Fight Finance

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

According to the CAPM, high systematic risk investments deserve high returns. This is because systematic risk is the only risk that can't be diversified away, it affects all systematically risky assets at the same time. Therefore it's the more frightening risk for investors and those who are willing to put up with it deserve higher total returns.

Note that systematic risk is also known as market risk or un-diversifiable risk.

Beta ##(\beta)## measures systematic risk. The higher the beta, the higher the required total return. This is reflected in the CAPM's security market line (SML) formula:

###r_i = r_f + \beta_i (r_m - r_f) ###

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

Starting with the CAPM's SML equation we can find the required return from the stock's beta, the market return and the risk free rate:

###\begin{aligned} \mu_E &= r_f + \beta_E(\mu_m-r_f) \\ &= 0.03 + 1.2(0.07-0.03) \\ &= 0.078 \\ \end{aligned} ###

Now that we have the required return on equity we can discount the dividends on equity using the perpetuity with growth formula, also known as the Gordon growth model.

###\begin{aligned} P_0 =& \frac{C_1}{\mu_{E} - g} \\ =& \frac{20}{0.078-0.015} \\ =& 317.46031746 \end{aligned}###

Answer: Good choice. You earned $10. Poor choice. You lost $10. ###\begin{aligned} \mu_i &= r_f + \beta_i(\mu_m-r_f) \\ &= 0.05 + 1.5 \times (0.1-0.05) \\ &= 0.125 \\ \end{aligned} ###

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

Over the last 5 minutes, the return on the risk free rate would be close to zero ##(r_{f\text{ 5 min}} \approx 0)## since it's only 5% per year. The historical market return ##(r_{m\text{ 5 min}})## over the last 5 minutes was -1%. Note that in the CAPM equation below all returns are effective 5 minute historical returns. Substituting this into the CAPM equation: ###\begin{aligned} r_{i\text{ 5 min}} &= r_{f\text{ 5 min}} + \beta_i(r_{m\text{ 5 min}}-r_{f\text{ 5 min}}) \\ &= 0 + 1.5 \times (-0.01-0) \\ &= -0.015 \\ \end{aligned} ###

Discussion of why the 5 minute risk free rate is close to zero

To find the exact 5 minute risk free rate and show that it is truly close to zero, let's convert this 5% effective annual risk free rate into an effective 5 minute risk free rate. Assume that there are 365 days per year, 24 hours per day, 60 minutes per hour and therefore 12 (=60/5) five minute periods per hour.

###(1+r_{f\text{ 5 min}})^\text{number of 5 min periods in a year} = (1+r_{f\text{ annual}})^1### ###(1+r_{f\text{ 5 min}})^{365 \times 24 \times 60 / 5} = (1+r_{f\text{ annual}})### ###\begin{aligned} r_{f\text{ 5 min}} &= (1+r_{f\text{ annual}})^{1/(365 \times 24 \times 60 / 5)}-1 \\ &= (1+0.05)^{1/(365 \times 24 \times 60 / 5)}-1 \\ &= 0.000000464137895 \text{ pa} \\ &= 0.0000464137895 \text{% pa} \\ &\approx 0\text{% pa} \end{aligned}###

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

Over the last year, the historical effective return on the risk free rate was 5% ##(r_f = 0.05)##. The historical market return ##(r_m)## over the last year was -1% ##(r_m = -0.01)##. Note that in the CAPM equation below all returns are effective annual historical returns. Substituting this into the CAPM equation: ###\begin{aligned} r_i &= r_f + \beta_i(r_m-r_f) \\ &= 0.05 + 1.5 \times (-0.01-0.05) \\ &= -0.04 \\ \end{aligned} ###

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

According to the CAPM, a stock's required total return depends on its systematic variance or beta. For some stock ##i##:

###r_\text{total i} = r_f + \beta_i (r_m - r_f)###

A stock's total variance is the sum of the systematic and diversifiable variances. For some stock ##i##:

###\text{TotalVariance} = \text{SystematicVariance} + \text{DiversifiableVariance}### ###\begin{aligned} \sigma_\text{total i}^2 &= \sigma_\text{systematic i}^2 + \sigma_{\text{diversifiable i}}^2 \\ &= \beta_i^2\sigma_\text{m}^2 + \sigma_{\epsilon\text{ i}}^2 \\ \end{aligned}\\###

Although the total return and total variance have the same title 'total', they are not closely related since total required returns depend on systematic variance only. This is because investors ignore diversifiable variance since it's easily reduced to zero by holding a variety of stocks.

The sum of the capital and dividend returns will equal the stock's total return.

###\begin{aligned} r_\text{total} &= r_\text{capital} + r_\text{dividend} \\ &= \dfrac{\text{Price}_\text{end} - \text{Price}_\text{start}}{\text{Price}_\text{start}} + \dfrac{\text{Dividend}_\text{end}}{\text{Price}_\text{start}} \\ \end{aligned}###

The split between capital and dividend yields will depend on the company's payout policy. If the company chooses to:

• Not to pay any dividends and instead re-invests, the dividend yield will be zero and the capital yield will be high and equal to the total return.
• Pay out all of its cash flows as dividends, the dividend yield will be high and equal to the total return and the capital yield will be zero since there's no re-investment back into the company.

See Question 455 for a discussion of how dividend policy and re-investment affects capital returns.

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

Over the last 5 minutes, the return on the risk free rate would be close to zero ##(r_{f\text{ 5 min}} \approx 0)## since it's only 5% per year. The historical market return ##(r_{m\text{ 5 min}})## over the last 5 minutes was -3%. Note that in the CAPM equation below all returns are effective 5 minute historical returns. Substituting this into the CAPM equation: ###\begin{aligned} r_{i\text{ 5 min}} &= r_{f\text{ 5 min}} + \beta_i(r_{m\text{ 5 min}}-r_{f\text{ 5 min}}) \\ &= 0 + 0.5 \times (-0.03-0) \\ &= -0.015 \\ \end{aligned} ###

Discussion of why the 5 minute risk free rate is close to zero

To find the exact 5 minute risk free rate and show that it is truly close to zero, let's convert this 5% effective annual risk free rate into an effective 5 minute risk free rate. Assume that there are 365 days per year, 24 hours per day and 60 minutes per hour. Expressing a 5-minute period as a fraction of a year:

###\begin{aligned} \text{5 minutes } &= \frac{1}{365} \times \frac{1}{24} \times \frac{5}{60} \text{ years} \\ &= \frac{5}{365 \times 24 \times 60} \text{ years} \\ \end{aligned}### ###\begin{aligned} r_{f\text{ 5 min}} &= (1+r_{f\text{ annual}})^{5/(365 \times 24 \times 60)}-1 \\ &= (1+0.05)^{5/(365 \times 24 \times 60)}-1 \\ &= 0.000000278483 \text{ pa} \\ &= 0.0000278483 \text{% pa} \\ &\approx 0\text{% pa} \end{aligned}###

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

No explanation provided.

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

All statements are true except the last phrase in answer (c). The market has a beta of one by definition, and so does stock B. Therefore they have the same level of systematic risk.

But total risk is comprised of systematic and diversifiable risk. Let a stock be represented by 'i'. When risk is measured using variance, then the following equation breaks any stock i's total variance into its systematic and idiosyncratic variances:

###\begin{aligned} \sigma_\text{i, total}^2 &= \beta_i^2.\sigma_m^2 + \sigma_\text{i, diversifiable}^2 \\ &= \sigma_\text{i, systematic}^2 + \sigma_\text{i, diversifiable}^2 \\ \end{aligned} ###

The market portfolio has no diversifiable variance since it is fully diversified. Stock B, on the other hand, is likely to have significant diversifiable variance since it is a single stock. Therefore stock B is likely to have higher total variance than the market portfolio since it has more diversifiable variance.

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

According to the CAPM:

• beta is measured as:

  ###\begin{aligned} \beta_{i} &= \frac{\sigma_{i,m}}{\sigma_m^2} = \frac{cov(r_i, r_m)}{var(r_m)} \\ \end{aligned} ###

• total variance is the sum of systematic and idiosyncratic variances:

  ###\begin{aligned} \sigma_\text{i,total}^2 &= \beta_i^2.\sigma_m^2 + \sigma_\text{i,diversifiable}^2 \\ &= \sigma_\text{i,systematic}^2 + \sigma_\text{i,diversifiable}^2 \\ \end{aligned} ###

The risk free rate ##r_f## does not vary, it is a constant, therefore it has zero systematic and idiosyncratic variance and its beta is zero:

###\begin{aligned} \beta_{f} &= \frac{\sigma_{f,m}}{\sigma_m^2} = \frac{0}{\sigma_m^2} = 0 \\ \end{aligned} ###

The market portfolio is a portfolio of risky assets so it has a variance, but by definition its beta is one since the covariance of an asset with itself is equal to its variance:

###\begin{aligned} \beta_{m} &= \frac{\sigma_{m,m}}{\sigma_m^2} = \frac{\sigma_m^2}{\sigma_m^2} = 1 \\ \end{aligned} ###

The market portfolio has no idiosyncratic variance because it is fully diversified, and this makes sense since, using the total variance equation:

###\begin{aligned} \sigma_\text{i,total}^2 &= \beta_i^2.\sigma_\text{m,total}^2 + \sigma_\text{i,diversifiable}^2 \\ \sigma_\text{m,total}^2 &= \beta_m^2.\sigma_\text{m,total}^2 + \sigma_\text{m, diversifiable}^2 \\ &= 1^2.\sigma_\text{m,total}^2 + \sigma_\text{m,diversifiable}^2 \\ &= \sigma_\text{m,total}^2 + \sigma_\text{m,diversifiable}^2 \\ \sigma_\text{m,diversifiable}^2 &= \sigma_\text{m,total}^2 - \sigma_\text{m,total}^2 \\ &= 0 \\ \end{aligned} ###

Therefore the risk free rate has no risk at all so it has zero systematic and diversifiable (or idiosyncratic) risk, and the market portfolio has zero idiosyncratic risk.

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

Other names for diversifiable risk include:

• idiosyncratic risk
• firm-specific risk
• unique risk
• residual risk
• non-market risk
• non-systematic risk

Other names for systematic risk include:

• market risk
• undiversifiable risk
• beta risk

According to the CAPM, total risk is comprised of systematic and diversifiable risk. Mathematically, for some asset 'i':

###\text{total variance} = \text{systematic variance} + \text{idiosyncratic variance}### ###\begin{aligned} \sigma_\text{total, i}^2 &= \sigma_\text{systematic, i}^2 + \sigma_{\text{idiosyncratic, i}}^2 \\ &= \beta_i^2\sigma_\text{m}^2 + \sigma_{\epsilon\text{, i}}^2 \\ \end{aligned}\\###

Idiosyncratic risk can be reduced by diversification but systematic risk can not.

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

Diversifiable events affect only a specific company or industry or area, such as a firm's poor earnings announcement. Systematic events affect a whole country or the world and can't be avoided.

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

Beta (##\beta##) is a measure of systematic risk.

Since the value of the firm's assets must equal the value of its debt and equity,###V=D+E### The firm's assets' beta must equal the weighted average beta of the debt and equity, so: ###\beta_V = \frac{D}{V}\beta_D + \frac{E}{V}\beta_E###

In this question, there is no change in the firm's assets. Therefore, all things remaining equal, there shouldn't be any change in the beta of the firm's assets (##\beta_V##).

Since the firm is issuing more debt by borrowing in the form of a loan or a bond, the amount of debt will increase (↑ D). The funds raised from the debt are being used to repurchase equity, so the amount of equity will decrease (↓ E).

Equity holders have a residual claim on the firm's assets, which means that they get paid last if the firm goes bankrupt. So shareholders get paid after debt holders. Therefore the increase in the amount of debt means that the equity holders are less likely to receive any money if the firm goes bankrupt. It also means that there will be a larger amount of interest payments that the firm must meet so there is a higher chance of going bankrupt. This means that equity must have more systematic risk, so it's beta will increase (↑##\beta_E##). This is the answer.

Also note that since there are more debt-holders, the larger amount of debt also has more systematic risk (↑##\beta_D##). This may appear impossible since how can the beta on debt and equity rise, while the beta on assets remain constant? But this is possible since the beta on debt is always less than the beta on equity (##\beta_D < \beta_E##), and while both betas rise, there is a larger weight on debt (↑##\frac{D}{V}##), and a lower weight on equity (↓##\frac{E}{V}##), so the asset beta stays the same.

To summarise: ###\begin{matrix} \cdot \\ \beta_V \\ \end{matrix} \begin{matrix} \phantom{1} \\ = \\ \end{matrix} \begin{matrix} \uparrow \\ \frac{D}{V} \\ \end{matrix} \begin{matrix} \uparrow \\ \beta_D \\ \end{matrix} \begin{matrix} \phantom{1} \\ + \\ \end{matrix} \begin{matrix} \downarrow \\ \frac{E}{V} \\ \end{matrix} \begin{matrix} \uparrow \\ \beta_E \\ \end{matrix}###

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

Statement (i) is true according to Miller and Modigliani's theories of capital structure and payout policy irrelevance.

Statement (ii) is true according to the CAPM because idiosyncratic risk is irrelevant since it can be diversified away when combined with a large portfolio of assets.

Statement (iii) is false according to the CAPM. Non-diversifiable systematic risk is rewarded with a higher return. This is the basis of the CAPM SML equation, expected return increases with beta:

###\begin{aligned} \mu_i &= r_f + \beta_i(\mu_m-r_f) \\ \end{aligned} ###

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

Systematic events affect a whole country and can't be avoided, such as an increase in a country's corporate tax rates. Diversifiable events affect only a specific company or industry or area, such as a firm's poor earnings announcement.

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

Beta (##\beta##) is a measure of systematic risk, along with variance (##\sigma^2##) and standard deviation (##\sigma##) .

Since the firm's assets (V) are funded by debt (D) and equity (E), the systematic risk of the firm's assets equals the weighted average beta of the debt and equity, so: ###\beta_V = \frac{D}{V}\beta_D + \frac{E}{V}\beta_E###

In this question, there is no change in the firm's assets. Therefore, all things remaining equal, there shouldn't be any change in the beta of the firm's assets (##\beta_V##).

Since the firm is issuing more equity (using a rights issue or private placement for example) and using the funds to repay debt (paying back the bond or loan-holders), the amount of equity will increase (↑ E) and the amount of debt will decrease (↓ D).

Equity holders have a residual claim on the firm's assets, which means that they get paid last if the firm goes bankrupt. So shareholders get paid after debt holders. Therefore the decrease in the amount of debt means that the equity holders are more likely to receive some payment if the firm goes bankrupt. It also means that there will be a smaller amount of interest payments that the firm must meet so there is a lower chance of going bankrupt. This means that equity must have less systematic risk, so it's beta will fall (↓##\beta_E##). This is the answer.

Also note that since there are less debt-holders, the smaller amount of debt also has less systematic risk (↓##\beta_D##). This may appear impossible since how can the beta on debt and equity fall, while the beta on assets remain constant? But this is possible since the beta on debt is always less than the beta on equity (##\beta_D < \beta_E##), and while both betas fall, there is a lower weight on debt (↓##\frac{D}{V}##), and a higher weight on equity (↑##\frac{E}{V}##), so the beta on assets stays the same.

To summarise: ###\overbrace{\beta_V}^{\cdot} = \overbrace{\frac{D}{V}}^{\downarrow} \overbrace{\beta_D}^{\downarrow} + \overbrace{\frac{E}{V}}^{\uparrow} \overbrace{\beta_E}^{\downarrow} ###

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

This question is related to the security market line (SML) in the capital asset pricing model (CAPM). Stocks that plot:

• Above the SML have a positive alpha (or positive abnormal return), are under-priced, and buying them is a positive NPV investment.
  Assets A and C are in this category.
• On the SML have a zero alpha (or zero abnormal return), are fairly-priced, and buying them is a zero NPV investment.
  Assets E, M and ##r_f## are in this category.
• Below the SML have a negative alpha (or negative abnormal return), are over-priced, and buying them is a negative NPV investment.
  Assets B and D are in this category.

Therefore, answer choices (a), (b), (c) and (e) are all correct.

But answer (d) is not correct since stock D has a higher beta (##\beta##) than the market portfolio (M) since it plots further to the right. Because beta is a measure of systematic risk, stock D must have more systematic risk than the market portfolio, not less.

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

Investment assets that plot above the SML are:

• Expected to over-perform, having a positive abnormal return or 'alpha';
• Under-priced, the price is cheap;
• Positive NPV decisions when bought by buyers, and negative NPV decisions when sold by sellers.

Note that the SML relates beta (a measure of systematic risk) to expected return, so it only accounts for systematic risk and can not be used to gauge diversifiable risk or total risk.

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

Investment assets that plot on the SML are:

• Expected to neither over- or under-perform, having a zero abnormal return or 'alpha';
• Fairly priced, since the price is fair for the buyer and the seller;
• Zero NPV decisions when bought or sold.

Note that the SML relates beta (a measure of systematic risk) to expected return, so it only accounts for systematic risk and can not be used to gauge diversifiable risk or total risk.

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

The net present value (NPV) of buying asset A would be positive since it plots above the security market line (SML), it has a positive Jensen's alpha, because it is under-priced.

The net present value (NPV) of buying asset B would be zero since it plots on the security market line (SML), it has a zero Jensen's alpha, because it is fairly priced.

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

The only correct statement is that the Gordon growth model (GGM) and capital asset pricing model (CAPM) both give an asset's total required return:

###r_\text{total, GGM} = r_\text{total, CAPM} ### ###\frac{c_1}{p_0}+r_\text{capital} = r_f + \beta(r_m - r_f) ### ###r_\text{income}+r_\text{capital} = r_\text{time value}+r_\text{risk premium} ###

The income return in the GGM is unrelated to the time value of money (##r_f##) or the market risk premium (##\beta (r_m - r_f)##) in the CAPM. This is mainly because the income cash flow from an asset is often discretionary, such as:

• Dividends on shares which are decided by the company's board of directors.
• Coupon payments on fixed-coupon debt which are often set equal to the yield when the bond is first issued (so it's issued at par), but after that as the yield (which is the total return of debt) goes up or down, the coupon stays the same since it's fixed.

Both of these are examples of how the income return component of total returns is often quite arbitrary and unrelated to the risk free rate or market risk premium in the CAPM. Therefore, the capital return is also unrelated to either the time value of money or market risk premium too.

On the other hand, it could be argued that the floating coupon income return on a floating rate bond are closely related to the risk free rate.

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

The house asset (V) is financed by the home loan debt (D) and the owners wealth or equity in the house (E).

###V = D + E###

Owning all of the debt and equity is equivalent to owning the house asset. Therefore the house asset can be seen as a portfolio of debt and equity.

Method 1: Use the CAPM Portfolio beta equation to solve for the beta of equity

Applying the portfolio beta equation, the beta of the asset must equal the weighted average of the betas on debt and equity.

###\beta_\text{portfolio} = \beta_1.x_1 + \beta_2.x_2 + ... + \beta_n.x_n ### ###\begin{aligned} \beta_V &= \beta_D.x_D + \beta_E.x_E \\ &= \beta_D.\frac{D}{V} + \beta_E.\frac{E}{V} \\ 1 &= 0.2 \times \frac{800,000}{1,000,000} + \beta_E.\frac{200,000}{1,000,000} \\ \end{aligned} ### ### \beta_E = 4.2 ###

Applying the CAPM,

###\begin{aligned} r_E &= r_f + \beta_E.(r_m - r_f) \\ &= 0.05 + 4.2 \times (0.1 - 0.05) \\ &= 0.26 \\ \end{aligned} ###

It may seem surprising that the equity's beta and required total return is so high. The reason is because of leverage. The debt-to-assets ratio (D/V) is 80% and the debt-to-equity ratio (D/E) is 400%. If the value of the house asset rose by 1%, the value of equity would rise by 5%.

Method 2: Use the WACC equation to solve for the cost of equity

Find the required return on debt ##(r_D)## and assets ##(r_V)## using the CAPM:

###\begin{aligned} r_D &= r_f + \beta_D.(r_m - r_f) \\ &= 0.05 + 0.2 \times (0.1 - 0.05) \\ &= 0.06 \\ \end{aligned} ### ###\begin{aligned} r_V &= r_f + \beta_V.(r_m - r_f) \\ &= 0.05 + 1 \times (0.1 - 0.05) \\ &= 0.1 \\ \end{aligned} ###

Using the weighted average cost of capital (WACC) equation (before tax since the question says ignore taxes), the cost of equity (also known as the required return on equity or opportunity cost of equity) can be found. ###\begin{aligned} r_V &= \text{WACC}_\text{before tax} \\ &= r_D.\dfrac{D}{V} + r_E.\dfrac{E}{V} \\ 0.1 &= 0.06 \times \dfrac{800,000}{1,000,000} + r_E \times \dfrac{200,000}{1,000,000} \\ \end{aligned} ### ###\begin{aligned} r_E &= \left(0.1 - 0.06 \times \dfrac{800,000}{1,000,000} \right) \times \dfrac{1,000,000}{200,000} \\ &= 0.26 \\ \end{aligned} ###

We can use the CAPM to find the beta of equity from this required return on equity:

###r_E = r_f + \beta_E.(r_m - r_f) ### ###0.26 = 0.05 + \beta_E.(0.1 - 0.05) ### ###\begin{aligned} \beta_E &= \dfrac{0.26 - 0.05}{0.1 - 0.05} \\ &= 4.2 \\ \end{aligned} ###

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

Strictly, cash flows in one month should be matched with required returns over the next one month, and cash flows in 10 years should be matched with required returns over the next ten years. This means that the required total return (same as the opportunity cost of capital), and the risk free rate and market portfolio rate used to calculate the required return, should always match the timing of the cash flows.

But in practice, since using multiple discount rates is too cumbersome, most analysts simply assume that required returns don't change significantly over time. Analysts tend to use the longest term government bond yield available. This makes some sense since most businesses are a 'going concern', meaning that they are supposed to continue forever. Therefore the cash flows are expected to go forever so the required total return should be measured over a long period.

Note that the current long term government bond yield is more useful than the historical government bond yield since we're trying to find the present value of future cash flows, not past cash flows.

The choice of which government bond yield to use as the risk free rate is a hotly debated topic, as is the existence of a truly risk-free rate at all.

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

Total required returns only depend on systematic risk since that's the only risk that can't be diversified. Investors are afraid of systematic risk and deserve a higher expected return when exposed to this undiversifiable risk. This relationship is expressed mathematically in the security market line (SML) equation from the capital asset pricing model (CAPM). The total required return ##(r_\text{i, total})## of some asset ##i## is a function of its beta ##(\beta_i)##, which measures systematic risk.

###r_\text{i, total} = r_f + \beta_i(r_m - r_f)###

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

If risk-averse investors were forced to invest all of their wealth in a single risky asset A or B (not M since it's a portfolio) then they cannot diversify so total risk is important to them, not just systematic risk. Total risk is shown on the left graph's x-axis. Total variance equals systematic variance plus diversifiable variance:

###\text{TotalVariance} = \text{SystematicVariance} + \text{IdiosyncraticVariance}### ###\begin{aligned} \sigma_\text{total i}^2 &= \sigma_\text{systematic i}^2 + \sigma_{\text{idiosyncratic i}}^2 \\ &= \beta_i^2\sigma_\text{m}^2 + \sigma_{\epsilon\text{ i}}^2 \\ \end{aligned}\\###

People who prefer low risk will choose asset B instead of A since ##\sigma_\text{B total} = 0.1## is less than ##\sigma_\text{B total}= 0.4##.

They're the sort of people who might carry an umbrella in their bag even when it's sunny, just in case it might rain. They suffer carrying it around but occasionally it helps them avoid getting drenched and sick. Or perhaps they purchase comprehensive car insurance. They're afraid of risk and are prepared to suffer low expected (average) returns to avoid large losses.

People who prefer high returns will choose asset A instead of B since ##\mu_A = 0.12## is greater than ##\mu_B = 0.075##.

They're the sort of people who may not purchase comprehensive car insurance because they're comfortable with the possibility of crashing their car and losing a large sum to replace it, if it means that they will have more money otherwise. They're comfortable with the possibility of suffering large losses if it means that on average they could gain more.

Choosing asset A or B is a personal choice, there's no correct answer. It depends on your return versus risk preferences. Of course in reality, you're not restricted to choose between A or B, you can choose a bit of both by making a portfolio which is the best idea.

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

No explanation provided.

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

Using the capital asset pricing model's (CAPM) security market line (SML) formula:

###\begin{aligned} r_i &= r_f + \beta_i (r_m - r_f) \\ &= 0.05 + 0.7 (0.1 - 0.05) \\ &= 0.085 \\ &= 8.5 \text{% pa} \\ \end{aligned}###

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

While the risk free rate is 5% per annum, over the last 5 minutes it would be close to zero since that's such a short time.

Using the capital asset pricing model's (CAPM) security market line (SML) formula based on these returns over the last 5 minutes, the stock's return (some stock ##i##) over the last 5 minutes is expected to be:

###\begin{aligned} r_{i \text{ 5min}} &= r_{f \text{ 5min}} + \beta_i (r_{m \text{ 5min}} - r_{f \text{ 5min}}) \\ &= 0 + 0.7 (-0.02 - 0) \\ &= -0.014 \\ &= -1.4 \text{% pa} \\ \end{aligned}###

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

The sample covariance of the effective total annual returns between the S&P500 and Apple is actually 0.298181591. This makes Apple's equity beta equal 3.530767 (=0.298181591/0.084452) which is quite a lot higher than the market's.