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.

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.

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.

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.

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.

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.\sigma_{1,2} \\ =& x_1^2.\sigma_1^2 + x_2^2.\sigma_2^2 + 2.x_1.x_2.\rho_{1,2}.\sigma_{1}.\sigma_{2} \\ =& \left( \frac{60}{60+140} \right) ^2 \times 0.4^2 + \left( \frac{140}{60+140} \right) ^2 \times 0.6^2 + \\ & 2 \times \left( \frac{60}{60+140} \right) \times \left( \frac{140}{60+140} \right) \times 0.5 \times 0.4 \times 0.6 \\ =& 0.2412 \\ \sigma_p =& (0.2412)^{1/2} \\ =& 0.491121166 \\ \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.

Corporate debt has the lowest risk so it should have the greatest weight for a person trying to minimise risk.

While real estate is more risky than corporate debt, having at least a small weight in real estate can reduce portfolio risk since the correlations between real estate and corporate debt is less than one.

For the same reasons, having a small weight in equity can also reduce portfolio risk. It may seem as though equity provides no additional advantage over real estate in terms of diversification since they both have the same correlation with corporate debt. And equity has high risk. But since equity and real estate have a correlation less than one, the equity not only helps diversify the debt, but also the real estate as well, reducing portfolio risk.

Being able to invest in 3 assets rather than 2 can only lead to improvements in diversification. This can be portrayed graphically on the expected standard deviation and return axes by the Markowitz bullet shifting to the left as more assets are made available for investment.

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.