Risk and Return.
Measuring Return and Risk.
Past vs. Future.
One.
$120.
21.2%.
Yes, if the division has the same risk profile as the firm's overall operations.
Treynor Ratio = (Expected Return - Risk-Free Rate) / Beta.
$1,050.
D_1 = D_0 (1 + g)
Expected return = Risk-free rate + Risk premium.
It reduces tax liability, effectively lowering the after-tax cost of debt.
$1,040 million.
Expected return = Risk-free rate + (Beta ร Market risk premium) = 3% + (1.1 ร 10%) = 14%.
$201.68 million.
5.1%.
$10.40.
9%.
The relation between Return and Risk.
25 years.
11.1%.
$100 per share.
50%.
Lowly correlated stocks.
It is a statistical representation of the actual returns observed over a specific period.
About 2/3 of the time.
It helps investors understand the historical performance and volatility of an asset or portfolio.
The effective rate that a company pays on its borrowed funds.
Using the formula ๐ฝโ = ฯแตข ร ๐ฅแตข ร ๐ฝแตข.
Higher expected returns are associated with greater risk of doing badly in bad times.
The expected squared deviation from the mean.
(Portfolio Return - Risk-Free Rate) / Beta.
The expected returns of the individual assets and their respective weights in the portfolio.
$25.
Weighted Average Cost of Capital.
9%.
It describes the likelihood of different outcomes for investment returns.
It can only be used if r (required return) is greater than g (growth rate).
4.3.
$1,000.
Using the average return estimate based on realized returns.
It helps in making investment decisions and evaluating projects.
It can indicate the likelihood of various return outcomes based on historical data.
It helps investors assess whether an asset is fairly valued based on its risk and expected return.
Debt and equity.
10%.
The realized return of a security in year t.
CAPM does not have to deal with the difficulty of predicting the growth rate 'g'.
Conversion ratio = Bond face value / Conversion price.
The volatility of the portfolio declines.
Preference share dividends are fixed.
No, because it can be eliminated relatively costlessly by holding a portfolio.
The company must pay a meaningful dividend.
It is the relationship between systematic risk and expected return.
To measure the risk-adjusted return of a portfolio.
The average rate of return a company is expected to pay its security holders to finance its assets.
The assetโs risk premium, E(Ri) โ rf.
It determines the required return for those projects.
The present value (PV) of future cash flows.
Interest expense for the year / Interest-bearing debt.
The horizontal axis.
When evaluating projects that have the same risk as the firm's current operations.
$1 billion.
Risk refers to the potential for loss or the uncertainty regarding the return on an investment.
The degree of variation in stock prices over a specific period.
Calculating the cost of equity.
(r e โ r f) / ฮฒ.
A measure of the price fluctuations of an investment over time.
E(Ri) = Rf + ฮฒi(E(RM) - Rf)
The unpredictability of investment returns over time.
It helps in assessing the potential future performance of an investment.
Yes, it suggests a potentially favorable return.
Convertible bonds and multiple types of debt.
$75.
By diversifying their portfolio.
To lower yields on bonds.
It helps investors assess the potential profitability of their investment choices.
Coupon payments and principal payment.
Long-dated expiration.
Using the formula โ[0.25(0 - 0.10)ยฒ + 0.50(0.10 - 0.10)ยฒ + 0.25(0.20 - 0.10)ยฒ].
It marks a period before significant market events like the Great Depression.
From 7.7% to 16.9%.
The return a company needs to generate to cover the cost of financing its operations.
1926 to 2004.
50 million shares.
The growing perpetuity formula: Vโ = D(1+g)/(1+r) + D/(1+g)ยฒ/(1+r)ยฒ + ... + D/(1+g)แต/(1+r)แต.
The reward-to-risk ratio.
Dividends must grow at a constant rate.
Investors would hold the market portfolio, which includes all assets.
Unsystematic risks for each stock will average out and be diversified.
The cost of equity calculated is highly sensitive to the estimated growth rate.
21.2%
No, systematic risk will affect all firms and cannot be diversified.
Typically, higher historical average returns are associated with higher volatility.
Because there must be some economic rationale behind the risk-reward relationship.
You can derive the CAPM formula.
To calculate the cost of equity.
๐ ๐ = ๐ฅ1๐ 1 + ๐ฅ2๐ 2 + ... + ๐ฅ๐๐ ๐.
The value would depend on the specific investment vehicle and its performance over the years.
It reduces the overall risk, leading to a portfolio standard deviation that is less than the weighted average of individual stock standard deviations.
Systematic risk and unsystematic risk.
As a percentage of the par value of the share.
It is a graphical representation of the relationship between expected return and systematic risk (beta) of assets.
Systematic risk is the risk inherent to the entire market or market segment.
It helps assess the risk associated with individual stocks.
How the market views the risk of the firm's assets.
By collecting historical return data and plotting the frequency of different return levels.
It measures the volatility or risk of the investment returns.
Single vs. Multiple Assets.
4.28%.
7.07%
For the firm or project.
$1,050 (calculated as 14 shares * $75).
Yes, if the expansion has the same risk as current operations.
0.045.
Weighted average of individual stockโs return.
E(R) = 25%(-0.20) + 50%(0.10) + 25%(0.40) = 10%
14 shares.
95%.
It is the return to an investor, which equals the cost to the firm.
To compensate investors for the financing provided.
Interest expense for the year divided by interest-bearing debt.
$80 per share.
Future dividends are highly uncertain and depend on estimating the growth rate.
Excess return earned per unit of risk taken.
Var(R) = E(R) - E(R)^2 = ฯ_sp ร R - E(R)^2.
E(RM) - Rf, which is the excess return of the market portfolio over the risk-free rate.
To compare different investment alternatives.
The square root of the variance.
The systematic risk of asset i.
It helps in making informed investment decisions and managing risk.
They are undesirable unless they have some redeeming virtue.
The weighted average of the possible returns, where the weights correspond to the state probabilities.
The cost of equity.
It can only be used if the company pays meaningful dividends.
$45.
Yes, as long as we can estimate beta.
Long-term debt.
Only systematic risks.
10%.
The portfolio standard deviation is less than the weighted average due to diversification benefits.
The average return of an investment over a specified period based on past performance.
The weighted average beta of the securities in the portfolio.
The weighted average of the expected returns of the individual assets within the portfolio.
Normal distribution, log-normal distribution, and binomial distribution.
It indicates the range of returns relative to the mean.
The growth of preference share dividend is zero.
5.1% per year.
The return on a risk-free asset.
A bond that can be converted into a specified number of shares of stock.
Higher volatility may indicate higher risk, influencing an investor's choice.
Beta measures the risk of a security in relation to market movements.
Market volatility, economic conditions, and company performance.
Var(R) = (1/(T-1))ฮฃ(Rt - Rฬ)ยฒ.
Higher risk typically leads to a higher Cost of Capital.
SD(R) = Var(R).
Unsystematic risks.
Diversification can reduce risk by spreading investments across various assets.
Major war, global recession, etc.
To measure the reward-to-risk ratio of an asset.
Business risk and financial risk.
If dividends aren't growing at a reasonably constant rate.
RATE(NPER, PMT, PV, FV).
Capital structure, exchange rate, interest rates, credit risks, and liquidity.
10%.
Market performance, interest rates, inflation, and the type of investment.
4.
Capital Asset Pricing Model.
Risk that affects all securities due to market-wide news.
1926 to 2004.
Risk that affects a particular security due to firm-specific news.
14 shares.
Unsystematic risk is specific to a particular company or industry.
Higher risk typically correlates with the potential for higher returns.
The vertical axis.
The return required by equity investors given the risk of the cash flows from the firm.
The fraction of the total investment in the portfolio held in each individual investment.
$71.43 (calculated as $1,000 / 14 shares).
You must be prepared to sacrifice returns.
20.68%.
The present value of a series of equal cash flows received at regular intervals.
It is above the current stock price.
He compares it to making money in Las Vegas.
Under-valued.
BFI: 0.25 (0.40), 0.50 (0.10), 0.25 (-0.20); BFJ: 0.00 (0.00), 0.10 (0.10), 0.20 (0.20).
Capital Asset Pricing Model.
Conversion value = Conversion ratio x Stock price.
The effective rate that a company pays on its borrowed funds.
7%.
It helps in making investment decisions and evaluating projects.
0%.
0.4 (or 40%).
4.5%.
By multiplying the number of shares by the stock price (14 x 75).
There is a positive relationship between volatility and average returns.
12%.
The return required by equity investors for their investment in a company.
Having many types of debt, convertible bonds, or unknown bond values.
$1.50.
Finding a method to accurately estimate the discount rate.
It assumes that returns are symmetrically distributed around the mean.
A Nobel-prize winning work.
The Treynor Ratio can be derived from the CAPM formula when considering a market portfolio.
1.15.
Debt can lower the overall Cost of Capital due to tax benefits.
Expected return = Risk-free rate + Beta * (Market return - Risk-free rate).
Constant dividend growth model (DGM) and Capital asset pricing model (CAPM).
It must equal that of the market.
5%.
Capital Market History.
The value of the bond if conversion does not take place, valued using cash flows of the bond.
It explicitly adjusts for systematic risk.
The weighted average of the expected return of the investments within the portfolio.
R D = 3.927(2) = 7.854%
5.0%.
If systematic risks are the same, the project is in the same line of business, and the project is not exceptionally large compared to the firm.
The required return (r) must be greater than the growth rate (g).
15 years.
Because it measures total risk.
10%.
Both the expected market risk premium and beta.
-5%.
Risk is often measured using statistical metrics such as standard deviation or beta.
It helps investors assess risk and make informed decisions.
An estimate of the corresponding discount rates.
The Fixed Income Portfolio.
By multiplying the expected return of each asset by its weight in the portfolio and summing the results.
Rฬ = (1/T)(R1 + R2 + ... + RT) = (1/T)ฮฃRt.
ฯแตข is the standard deviation of the security, ๐ฅแตข is the weight of the security in the portfolio, and ๐ฝแตข is the beta of the security.
Higher volatility often indicates higher potential returns and risks.
Assuming constant capital structure, ignoring market conditions, and not accounting for risk differences.
ฮฒi = Systematic risk of asset i / Systematic risk of market portfolio.
Stocks with higher historical volatility may offer higher returns but also come with greater risk.
R E = (1.50/25) + 0.051.
๐ฅ๐ = Value of investment i / Total value of portfolio.
The cost of equity.
Ordinary shares (E), preference shares (P), and debt (D).
The calculation is not provided in the text.
Diversification and Types of Risk.
The required return on a firm's debt.
The proportion of money invested in each stock.
๐ธ๐ ๐ = ฯ๐๐ฅ๐๐ธ๐ ๐.
NPER = 30; PV = -1,100; PMT = 45; FV = 1,000.
10%.
=((0.25*(0.4 - 0.1)^2)+(0.5*(0.1 - 0.1)^2)+(0.25*(-0.2 - 0.1)^2))^0.5*100
$100.84.
$75.
9%.
WACC = wE * RE + wD * RD * (1 - TC).
12 years.
6.1%.
30%.
WACC = 0.7843 (15.35%) + 0.2157 (4.712%).
The greater the systematic risk, the higher the reward.
Expected Return = ฮฃ (Probability ร Return)
The total number of years for which returns are realized.
It is easy to understand and use.
$908.72 per $1,000 bond.
The return required by equity investors given the risk of the investment.
The price at which the bond can be converted to stock.
$5,000.
Over-valued.
$71.43.
Market values, not book values.
Lower bound for CB value = Max (straight bond value, conversion value).
V = E + P + D.
CB value > lower bound.
Systematic risk only.
2%.
Next year's dividend.
$1 per share.
13.06%.
Portfolio SD < Weighted average of component stocksโ SDs due to diversification benefit.
The Fixed Income Portfolio with a Treynor Ratio of approximately 4.29%.
$2000.
Higher beta indicates higher expected return due to increased risk.
7.3%.
83.8%.
Conversion premium = (Conversion price โ Stock price) / Stock price.
By 'beta'.
E(Ri) - rf / ฮฒi = E(RM) - rf / 1.
$1,000.
The current worth of a future sum of money or stream of cash flows given a specified rate of return.
If Conversion Price > Stock Price throughout the conversion period.
Using the Capital Asset Pricing Model (CAPM).
15%.
$80.
It suggests that the stock may have potential for higher returns compared to its current price.
Find the appropriate beta for the project.
They may be priced higher than their intrinsic value, indicating potential risk.
R_E = Risk-free rate + Beta * Market risk premium.
Risk.
8.7%.
E(Ri) = rf + ฮฒi(E(RM) - rf)
Treynor Ratio = (5% - 2%) / 0.70 โ 4.29%.
6.1%.
The value of a stock using the growing perpetuity formula.
10%, regardless of the portfolio weights.
5.1%.
The market value of the firm, which is D + E.
R E = 5 + 1.15(9) = 15.35%
It is the weighted average of all financing sources.
110.
wE = E/V, where E is equity and V is total value.
9%, semiannual coupons.
It requires estimating the expected market risk premium, which varies over time.
No, the cost of debt is not necessarily the coupon rate.
2.5%.
Weighted Average Cost of Capital.
R_E = D_1 / (P_0 + g)
4%.
=((0.25*(0 - 0.1)^2)+(0.5*(0.1 - 0.1)^2)+(0.25*(0.2 - 0.1)^2))^0.5*100
10%.
7.5 percent.
8.6%.
Number of years to conversion and conversion value (not par value).
62.5.
4.6%.
0.25 probability for 0.40 return, 0.50 probability for 0.10 return, and 0.25 probability for 0.00 return.
Three years.
Yes, if the new product has the same risk as current operations.
6.25%.
No, unsystematic risk is not rewarded.
The weights correspond to the state probabilities.
21.2%
$6 (5% of $120).
By dividing the bond face value by the conversion ratio (1000 / 14).
An increase in g of 1% increases the cost of equity by at least 1%.
By computing the Yield to Maturity (YTM) on existing debt.
75%.
The percent financed with equity (E/V).
3%.
$41.20 per share.
11.1%.
30%.
$200 million.
$1 par.
The dollar amount of each source.
Product/service offering and variability in profits due to operating leverage.
4 billion.
5%.
1.1 billion.
40%.
5.1 billion.
Treynor Ratio = (7% - 2%) / 1.25 = 4%.
7.0%.
1.23.
$100.84.
14%.
Year 0: -1750, Year 1: 50, Year 2: 50, Year 3: 2050.
The marketโs perception of the risk of the firmโs assets.
WACC = (E/V * Re) + (D/V * Rd * (1 - Tc))
25%.
38.50%.
6 percent.
1.1.
7.07%
3%.
Because it is more appropriate for equity rather than debt.
0.2157.
Alternative methods such as using industry averages or estimating based on similar firms can be used.
It is the same as non-convertible debt.
Dividends grow at a constant rate.
It reduces the overall risk compared to individual stocks.
Yes, but they may not always do so.
$200.
Over-valued (expected return 10% vs actual 15%).
16.2%.
R D (1 - T C ) = 7.854(1 - .4) = 4.712%
Cost of preferred stock = Dividend / Price.
By using the beta of a listed company in that business.
6% (calculated as $6 / $100).
0.58.
7.07%
17%.
10%.
5.7%.
It is linear, as represented by the CAPM formula.
Evaluate the assumptions and inputs used in both models.
4.9%.
12.7%.
It is highly sensitive to the choice of market risk premium and beta.
0.70.
Semi-annually.
The capital structure must be the same; otherwise, it needs to be adjusted for capital structure.
40%.
Using the historical average of percent changes in dividends.
The percent financed with debt (D/V).
0.7.
Expected return = Risk-free rate + (Beta ร Market risk premium) = 3% + (0.7 ร 10%) = 10%.
$2,575.
Expected return = Risk-free rate + (Beta ร Market risk premium) = 3% + (1.4 ร 10%) = 17%.
Under-valued (expected return 14% vs actual 10%).
ฮฒi (beta).
After-tax cost of debt = RD * (1 - TC).
Because there is no tax impact on the cost of equity.
0.25 probability for 0.20 return, 0.50 probability for 0.10 return, and 0.25 probability for -0.20 return.
Under-valued (expected return 17% vs actual 12%).
Expected return on BFJ = 0.25(0) + 0.50(0.10) + 0.25(0.20) = 10%.
If fairly valued, V_0 = P_0.
Beta measures the sensitivity of the asset's returns to the returns of the market.
$1,242 million.
