Investments:
Risk and Return
Business Administration 365
Professor Scott Hoover
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Returns
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Return  measure of the benefit received from an investment
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holding period return  % change in value over the period

effective annual return (EAR)  1-year holding period return
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includes the effects of compounding.
annual percentage rate (APR)  per period rate × # of periods
per year
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ignores the effects of compounding
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If rp is the per period rate and m is the # of compounding
periods per year,
APR  m  rp
EAR  (1  rp ) m - 1
Putting those together gives
m
 APR 
EAR  1 
 -1
m 

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The distinction is important because…
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Annual discount rates must be EARs
Banks quote APRs
Bond yield-to-maturities (YTMs) are APRs
Credit cards and option valuation models use continuouslycompounded interest rates

APRs with m = ∞
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A few examples…
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A bank quotes 9% APR, compounded monthly.
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A bond pays 4% interest every six months.
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Per period rate 9%/12 = 0.75%
EAR = 1.007512 - 1 = 9.38%
YTM = 4%×2 = 8%
EAR = 1.042 - 1 = 8.16%
The continuously-compounded interest rate is 6% per year.

What happens as m gets bigger?

m = 10 → EAR = 6.165%

m = 100 → EAR = 6.182%

m = 1000 → EAR = 6.183%

m = ∞ → EAR = 6.184%
m
 APR 
APR
EAR  lim 1 
 - 1  e 1
m 
m  
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
Risk

Risk  measure of the potential for loss in an
investment

How should we measure risk?
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Reality: we really don’t know how to best measure risk.
What do we know about risk?
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Some risk is all but eliminated in well-diversified
portfolios
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 diversifiable risk (aka, idiosyncratic risk, firm-specific
risk)
Some risk remains no matter how well-diversified we are
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 non-diversifiable risk (aka, systematic risk, market risk)
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Capital Asset Pricing Model (CAPM)
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Intuition: Investors will only be compensated for the level
of non-diversifiable risk they take on. Why?
Definition: market portfolio ≡ portfolio of all assets
weighted according to their market values.
Result
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Expected return on a well-diversified portfolio is a linear
function of the standard deviation of the portfolio’s returns
 Why is std. dev. a reasonable measure here?
The line is called the Capital Market Line (CML)
 E(R) = Rf + (/ m)(E(Rm)-Rf)
 R  return on portfolio
 Rm  return on market portfolio
 Rf  risk-free return
   std. dev. of portfolio
 m  std. dev. of market portfolio
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Expected
Return
Capital Market Line
E(Rm)
Rf
m
Risk (standard deviation)
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The CML is useful for well-diversified portfolios, but not for
individual assets.
To eliminate diversifiable risk, we extract the portion of the
standard deviation that is correlated with the market.
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This leaves b, which is our measure of the non-diversifiable risk
of an asset.
COV R, Rm  r m r
b 


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VARRm 
m
m
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r ≡ correlation between asset returns and market returns
Result: expected return on any asset is a linear function of b.
This line is called the Security Market Line (SML)
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E(R) = Rf + b(E(Rm)-Rf)
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Expected
Return
Security Market Line
E(Rm)
Rf
1
Risk (beta)
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Roughly speaking, beta tells us how the asset price tends
to move when the market moves.
 e.g., Suppose b=1.5 for some asset. The asset will tend
to move by 15% whenever the market moves by 10%.
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Do assets with negative betas exist? After all, E(R) < Rf
 Yes! Such assets provide insurance on the portfolio
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Miscellaneous Notes
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If markets are efficient, investing is nothing more than choosing a
b and a method to achieve it.

 invest in a combination of the risk-free and a market index
Portfolio betas

The beta of a portfolio is the weighted average of the betas of
the individual assets in the portfolio:
bp = wAbA + wBbB + …
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The weights are the fraction of our money we have invested in
each, with short positions having negative weights
example: Suppose we have $10,000 to invest and that we
short $6,000 of one asset. We then invest $16,000 into a
second asset.
w1 = -$6,000/$10,000 = -0.6
w2 = $16,000/$10,000 = 1.6
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Problems with using the CAPM
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What is the market portfolio?
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can include every conceivable asset (stocks, bonds,
baseball cards, ostrich eggs, etc.).
difficult to measure, so we proxy by using a market index
(such as the S&P500 or the Russell 3000).
expected return on the market portfolio is difficult to
estimate.
 def’n: market risk premium (MRP) = E(Rm)-Rf
 One study shows that the MRP was about 7.4% from
1926-1999.
 Recent evidence suggests that it should be around 4%.
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Other factors may be important.
Historical returns may not be representative of future
returns.
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What if we don’t have historical data for the asset?
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See spreadsheet example
find comparables
use common sense
Bottom Line: use the CAPM and common sense
to estimate the appropriate discount rate
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Applying the CAPM
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gives the appropriate rate for discounting cash
flows to shareholders
used as part of the WACC calculation
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WACC = wdRd(1-T) + wpsRps + weRe
Recall that
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wi  fraction of firm (using mkt values) financed with type i
T  tax rate
Re  required return on equity
 estimated using CAPM
Rps  required return on preferred stock
 estimated using D/P
Rd  required return on debt
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Techniques to estimate cost of debt
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#1: Find the yield-to-maturity on the company’s
outstanding debt
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Potential problems
 1. YTM depends on the maturity of the debt.
If the company’s debt has a very long or very short
maturity, our estimate may be biased.
 2. The YTM does not reflect the expected return to
investors.
 3. The company’s debt may not be publicly-traded
 4. Bonds with embedded options are problematic.
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#2: Use the company’s debt rating in conjunction with
yield spreads to estimate the company’s cost of debt.
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yield spread: the difference between the yield on a bond (or
class of bonds) and a corresponding Treasury bond with
the same maturity
example: Suppose a company’s debt is rated Baa1 by
Moody’s. What is our best estimate of the company’s cost
of debt?
 Suppose we choose to use a ten-year maturity for our
estimates.
 From www.bondsonline.com, we see that the yield
spread is 107 basis points, or 1.07%. (Note that this is
an old estimate).
 From www.cnnfn.com, we see that ten-year Treasuries
have a yield of 3.89%
 Cost of debt estimate = 3.89%+1.07% = 4.96%
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Potential problems
 The company’s debt may not be rated
 The bond may not be an average bond within its ratings
class.
#3: Find the cost of debt on comparable companies
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