Assignment #3 – 10 marks
Question 1 – 5 marks
a) If a company has an expected return of 20% while the risk-free rate is 4% and the expected return on the market is 16%, what is the company’s Beta? (1 mark)
) Do the weights or components of the market portfolio change as the risk-free rate changes? Explain (a diagram may be helpful) (2 marks)
c) Describe a risk which is not priced by the CAPM? (1 marks) Explain why bearing this risk offer no return in the CAPM? (1 marks)
Question 2 – 5 marks
Wilson Inc. uses bonds and common shares to finance its investments and operations. The firm's debt consists of 20,000 bonds that are cu
ently trading at a price of $ XXXXXXXXXXeach and yielding a 2.75% spread above federal government debt of the same maturity. Wilson's 500,000 common shares are cu
ently trading at a price of $48.85 each. The firm just paid a dividend of $3.20 per share and it expects the dividend to grow by five percent per year for the foreseeable future. The risk-free rate of interest is cu
ently 2.95% and the market risk premium is estimated to be 6.45%. The company's average tax rate is 38%.
a) What is Wilson's D/E ratio? (2 marks)
) Calculate Wilson's WACC (3 marks)
Business Stats: An Applied Approach
Intermediate Finance:
Portfolios, CAPM, and the Cost of Capital
Fall 2021
Blended Learning Part-Time MBA
Note: The hex code for McMaster Maroon is #6A1A41 or (106,26,65) in RGB
1
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 1 Instructor Notes 1-‹#›
Risk, Return and Financial Markets
To help us determine the appropriate risk-adjusted returns on non-financial assets, we can examine the returns of the securities into which their cash flows have been repackaged.
These provide benchmarks for the required rate of return on various forms of capital
There are a few basic lessons from capital market history:
There is a reward for bearing risk (heavier discounts create greater opportunities)
The greater the potential reward, the greater the risk (the market prices risk efficiently)
While there is some debate about the degree of market efficiency, the consensus view is that markets are “generally efficient most of the time”.
There is a lot of noise in the system however so large sample sizes are important.
10.2
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 1 Instructor Notes 1-‹#›
Measuring Returns
Total dollar return = income from investment + capital gain (loss)
You bought a bond for $950 1 year ago. You have received two coupons of $30 each. You can sell the bond for $975 today. What is your total dollar return?
Income = XXXXXXXXXX = 60; Capital gain = 975 – 950 = 25; Total dollar return = XXXXXXXXXX = $85
This is only so practical though (no scaling for investment). It is generally more intuitive to think in terms of percentages than dollar returns
Total percentage return = dividend yield + capital gains yield
Dividend yield = income / beginning price
Capital gains yield = (ending price – beginning price) / beginning price
You bought a $35 stock and received dividends of $1.25. The stock is now selling for $40.
Dollar return = XXXXXXXXXX – 35) = $6.25
Percentage return = 3.57% (div XXXXXXXXXX% (cap gains) = 17.86%
Dividend yield = 1.25 / 35 = 3.57%; Capital gains yield = (40 – 35) / 35 = 14.29%
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10.3
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 1 Instructor Notes 1-‹#›
Estimating Future Returns
While past returns might be interesting, investor’s are most concerned with future returns. Historical average returns might not be realized in the future.
Developing an independent estimate of ex ante returns usually involves use of forecasting discrete scenarios with outcomes and probabilities of occu
ence.
Where:
ER = the expected return on an investment
i = the estimated return in scenario i
Probi = the probability of state i occu
ing
Estimating Future Returns
A three scenario estimate of returns given different economic conditions.
Analysts forecast the impact of state variables (growth, inflation, etc) for the firm by building a financial model which considers their impact on sales, prices, costs, and profits.
These are transformed into estimates of returns under various conditions.
State of the Economy Probability of Occu
ence Return on Stock A in that State
Economic Expansion 25.0% 30%
Normal Economy 50.0% 12%
Recession 25.0% -25%
Expected Return on the Stock =
Weighted Possible Returns on the Stock
7.50%
6.00%
-6.25%
7.25%
Measuring Average Returns
There two different ways of calculating returns over multiple periods. Which to use depends on the application.
The Arithmetic Mean: a simple average. Used when we want to estimate what to expect from a single observation or calculate statistics like standard deviation.
You invested $100 in a stock five years ago. Over the last five years, annual returns have been 15%, -8%, 12%, 18% and -11%. The average return was 5.2%.
The Geometric Mean: a multiplicative average. Used to estimate a single rate that approximates the impact of different compounding rates over time.
Same numbers as above. In five years you earned a total compound return of 24.444%.
(1.15)*(0.92)*(1.12)*(1.18)*(.89) = XXXXXXXXXX
This is equivalent to earning: XXXXXXXXXX/5) – 1 each year = 4.47%/yea
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Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 1 Instructor Notes 1-‹#›
Rates of Return, XXXXXXXXXXGeometric Return in Brackets)
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S&P500 = 11.73% (10.42%)
TSX = 10.30% (9.01%)
Long CDN Bond = 8.57% (8.15%)
T-BILLS = 5.62% (5.51%)
CPI = 3.82% (3.77%)
10.7
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 1 Instructor Notes 1-‹#›
Risk and Standard Deviation
One of the most commonly used measures of risk is the standard deviation / volatility of asset returns (The greater the volatility, the greater the uncertainty).
This arises from the (near) normality of many distributions, and that normal distributions are defined in terms of their mean and variance (an M-V framework).
People have a tendency towards risk aversion – a “disutility” arising from the uncertainty associated with future outcomes independent of expected payoff ($8 for sure vs. $8 +/- $2).
Measuring the historical standard deviation is easy (particularly with a spreadsheet!)
But if you like math…
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10.8
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 1 Instructor Notes 1-‹#›
Scenario Based Estimates of Standard Deviation
Discrete scenarios might be the best way to make estimates of the future (past conditions aren’t representative of what we expect going forward).
Note that deviations are weighted by their probabilities, and not equally.
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10.9
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 1 Instructor Notes 1-‹#›
Compensation for Volatility
A government free to issue its own cu
ency can issue debt which is effectively risk-free.
This is not a claim about the value of cu
ency used to repay - only that such issuers default by choice.
Thus government debts are often treated as “risk-free” (domestically) and are used as the baseline for the required rate of return on all other assets.
A risk premium is the incremental compensation demanded by investors (above the risk-free rate) in return for bearing some degree of uncertainty
Ideally, we would prioritize investments which offer the highest rate of return per unit of risk.
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10.10
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 1 Instructor Notes 1-‹#›
Two Sides of the Same Coin
Always be sure to consider both the risk and reward of an investment opportunity
RETURN: A returns 8% 100% of the time; B returns 1000% (10%) or -100% otherwise (90%).
RISK: A returns 5% each year +/- 5%; B returns -2% each year +/- 0%
Rather than isolated consideration, we’re better off thinking in terms of the return we earn for each unit of risk endured (the slope of the line on the last slide).
We look at the incremental return per standard deviation as we can earn Rf without taking risk.
This allows us to rank investments by their risk-return efficiency!
The Risk-Reward Connection
While this approach is intuitively appealing, empirical data on returns and volatility from stock markets suggests that something more complex is happening.
There is little direct connection between volatility and returns for individual assets
In a liquid market where we can split capital across investments, we have to consider the risk of the portfolio, not the components (they interact in beneficial ways).
Note that diversified portfolios like the S&P500 and TSX index fall almost exactly on a linear estimate of a volatility premium.
Also note how they earn a better return per unit of volatility than almost all individual stocks!
Portfolios = Free Gains!
A portfolio is a collection of different assets combined to achieve diversification – a reduction in volatility resulting from the neutralization of certain risks.
Many of the approaches we use for individual assets apply to portfolios too, allowing us to conceptually think of a portfolio as a single asset itself.
The risk-return trade-off of portfolios is demonstrably different than what is earned by the individual assets making up that portfolio however.
Portfolio returns are the weighted average return of its components but risk is another matter.
The assets comprising a portfolio have imperfect co
elation, so their volatilities (which won’t all pull in the same direction at the same time) neutralize one another to some extent.
This improvement in the risk profile of a portfolio relative to its components is possibly the only “free lunch” in finance.
Portfolios With Discrete States
The expected return on a portfolio is the weighted average of the returns of the individual assets that make up the portfolio:
Similarly, its expected volatility is
Of Eggs and Baskets
Imagine you are a merchant with 10,000 ducats worth of goods to send a
oad for resale. Any trade mission returns 40% but has a 20% chance of a 100% loss.
Assuming the probabilities of ruin are independent, should you send out all your goods on one mission or spread them over two?
Return to Risk (one boat): 12% / 56% = .2143; (two boats): 12% / 39.6% = .3030
By spreading our investments, we not only reduce the chance of ruin (-100% loss) from 20% to 4%, but we also improve the rate of return we earn per unit of risk!
This procedure can be relatively inefficient and cumbersome however.
How to incorporate all risks impacting individual securities in our model?
How can we identify which other assets might improve the portfolio’s performance?
Thankfully, statistics offers us a way forward
A Statistical Approach to Portfolios
Instead of thinking about asset returns as realizations of state variable scenarios, think of them as processes with outcomes defined in terms of their expected mean (return) and the standard deviation of outcomes (risk).
In this portfolio model, each asset return process is