Answer: Good choice. You earned $10. Poor choice. You lost $10.

###\begin{aligned} r_\text{total} =& r_\text{capital} + r_\text{income} \\ =& \frac{P_1 - P_0}{P_0} + \frac{C_1}{P_0} \\ =& \frac{16 - 20}{20} + \frac{3}{20} \\ =& \frac{-4}{20} + \frac{3}{20} \\ =& -0.2 + 0.15 \\ \end{aligned}### So the capital return is -0.2 and the income return is 0.15. The total return is the sum, so: ### r_\text{total} = -0.05 ###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

Standard deviation measures the volatility of a variable above and below its mean.

The higher the asset’s standard deviation of returns, the less predictable are the asset’s returns and future prices.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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} ###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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} ###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

As correlation rises, there's less diversification, so variance (risk) rises, so long as both stocks have positive weights so neither are sold short.

Mathematically, this can be seen in the two-stock portfolio variance formula:

###\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###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

When the correlation of returns between assets A and B are perfectly negative (-1), every time asset A's return rises, asset B's return falls, and vice versa. They will always move in opposite directions in the same ratio. They always tend to cancel each other out.

Therefore a risk-free portfolio of A and B is possible with a certain weighting in each. A risk free portfolio has all risk diversified away which is the ideal situation.

In common sense terms, if the correlation is negative one then when stock A falls, stock B will rise, which means you've overall lost nothing, the risks have offset and this is ideal. What you don't want is a correlation of positive one since that means that when stock A falls, stock B also falls, causing losses in your portfolio.

Mathematically, when the correlation between two stocks is negative, the portfolio variance (##\sigma_p^2## and standard deviation ##\sigma_p##) will be lower since the last term ##(2.x_A.x_B.\rho_{A,B}.\sigma_{A}.\sigma_{B})## will be negative when the correlation (##\rho_{A,B}##) is negative and the weights (##x_A## and ##x_B##) are positive:

###\begin{aligned} \sigma_p^2 =& x_A^2.\sigma_A^2 + x_B^2.\sigma_B^2 + 2.x_A.x_B.\rho_{A,B}.\sigma_{A}.\sigma_{B} \\ \end{aligned} ###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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} ###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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}###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The 90% confidence interval bounds correspond to the mean return plus and minus the standard deviation of returns multiplied by the 95% one-tail Z-statistic:

###\begin{aligned} & \text{90PercentConfidenceIntervalUpperBound} \\ &=\text{95thPercentile} \\ &= \mu + \text{Z}_\text{95% one tail} \times \sigma \\ &= 0.1 + 1.645 \times 0.2 \\ &= 0.429 \\ &= 42.9\%\text{ pa} \\ \end{aligned}### ###\begin{aligned} & \text{90PercentConfidenceIntervalLowerBound} \\ &=\text{5thPercentile} \\ &= \mu - \text{Z}_\text{95% one tail} \times \sigma \\ &= 0.1 - 1.645 \times 0.2 \\ &= -0.229 \\ &= -22.9\%\text{ pa} \\ \end{aligned}###

Note that confidence intervals are measured between the two-tails of the normal distribution, so we need the one-tail 95% Z-statistic for the 5% tail on the left and the 5% tail on the right of the distribution, with the 90% confidence interval in the middle, adding to 100%.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The 95% confidence interval bounds correspond to the mean return plus and minus the standard deviation of returns multiplied by the 97.5% one-tail Z-statistic:

###\begin{aligned} & \text{95PercentConfidenceIntervalUpperBound} \\ &=\text{97.5thPercentile} \\ &= \text{Mean} + \text{Z}_\text{97.5% one tail} \times \text{StandardDeviation} \\ &= \mu + \text{Z}_\text{97.5% one tail} \times \sigma \\ &= 0.1 + 1.96 \times 0.2 \\ &= 0.492 \\ &= 49.2\%\text{ pa} \\ \end{aligned}### ###\begin{aligned} & \text{95PercentConfidenceIntervalLowerBound} \\ &=\text{2.5thPercentile} \\ &= \text{Mean} - \text{Z}_\text{97.5% one tail} \times \text{StandardDeviation} \\ &= \mu - \text{Z}_\text{97.5% one tail} \times \sigma \\ &= 0.1 - 1.96 \times 0.2 \\ &= -0.292 \\ &= -29.2\%\text{ pa} \\ \end{aligned}###

Note that confidence intervals are measured between the two-tails of the normal distribution, so we need the one-tail 97.5% Z-statistic for the 2.5% tail on the left and the 2.5% tail on the right of the distribution, with the 95% confidence interval in the middle, adding to 100%.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The Z-stat corresponding to a two-tail 90% probability is 1.645 since there will be 5% in the left tail and 5% in the right tail. In other words, plus and minus 1.645 standard deviations above and below the mean captures 90% of the distribution in the middle.

The standard deviation of returns can be calculated based on the 90% confidence interval between -15% and 35% which is the mean of 10% plus or minus 25 percentage points. Make a formula for the upper bound of the confidence interval and solve for the standard deviation sigma ##(\sigma)##:

###\begin{aligned} &\text{90PercentConfidenceIntervalUpperBound} = \\ &\text{95thPercentile} = \mu + \text{Z}_\text{95% one tail} . \sigma \\ &0.35 = 0.1 + 1.645 \times \sigma \\ \end{aligned}### ###\begin{aligned} \sigma &= \dfrac{0.35 - 0.1}{1.645} \\ &= 0.15197568389 \\ &= 15.2\text{ pp pa} \\ \end{aligned}###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The covariance between variables X and Y is represented by ##\sigma_{X,Y}##.

Unfortunately, statisticians commonly use lower case sigma ##(\sigma)## to represent covariance and standard deviation. The only way to distinguish between them is to look at the subscript. The covariance will have a comma between the variables listed in the subscript but the standard deviation won't. Standard deviation is of a single variable such as X ##(\sigma_X)##, while covariance is between two variables such as X and Y ##(\sigma_{X,Y})##.

To avoid the confusion caused by the clashing symbol notation, some people prefer to use the function notation ##cov(X,Y)## for ##\sigma_{X,Y}## and ##sd(X)## for ##\sigma_{X}##.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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}###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The correlation of a variable with itself is positive one, which is perfect correlation.

Since the covariance of a variable with itself is its variance, and the correlation of two variables is the covariance divided by the product of each variables' standard deviation, and standard deviation squared is variance, then:

###\begin{aligned} corr(X,Y) &= \dfrac{cov(X,Y)}{sd(X).sd(Y)} \\ corr(X,X) &= \dfrac{cov(X,X)}{sd(X).sd(X)} \\ &= \dfrac{var(X)}{(sd(X))^2} \\ &= \dfrac{var(X)}{var(X)} \\ &= 1 \\ \end{aligned}###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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}###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The correlation of a variable with a constant is undefined due to a divide-by-zero problem.

Since the covariance of a variable with a constant is zero, and the correlation of two variables is the covariance divided by the product of each variables' standard deviation, and the standard deviation of a constant is zero, then:

###\begin{aligned} corr(X,C) &= \dfrac{cov(X,C)}{sd(X).sd(C)} \\ &= \dfrac{0}{sd(X) \times 0} \\ &= \dfrac{0}{0} \\ &= \text{mathematically undefined} \\ \end{aligned}###