\ \bar{r}_B &= \frac{0.4 + -0.2 + -0.3 + 0.5}{4} = 0.1 \\ \end{aligned} ###
### \begin{aligned} \hat{\sigma}_{A,B} =& \frac{ \displaystyle\sum\limits_{t=1}^T{\left( \left( r_{A,(t-1)\rightarrow t} - \bar{r}_A \right)\left( r_{B,(t-1)\rightarrow t} - \bar{r}_B \right) \right)} }{T-1}\\ =& \frac{\left( \begin{aligned} & {(0.2-0.08)(0.4-0.1)} + \\ &{(0.04-0.08)(-0.2-0.1)} + \\ &{(-0.1-0.08)(-0.3-0.1)} + \\ &{(0.18-0.08)(0.5-0.1)} \\ \end{aligned} \right)\\ }{4-1} \\ =& 0.053333333 \\ \end{aligned} ###
To find the correlation there are another 2 steps which is to calculate each stock's standard deviation, then convert the covariance to a correlation.
For stock A,
###\begin{aligned} \hat{\sigma}^2 &= \dfrac{ \displaystyle\sum\limits_{t=1}^T{\left( \left( r_{(t-1)\rightarrow t} - \bar{r} \right)^2 \right)} }{T-1}\\ \hat{\sigma}_A^2 &= \frac{\left( \begin{aligned} &{(0.2-0.08)^2} + \\ &{(0.04-0.08)^2} + \\ &{(-0.1-0.08)^2} + \\ &{(0.18-0.08)^2} \\ \end{aligned} \right)\\ }{4-1} \\ &= 0.019466667 \\ \hat{\sigma}_A &= \left( \sigma_A^2 \right)^{1/2} \\ &= \left( 0.019466667 \right)^{1/2} \\ &= 0.139522997 \\ \end{aligned} ###
For stock B,
### \begin{aligned} \hat{\sigma}_B^2 &= \frac{\left( \begin{aligned} &{(0.4-0.1)^2} + \\ &{(-0.2-0.1)^2} + \\ &{(-0.3-0.1)^2} + \\ &{(0.5-0.1)^2} \\ \end{aligned} \right)\\ }{4-1} \\ &= 0.1666666 \\ \hat{\sigma}_B &= \left( 0.1666666 \right)^{1/2} \\ &= 0.40824829 \\ \end{aligned} ###
For the correlation,
###\begin{aligned} \rho_{A,B} =& \frac{ \hat{\sigma}_{A,B} }{\hat{\sigma}_{A}.\hat{\sigma}_{B}} \\ =& \frac{ 0.053333333 }{ 0.139522997 \times 0.40824829 } \\ =& 0.936329 \\ \end{aligned} ###
Stock A and B's returns have a correlation of 0.3. Which statement is NOT correct?
There will always be at least some diversification when two assets are combined in a portfolio for all correlations less than one (##\rho=1## is also known as perfectly positive correlation). If the correlation between asset A and B is less than one, then sometimes when the price of asset A falls, the price of asset B will rise, causing the portfolio's risk to be lower than asset A on its own, which is diversification.
| Portfolio Details | ||||||
| Stock | Expected return |
Standard deviation |
Correlation | Dollars invested |
||
| A | 0.1 | 0.4 | 0.5 | 60 | ||
| B | 0.2 | 0.6 | 140 | |||
What is the expected return of the above portfolio?
Using the portfolio return equation,
###\begin{aligned} \mu_p &= \mu_A.x_A + \mu_B.x_B \\ &= 0.1 \times \frac{60}{60+140} + 0.2 \times \frac{140}{60+140} \\ &= 0.17 \\ \end{aligned} ###
| Portfolio Details | ||||||
| Stock | Expected return |
Standard deviation |
Covariance ##(\sigma_{A,B})## | Beta | Dollars invested |
|
| A | 0.2 | 0.4 | 0.12 | 0.5 | 40 | |
| B | 0.3 | 0.8 | 1.5 | 80 | ||
What is the standard deviation (not variance) of the above portfolio? Note that the stocks' covariance is given, not correlation.
Using the two-asset portfolio variance formula,
###\begin{aligned} \sigma_p^2 =& x_1^2.\sigma_1^2 + x_2^2.\sigma_2^2 + 2.x_1.x_2.\rho_{1,2}.\sigma_{1}.\sigma_{2} \\ =& x_1^2.\sigma_1^2 + x_2^2.\sigma_2^2 + 2.x_1.x_2.\sigma_{1,2} \\ =& \left( \frac{40}{40+80} \right) ^2 \times 0.4^2 + \left( \frac{80}{40+80} \right) ^2 \times 0.8^2 + \\ &2 \times \left( \frac{40}{40+80} \right) \times \left( \frac{80}{40+80} \right) \times 0.12 \\ =& 0.355555556 \\ \sigma_p =& (0.355555556)^{1/2} \\ =& 0.596284794 \\ \end{aligned} ###
Question 282 expected and historical returns, income and capital returns
You're the boss of an investment bank's equities research team. Your five analysts are each trying to find the expected total return over the next year of shares in a mining company. The mining firm:
- Is regarded as a mature company since it's quite stable in size and was floated around 30 years ago. It is not a high-growth company;
- Share price is very sensitive to changes in the price of the market portfolio, economic growth, the exchange rate and commodities prices. Due to this, its standard deviation of total returns is much higher than that of the market index;
- Experienced tough times in the last 10 years due to unexpected falls in commodity prices.
- Shares are traded in an active liquid market.
- The analysts' source data is correct and true, but their inferences might be wrong;
- All returns and yields are given as effective annual nominal rates.
Alice is wrong because a risky mining company that's highly sensitive to changes in the market portfolio should have a higher expected return than risk-free government bonds.
Bob is wrong for the same reason as above. A risky stock which is highly correlated and sensitive to the market portfolio should have a higher expected return than risk-free government bonds. Also, while the average historical return is often a good estimate of the future expected return, in this case commodity prices unexpectedly fell over the past 10 years leading to lower than expected historical stock returns. But unexpected commodity future price falls by definition can't be expected, and are just as likely as unexpected rises, so using the stock's historical average return is not a good proxy for the expected future return which we expect should be higher.
Cate is also wrong since the expected return should be at least as high as the risk free rate. What's more, she has found the average price growth or 'capital return'. It is only a part of the total return which also includes the dividend return.
Dave is wrong for the same reason as Cate. He has found the market index's average price growth which is only the capital return, it is not the total total return since it excludes dividends. If he used the accumulation index (rather than the price index) which re-invests dividends then the historical average return would be higher. Another problem is that the mining stock is highly sensitive to the market index, meaning it is more risky, so it should have a higher expected total return than the market index. But at least Dave forecasts a return higher than the risk free rate.
Eve's answer is the most plausible. She is the only person who has tried to find the expected future return rather than the historical average return. Since she used the dividend discount model (DDM) to find the expected return, her forecast total expected return depends on all of the DDM's assumptions such as a constant perpetual growth rate of dividends and a constant level of risk. But her inputs into the model appear reasonable. Using next year's forecast dividend is correct. Since the firm is mature and is not fast-growing it is suited to DDM valuation. Using the inflation rate as the dividend growth rate, which is also the capital return, is a plausible assumption.
Eve should check that the forecast dividend is not a one-off dividend higher than the others and that it is expected to be paid every year into the future. Constructing pro-forma income statements and balance sheets 10 years into the future would also be beneficial since she could see what level of capital expenditure on new assets would be required to sustain the 3% growth rate and if there will be any cash flow shortfalls that will make the 3% growth rate unsustainable. She could also cross-check the expected return predicted by the DDM with the expected return given by the capital asset pricing model (CAPM) to see if they are approximately the same which would be re-assuring. Doing all of the above for other similar mining firms would also give an idea about whether the valuation and expected return of this mining firm is reasonable and consistent with its peers.
Return is the change in price and stock prices can change considerably. A mutual fund combines a number of stocks in a portfolio and sometimes when one stock's price rises, another's falls, leading to less overall portfolio price changes. This is called diversification and it leads to lower variation prices and therefore returns, meaning that a mutual fund's portfolio return will vary less than a single stock's return, so the mutual fund return is safer.
For the mathematically inclined, the total return of a single stock over one year is:
### r = \frac{p_1 - p_0 + d_1}{p_0} = \frac{p_1 + d_1}{p_0} - 1 ###Where ##p_0## is price now, ##p_1## is price in one year and ##d_1## is the dividend in one year.
The total return of a portfolio (##r_p##) of two stocks A and B over one year is the weighted average of their returns:
###\begin{aligned} r_{p} &= r_{A} . \frac{p_{A,0}}{p_{A,0}+p_{B,0}} + r_{B} . \frac{p_{B,0}}{p_{A,0}+p_{B,0}} \\ \end{aligned}###
Because portfolio return is an average it is clear how large negative returns of one stock can be dampened if the other stocks' returns are positive or close to zero.
Commentary
This question was used in the '2004 Health and Retirement Survey' of Americans over the age of 50. The survey results were as follows:
- 52.3% of respondents answered it correctly,
- 13.2% were incorrect,
- 33.7% answered "don't know" and
- 0.9% refused to answer.
This question tests knowledge of diversification and was used in the research paper 'Financial Literacy and Planning: Implications for Retirement Wellbeing' by Annamaria Lusardi and Olivia S. Mitchell in 2011.
Which of the following statements about short-selling is NOT true?
An investor who short-sells an asset is 'short' that asset, not long.
Question 558 portfolio weights, portfolio return, short selling
An investor wants to make a portfolio of two stocks A and B with a target expected portfolio return of 16% pa.
- Stock A has an expected return of 8% pa.
- Stock B has an expected return of 12% pa.
What portfolio weights should the investor have in stocks A and B respectively?
Use the portfolio return equation and the 'sum of the weights equals one' equation to solve simultaneously for the weights ##x_A## and ##x_B##.
###\mu_p = \mu_A.x_A + \mu_B.x_B### ###0.16 = 0.08 x_A + 0.12 x_B### ###0.16= 0.08 x_A + 0.12 (1-x_A)### ###0.16 = 0.08 x_A + 0.12 -0.12 x_A### ###-0.04 x_A = 0.04### ###\begin{aligned} x_A &= -0.04 / 0.04 \\ &= -1 = -100\% \\ \end{aligned}### ###\begin{aligned} x_B &= 1 - x_A \\ &= 1 - -1 \\ &= 2 = 200\% \\ \end{aligned}###Therefore the investor should short stock A and use the proceeds and his own money to long stock B. He should buy the stocks in the ratio -1:2, so for every $100 he intends to invest he should (short) sell $100 of stock A and buy $200 of stock B.
What is the covariance of a variable X with itself?
The cov(X, X) or ##\sigma_{X,X}## equals:
The covariance of a variable with itself is the variable's variance.
###\begin{aligned} cov(X,Y) &= \dfrac{ \displaystyle\sum\limits_{t=1}^T{\left( (X_{t} - \bar{X}).(Y_{t} - \bar{Y}) \right)} }{T-1}\\ cov(X,X) &= \dfrac{ \displaystyle\sum\limits_{t=1}^T{\left( (X_{t} - \bar{X}).(X_{t} - \bar{X}) \right)} }{T-1}\\ &= \dfrac{ \displaystyle\sum\limits_{t=1}^T{\left( (X_{t} - \bar{X})^2 \right)} }{T-1}\\ &= var(X) \\ \end{aligned}###What is the covariance of a variable X with a constant C?
The cov(X, C) or ##\sigma_{X,C}## equals:
A constant is a fixed number. Constants never change and therefore they have no standard deviation or variance. Therefore constants do not co-vary with a variable, so the covariance is zero. Mathematically, this is because the average of a constant is equal to the constant itself so ##C_{t} = \bar{C}##:
###\begin{aligned} cov(X,C) &= \dfrac{ \displaystyle\sum\limits_{t=1}^T{\left( (X_{t} - \bar{X}).(C_{t} - \bar{C}) \right)} }{T-1} \\ &= \dfrac{ \displaystyle\sum\limits_{t=1}^T{\left( (X_{t} - \bar{X}).(\bar{C} - \bar{C}) \right)} }{T-1} \\ &= \dfrac{ \displaystyle\sum\limits_{t=1}^T{\left( (X_{t} - \bar{X}) \times 0 \right)} }{T-1} \\ &= 0 \\ \end{aligned}###The standard deviation and variance of a stock's annual returns are calculated over a number of years. The units of the returns are percent per annum ##(\% pa)##.
What are the units of the standard deviation ##(\sigma)## and variance ##(\sigma^2)## of returns respectively?
Hint: Visit Wikipedia to understand the difference between percentage points ##(\text{pp})## and percent ##(\%)##.
The variance of returns is measured in percentage points per annum all squared ##\left( (\text{pp pa})^2 \right)## and the standard deviation is measured in percentage points per annum ##(\text{pp pa})##. This is why many people prefer to be shown standard deviations rather than variances, because the units are the same as the variable being measured, they're not squared.
The way engineers and scientists figure out the units of a quantity is to examine the units of the inputs into the formula that generates the quantity. The units of each random return ##(r_t)## and the average return ##(\bar{r})## are in percent per annum. The difference between two percentages is measured in percentage points ##(\text{pp})##. The number of years ##(T)## is obviously measured in years (annums). But the number of years is actually used to calculate the probability ##1/T## which is the probability of finding one particular year in the population of T years. This probability is a pure number with no units since it's years divided by years.
Since the only operation that changes the units of the annual returns (in percent per annum) is the power of 2, the units of variance turn into percent per annum all squared:
###\begin{aligned} \text{variance} &= \sigma^2 = \dfrac{ \displaystyle\sum\limits_{t=1}^T{\left( \left( r_t - \bar{r} \right)^2 \right)} }{T} = \dfrac{1}{T}.\displaystyle\sum\limits_{t=1}^T{\left( \left( r_t - \bar{r} \right)^2 \right)}\\ \end{aligned}### ###\begin{aligned} \text{units of variance} &= \dfrac{1 \text{ annum}}{T \text{ annum}}. \displaystyle\sum\limits_{t=1}^T{\left( \left( \%pa - \%pa \right)^2 \right)} \\ &= \left( \%pa - \%pa \right)^2 \\ &= \left( \text{pp pa} \right)^2 \\ \end{aligned}###Let the variance of returns for a share per month be ##\sigma_\text{monthly}^2##.
What is the formula for the variance of the share's returns per year ##(\sigma_\text{yearly}^2)##?
Assume that returns are independently and identically distributed (iid) so they have zero auto correlation, meaning that if the return was higher than average today, it does not indicate that the return tomorrow will be higher or lower than average.
Annual variance is equal to monthly variance multiplied by twelve. This will occur so long as there is no auto-correlation of monthly returns.
There are formulas to find the variance of a sum of variables. Let the first month's return be ##r_1##, the second ##r_2## and so on. Below are the equations for the variance of returns over a two month period, three month period, and four month period respectively:
###var(r_1+r_2) = var(r_1) + var(r_2) + 2.cov(r_1,r_2) ### ###\begin{aligned} var(r_1+r_2+r_3) &= var(r_1) + var(r_2) + var(r_3) + \\ &+2.cov(r_1,r_2) + 2.cov(r_1,r_3) + \\ &+2.cov(r_2,r_3) \\ \end{aligned}### ###\begin{aligned} var(r_1+r_2+r_3+r_4) &= var(r_1) + var(r_2) + var(r_3) + var(r_4) + \\ &+2.cov(r_1,r_2) + 2.cov(r_1,r_3) + 2.cov(r_1,r_4) + \\ &+2.cov(r_2,r_3) + 2.cov(r_2,r_4) + \\ &+2.cov(r_3,r_4) \\ \end{aligned}###The variance of the sum of twelve months of returns would equal the variance of each month's returns plus the two times the covariance of every month's return with every other month's return.
###\begin{aligned} var(r_\text{year}) &= var(r_\text{month 1}+r_\text{month 2}+...+r_\text{month 12}) \\ &= var(r_\text{month 1})+var(r_\text{month 2})+...+var(r_\text{month 12}) + \\ &+ 2.cov(r_\text{month 1}, r_\text{month 2})+...+ 2.cov(r_\text{month 11}, r_\text{month 12}))\\ \end{aligned}###Auto-correlation is zero, which means that if returns are high in one month, they will not necessarily be high in the next month. There is no correlation between returns across time. Therefore the covariance between the months' returns are zero, so the annual variance is simply the sum of the monthly variances.
###\begin{aligned} var(r_\text{year}) &= var(r_\text{month 1})+var(r_\text{month 2})+...+var(r_\text{month 12}) \\ \end{aligned}###Since the monthly variances are all the same, the annual variance is twelve times the monthly variance.
###\begin{aligned} var(r_\text{yearly}) &= 12.var(r_\text{monthly}) \\ \end{aligned}###