Risk & Return Introduction

RWJ Chapter 12
Learning Outcomes

1. Appreciate the concept of risk and return by

examining how various investments have performed
in the past
2. Appreciate how inflation can affect return
3. Estimate return
4. Appreciate the concept of stand alone risk

2
 A First Look at Risk and Return
Let’s begin by looking at historical return and risk (volatility) experienced by
various investments.

3
 What Are Investment Returns?
• Investment returns measure the financial results of an
investment.
• Returns may be historical or prospective (anticipated).
• Returns can be expressed in:

 Dollar terms: Amount received – Amount invested

 Percentage terms: Am ount received – Am ount invested

Am ount invested

4
 Dollar Returns

• When an investor buys a stock or a bond, his return comes in 2 forms:

1. Any dividend income or interest income received, and
2. A capital gain or a capital loss (due to change in price)
Total dollar return = income+ capital gain (loss) due to change in price

Examples:
 You bought PepsiCo stock at $43 a share. By the end of the year, the value of
each share has appreciated to $49. In addition, PepsiCo paid a dividend of
$0.56 a share.
Total dollar return = dividend income + capital gain = $0.56 + $6 = $6.56

 You bought a bond for $950 one year ago. You have received two coupons of
$30 each. You can sell the bond for $975 today. What is your total dollar return?
Total dollar return = interest income + capital gain = $60+25 = $85
5

12-5
 Percentage Returns
• It is generally more intuitive to think in terms of percentage, rather than
dollar, returns
– Dividend yield = dividend / beginning price
– Capital gains yield = (ending price – beginning price) / beginning price
– Total percentage return = dividend yield + capital gains yield

Div 0.56
Dividend Yield =   0.013 or 1.3%
Initial Share Price 43
Capital Gain 6
Capital Gain Yield =   .140 or 14.0%
Initial Share Price 43
Dividend  Capital Gain 0.56  6
Total Percentage Return =   .153 or 15.3%
Initial Share Price 43 6

12-6
Learning Outcomes

1. Appreciate the concept of risk and return by

examining how various investments have performed
in the past
2. Appreciate how inflation can affect return
3. Estimate return
4. Appreciate the concept of stand alone risk

7
 What is the Impact of Inflation?
• What we have calculated earlier is a nominal return.

• With inflation, the ending dollars may not be able to buy the same basket of goods.

• To find out the return in terms of the increase in purchasing power relative to initial
investment purchasing power, we need the real rate of return
– To find out how much more can be bought with the money at end of year.

• Fisher equation describes the relationship between interest rates and inflation:

1  rnominal  (1  rreal rate)(1  inflation rate)

1+rnominal = 1 + rreal rate + inflation rate + (rreal rate )(inflation rate)

• An approximation for the relationship:

rnominal  rreal rate  inflation rate 8

 Example: Inflation
• Let’s go back to the PepsiCo example. If the inflation for the year was
2.8%, what was the real rate of return?
Dividend  Capital Gain 0.56  6
Total Percentage Return =   .153 or 15.3%
Initial Share Price 43

1  rnominal  (1  rreal rat e)(1  inflation rate)

1  0.153  (1  rreal rat e)(1  0.028)

(1  rnominal)
rreal rat e   1  12.2%
(1  inflation rate)

Approximation:
rreal rat e  rnominal  inflation rate  12.5%
9
Learning Outcomes

1. Appreciate the concept of risk and return by

examining how various investments have performed
in the past
2. Appreciate how inflation can affect return
3. Estimate return
4. Appreciate the concept of stand alone risk

10
 How to estimate return?
• There are 2 ways of doing so
1. If given possible returns and the probabilities, calculate expected return:

n
rˆ   ri Pi
i 1
where
ri = ith possible return
Pi = probability of ith possible return
n= number of possible returns

Expected return takes into account uncertainties that are present in

different scenarios.

11
 Example: Expected Return

Probability of Possible
Return Return
An asset has a
• 30% probability of a 10% return 0.3 10%

• 10% probability of a -10% return 0.1 -10%

• 60% probability of a 25% return 0.6 25%

What is the expected return?

Expected Return = (0.30)(10%) + (0.10)(-10%) + (0.60)(25%) = 17%

2. If using historical data on an asset, calculate the average return which can
either be the arithmetic average return or the geometric average return

Arithmetic average return also called Arithmetic mean

T

r t
r t 1

Geometric average return also called Geometric mean

r  T ( 1  r1 )( 1  r2 )( 1  r3 )( 1  r4 )......(1  rT )  1

where rt are the actual nominal returns in year 1, year 2, ...

 Geometric vs. Arithmetic
• If an investor buys an asset at time 0 and holds it till time T:

‒ the geometric mean is what he actually earned per year on average

compounded annually. It is also known as the mean holding period
return or average compound return earned per year over a multi-
year period.

‒ the arithmetic mean is what he earned in a typical year.

14
 Geometric Mean
𝑃1 𝑃2 𝑃3 𝑃4 𝑃4
× × × =
𝑃0 𝑃1 𝑃2 𝑃3 𝑃0

1 + 𝑟1 1 + 𝑟2 1 + 𝑟3 1 + 𝑟4 = 1 + ℎ𝑜𝑙𝑑𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑 𝑟𝑒𝑡𝑢𝑟𝑛

1 + 𝑟1 1 + 𝑟2 1 + 𝑟3 1 + 𝑟4 = 1 + 𝑟 4

where r is the geometric mean

4
r= 1 + 𝑟1 1 + 𝑟2 1 + 𝑟3 1 + 𝑟4 - 1
15
 Example: Geometric Mean
Suppose you bought an investment and held it for 4 years. It provided the
following returns over the 4-year period:

Year Return Holding period return 

1 10%  (1  r1 )(1  r2 )(1  r3 )(1  r4 )  1
2 -5%  (1.10)(. 95)(1.20)(1.15)  1
3 20%  .4421  44.21%
4 15%
Geometric mean return
4
r= 1 + 𝑟1 1 + 𝑟2 1 + 𝑟3 1 + 𝑟4 - 1
4
= 1.10 0.95 1.20 1.15 - 1
= 0.0958 = 9.58%
So, you realized 9.58% annual return on your money for 4 years or
equivalently a holding period return of 44.21% 16
 Example: Arithmetic Mean

Year Return r1  r2  r3  r4
Arithmetic average return 
1 10% 4
2 -5% 10%  5%  20%  15%
  10%
3 20% 4
4 15%
 Benchmark

So, let’s say we have estimated a return:

• How does one determine whether the return is adequate?
– By comparing to a benchmark

• What should be the benchmark?

– The required rate of return

• What determines the required return?

– The risk of the investment
– What type of risk?
– Systematic risk – the risk that you cannot diversify away

18
Learning Outcomes

1. Appreciate the concept of risk and return by

examining how various investments have performed
in the past
2. Appreciate how inflation can affect return
3. Estimate return
4. Appreciate the concept of stand alone risk

19
 Stand-Alone Risk

• Stand-alone risk is the risk an investor faces if he holds only this one
asset.

• Measured by the standard deviation.

• The larger the standard deviation, the higher the probability that actual
returns will be far away from the expected or average return.

20
 Standalone Risk

Stock X

Stock Y

Rate of
-20 0 15 50 return (%)

Stocks X and Y both give the same expected return.

Stock Y has higher standalone risk. 21
But Stock Y may not have higher systematic risk.
Risk & Return Part 1
RWJ Chapter 13
 Risk and Return: Portfolio Theory & CAPM
Learning Outcomes:
1. Know how to compute the expected return and variance of
individual securities, as well as the covariance and
correlation between 2 securities.
2. Know how to compute portfolio risk and return of 2
securities
3. Appreciate how portfolios can reduce risk and be able to
identify the efficient set of a 2 security portfolio.
4. Apply the same reasoning to a portfolio of many securities
5. Understand the rationale behind the CAPM

23
 Individual Securities
• Characteristics of individual securities that are of interest:
– Expected Return
– Variance and Standard Deviation
– Covariance and Correlation
• Consider the following 2 risky security world with 3 possible
states of the economy. The 2 securities are a stock fund and a
bond fund. The 3 states each has 1/3 chance of occurring.
Scenario Probability Return of Return of
Stock fund Bond fund
Recession 1/3 -7% 17%
Normal 1/3 12% 7%
Boom 1/3 28% -3%
The table contains the future possible returns of the stock fund and the bond fund in
the 3 states. Note that these are future possible returns, not historical returns. 24
 Expected Returns, Variance, and Covariance
Scenario Probability Stock fund Bond fund
Return Squared Return Squared
Deviation Deviation
Recession 1/3 -7% 17%
Normal 1/3 12% 7%
Boom 1/3 28% -3%
Expected return 𝐸 𝑟
Variance 𝜎 2
Standard Deviation 𝜎
𝑁
• Expected return: 𝐸 𝑟 = 𝑖=1 𝑝𝑖 𝑟𝑖 (where i means the i-th economy state)
𝑁 2
• Variance: 𝜎 2 = 𝑝
𝑖=1 𝑖 𝑖 𝑟 − 𝐸 𝑟
• Covariance: 𝜎𝑆𝐵 = 𝑁 𝑖=1 𝑝𝑖 𝑟𝑖𝑆 − 𝐸 𝑟𝑆 𝑟𝑖𝐵 − 𝐸 𝑟𝐵
𝜎
• Correlation coefficient: 𝜌𝑆𝐵 = 𝑆𝐵
𝜎𝑆 𝜎𝐵
 Expected Returns
Scenario Probability Stock fund Bond fund
Return Squared Return Squared
Deviation Deviation
Recession 1/3 -7% 17%
Normal 1/3 12% 7%
Boom 1/3 28% -3%
Expected return 𝐸 𝑟 11.00% 7.00%
Variance 𝜎 2
Standard Deviation 𝜎

3 1 1 1
• Expected return: 𝐸 𝑟𝑆 = 𝑖=1 𝑝𝑖 𝑟𝑖 = −7% + 12% + 28% = 𝟏𝟏%
3 3 3

26
 Variance
Scenario Probability Stock fund Bond fund
Return Squared Return Squared
Deviation Deviation
Recession 1/3 -7% 324%𝟐 17% 100%2
Normal 1/3 12% 1%2 7% 0%2
Boom 1/3 28% 289%2 -3% 100%2
Expected return 𝐸 𝑟 11.00% 7.00%
Variance 𝜎 2
Standard Deviation 𝜎

• Squared deviation of each state:

𝑟𝑖 − 𝐸 𝑟 2 = −7% − 11% 2 = 𝟑𝟐𝟒%𝟐

27
 Variance
Scenario Probability Stock fund Bond fund
Return Squared Return Squared
Deviation Deviation
Recession 1/3 -7% 324%2 17% 100%2
Normal 1/3 12% 1%2 7% 0%2
Boom 1/3 28% 289%2 -3% 100%2
Expected return 𝐸 𝑟 11.00% 7.00%
Variance 𝜎 2 205%𝟐 66.67%2
Standard Deviation 𝜎

• Variance:
3
2 1 1 1
𝜎2 = 𝑝𝑖 𝑟𝑖 − 𝐸 𝑟 = 324% + 1% + 289%2 = 205%2
2 2
3 3 3
𝑖=1
28
 Standard Deviation
Scenario Probability Stock fund Bond fund
Return Squared Return Squared
Deviation Deviation
Recession 1/3 -7% 324%2 17% 100%2
Normal 1/3 12% 1%2 7% 0%2
Boom 1/3 28% 289%2 -3% 100%2
Expected return 𝐸 𝑟 11.00% 7.00%
Variance 𝜎 2 205%2 66.67%2
Standard Deviation 𝜎 14.31% 8.16%

2 2
• Standard deviation: 𝜎 = 𝜎2 = 205%2 = 14.31%

29
 In summary
Scenario Probability Stock fund Bond fund
Return Squared Return Squared
Deviation Deviation
Recession 1/3 -7% 324%2 17% 100%2
Normal 1/3 12% 1%2 7% 0%2
Boom 1/3 28% 289%2 -3% 100%2
Expected return 𝐸 𝑟 11.00% > 7.00%
Variance 𝜎 2 205%2 > 66.67%2
Standard Deviation 𝜎 14.31% > 8.16%

• Expected return: Stock > Bond;

• Standard deviation (risk): Stock > Bond.

30
 Covariance
Scenario Probability Stock fund Bond fund
Return Squared Return Squared
Deviation Deviation
Recession 1/3 -7% 324%2 17% 100%2
Normal 1/3 12% 1%2 7% 0%2
Boom 1/3 28% 289%2 -3% 100%2
Expected return 𝐸 𝑟 11.00% 7.00%
Variance 𝜎 2 205%2 66.67%2
Standard Deviation 𝜎 14.31% 8.16%

Covariance: 𝜎𝑆𝐵 = 3𝑖=1 𝑝𝑖 𝑟𝑖𝑆 − 𝐸 𝑟𝑆 𝑟𝑖𝐵 − 𝐸 𝑟𝐵

1 1
= −7% − 11% 17% − 7% + 12% − 11% 7% − 7%
3 3
1
+ 28% − 11% −3% − 7% = −116.67%2
3 31
 Correlation
Scenario Probability Stock fund Bond fund
Return Squared Return Squared
Deviation Deviation
Recession 1/3 -7% 324%2 17% 100%2
Normal 1/3 12% 1%2 7% 0%2
Boom 1/3 28% 289%2 -3% 100%2
Expected return 𝐸 𝑟 11.00% 7.00%
Variance 𝜎 2 205%2 66.67%2
Standard Deviation 𝜎 14.31% 8.16%

𝜎𝑆𝐵 −116.67%2
Correlation coefficient: 𝜌𝑆𝐵 = = = −0.9991 ≈ −1
𝜎𝑆 𝜎𝐵 14.31%∗8.16%

32
 Given Historical Data
If given historical data (rather than information on future possible
outcomes and probabilities), the formulas for average return, std
deviation and covariance are the following:

 r  r 
T 2
r t

t
t 1
r t 1

T T 1

arithmetic mean

 (r At  rA )(rBt  rB )
 AB  t 1
T 1
where rt are the actual returns
 Example
• Below are DEF Inc’s historical returns over the last 4 years. What is its
average return in a typical year?
Year Return
1 10%
2 -7%
3 28%
4 -11%

• The appropriate average return is the arithmetic mean, also known as

ex-post average return, observed average return and historical average
return.
r1  r2  r3  r4
Average return in a typical year 
4
10%  7%  28%  11% 20%
   5%
4 4
 Example (cont’d)
What is its standard deviation?
Year Return
1 10%
2 -7%
3 28%
4 -11%

4 2
𝑡=1𝑟𝑡 − 𝑟
𝜎=
4−1
10%−5% 2 + −7%−5% 2 + 28%−5% 2 + −11%−5% 2
=
3
5% 2 + −12 2 + 23% 2 + −16% 2
=
3
954%2
= =17.83%
3
Learning Outcomes:
1. Know how to compute the expected return and variance of
individual securities, as well as the covariance and
correlation between 2 securities.
2. Know how to compute portfolio risk and return of 2
securities
3. Appreciate how portfolios can reduce risk and be able to
identify the efficient set of a 2 security portfolio.
4. Apply the same reasoning to a portfolio of many securities
5. Understand the rationale behind the CAPM

36
 Portfolio Return and Risk
Scenario Probability Stock fund Bond fund
Return Squared Return Squared
Deviation Deviation
Recession 1/3 -7% 324%2 17% 100%2
Normal 1/3 12% 1%2 7% 0%2
Boom 1/3 28% 289%2 -3% 100%2
Expected return 𝐸 𝑟 11.00% > 7.00%
Variance 𝜎 2 205%2 > 66.67%2
Standard Deviation 𝜎 14.31% > 8.16%

• Let’s explore a portfolio that is 50% invested in the stock fund and 50%
invested in the bond fund.

37
 Portfolio Return
Scenario Probability Rate of Return
Stock Bond fund Portfolio Squared
fund 50% 50% Deviation
Recession 1/3 -7% 17% 5.0%
Normal 1/3 12% 7% 9.5%
Boom 1/3 28% -3% 12.5%
Expected return 𝐸 𝑟 11.00% 7.00%
Variance 𝜎 2 205%2 66.67%2
Standard Deviation 𝜎 14.31% 8.16%

• In each state, the rate of return on the portfolio is a weighted average of

the return on the stock fund and the bond fund.
• In the Recession state, the portfolio return is 5%:
𝑟𝑃 = 𝑤𝐵 𝑟𝐵 + 𝑤𝑆 𝑟𝑆 = 50% −7% + 50% 17% = 5%
38
 Portfolio Return
Scenario Probability Rate of Return
Stock Bond fund Portfolio Squared
fund 50% 50% Deviation
Recession 1/3 -7% 17% 5.0%
Normal 1/3 12% 7% 9.5%
Boom 1/3 28% -3% 12.5%
Expected return 𝐸 𝑟 11.00% 7.00% 9.0%
Variance 𝜎 2 205%2 66.67%2
Standard Deviation 𝜎 14.31% 8.16%

• The expected rate of return on the portfolio is:

1 1 1
𝐸 𝑟𝑃 = 3 5% + 3 9.5% + 3 12.5% = 9%
• Alternatively, expected rate of return on the portfolio can be computed as the
weighted average of the stock fund and the bond fund. The expected returns on
the stock fund and the bond fund are 11% and 7%:
𝐸 𝑟𝑃 = 𝑤𝑆 𝐸 𝑟𝑆 + 𝑤𝐵 𝐸 𝑟𝐵 = 50% 11% + 50% 7% = 9%
39
 Portfolio Risk
Scenario Probability Rate of Return
Stock Bond fund Portfolio Squared
fund 50% 50% Deviation
Recession 1/3 -7% 17% 5.0% 16%𝟐
Normal 1/3 12% 7% 9.5% 0.25%2
Boom 1/3 28% -3% 12.5% 12.25%2
Expected return 𝐸 𝑟 11.00% 7.00% 9.0%
Variance 𝜎 2 205%2 66.67%2
Standard Deviation 𝜎 14.31% 8.16%

• Squared deviation of each state:

𝑟𝑖𝑃 − 𝐸 𝑟𝑃 2 = 5% − 9% 2
= 16%2

40
 Portfolio Risk
Scenario Probability Rate of Return
Stock Bond fund Portfolio Squared
fund 50% 50% Deviation
Recession 1/3 -7% 17% 5.0% 16%2
Normal 1/3 12% 7% 9.5% 0.25%2
Boom 1/3 28% -3% 12.5% 12.25%2
Expected return 𝐸 𝑟 11.00% 7.00% 9.0%
Variance 𝜎 2 205%2 66.67%2 9.50%𝟐
Standard Deviation 𝜎 14.31% 8.16% 3.08%
• The variance of the portfolio is:
3 1 1 1
2
𝜎𝑃 = 𝑝𝑖𝑃 𝑟𝑖𝑃 − 𝐸 𝑟𝑃 2 = 16%2 + 0.25%2 + 12.25%2 = 9.5%2
𝑖=1 3 3 3
𝜎𝑃 = 3.08%
• Alternatively, the variance of the portfolio can be computed using:
𝜎𝑃2 = (𝑤𝑆 𝜎𝑆 )2 + (𝑤𝐵 𝜎𝐵 )2 + 2 𝑤𝑆 𝜎𝑆 𝑤𝐵 𝜎𝐵 𝜌𝑆𝐵 = 𝑤𝑆2 𝜎𝑆2 + 𝑤𝐵2 𝜎𝐵2 + 2𝑤𝑆 𝑤𝐵 𝜎𝑆 𝜎𝐵 𝜌𝑆𝐵 41
Portfolio Standard Deviation: Alternative Way

= w12 12 + w22 22 + 2 w1 w212

Variance terms Co-variance term

w1 w2
w1 w12 12 w1 w212
 2 2
W2 1 2 12 w2 2
w w
w2
 Portfolio Risk: 3-Asset Portfolio Formula
Variance terms
• Var( rp ) = w12 12 + w22 22 + w32 32
+ 2 w1 w212 + 2 w2 w3 23 + 2 w1 w3 13 Co-variance terms

w1 w2 w3
2
w1 w1  12 w1 w212 w1 w313
W2 w1 w212 2 2
w2 2 w2 w323
w2W3 w w  w w  2 2
1 3 13 2 3 23 w3  3
w3 are 3 variance terms and 3 -3 = 6 covariance terms
Note that there 2

For n asset portfolio, there will be n variance terms and n2-n covariance terms
 Portfolio Risk
Scenario Probability Rate of Return
Stock Bond fund Portfolio Squared
fund 50% 50% Deviation
Recession 1/3 -7% 17% 5.0% 16%2
Normal 1/3 12% 7% 9.5% 0.25%2
Boom 1/3 28% -3% 12.5% 12.25%2
Expected return 𝐸 𝑟 11.00% 7.00% 9.0%
Variance 𝜎 2 205%2 66.67%2 9.50%𝟐
Standard Deviation 𝜎 14.31% 8.16% 3.08%

Alternatively, the variance of the portfolio can be computed using:

𝜎𝑃2 = (𝑤𝑆 𝜎𝑆 )2 + (𝑤𝐵 𝜎𝐵 )2 + 2 𝑤𝑆 𝜎𝑆 𝑤𝐵 𝜎𝐵 𝜌𝑆𝐵 = 𝑤𝑆2 𝜎𝑆2 + 𝑤𝐵2 𝜎𝐵2 + 2𝑤𝑆 𝑤𝐵 𝜎𝑆 𝜎𝐵 𝜌𝑆𝐵
𝜎𝑃2 = 0.52 (205) +0.52 (66.67) + 2 0.5 0.5 (14.31)(8.16)(−1) = 9.5%2
𝜎𝑃 = 3.08%
44
 In summary
Scenario Probability Rate of Return
Stock Bond fund Portfolio Squared
fund 50% 50% Deviation
Recession 1/3 -7% 17% 5.0% 16%2
Normal 1/3 12% 7% 9.5% 0.25%2
Boom 1/3 28% -3% 12.5% 12.25%2
Expected return 𝐸 𝑟 11.00% 7.00% 9.0%
Variance 𝜎 2 205%2 66.67%2 > 9.50%2
Standard Deviation 𝜎 14.31% 8.16% > 3.08%

• bond return < portfolio return < stock return

• Portfolio risk < bond risk < stock risk

45
 In summary
Scenario Probability Rate of Return
Stock Bond fund Portfolio Squared
fund 50% 50% Deviation
Recession 1/3 -7% 17% 5.0% 16%2
Normal 1/3 12% 7% 9.5% 0.25%2
Boom 1/3 28% -3% 12.5% 12.25%2
Expected return 𝐸 𝑟 11.00% 7.00% 9.0%
Variance 𝜎 2 205%2 66.67%2 > 9.50%2
Standard Deviation 𝜎 14.31% 8.16% > 3.08%

• bond return < portfolio return < stock return

• Portfolio risk < bond risk < stock risk
– Does this contradict the famous quote “high risk high return”?

• Diversification reduces the risk. But the return is a weighted average. 46

Risk & Return Part 2
Learning Outcomes:
1. Know how to compute the expected return and variance of
individual securities, as well as the covariance and
correlation between 2 securities.
2. Know how to compute portfolio risk and return of 2
securities
3. Appreciate how portfolios can reduce risk and be able to
identify the efficient set of a 2 security portfolio.
4. Apply the same reasoning to a portfolio of many securities
5. Understand the rationale behind the CAPM

48
 The Efficient Set for Two Securities
% in stocks Risk (σ) Return
• We can combine the stock fund and the bond fund 0% 8.16% 7.00%
5% 7.04% 7.20%
using different weights, e.g., (0%, 100%), (5%, 95%), 10% 5.92% 7.40%
…… 15% 4.80% 7.60%
20% 3.68% 7.80%
25% 2.56% 8.00%
30% 1.44% 8.20%
35% 0.39% 8.40%
40% 0.86% 8.60%
45% 1.97% 8.80%
50% 3.08% 9.00%
55% 4.20% 9.20%
60% 5.32% 9.40%
65% 6.45% 9.60%
70% 7.57% 9.80%
75% 8.69% 10.00%
80% 9.81% 10.20%
85% 10.94% 10.40%
90% 12.06% 10.60%
95% 13.18% 10.80%
100% 14.31% 11.00%
105% 15.43% 11.20%
110% 16.55% 11.40%
115% 17.68% 11.60%
120% 18.80% 11.80%
49
125% 19.92% 12.00%
 The Efficient Set for Two Securities
% in stocks Risk (σ) Return
• We can combine the stock fund and the bond fund 0% 8.16% 7.00%
5% 7.04% 7.20%
using different weights, e.g., (20%, 80%), (0%, 10% 5.92% 7.40%
100%), …… 15% 4.80% 7.60%
20% 3.68% 7.80%
• What weights give the minimum variance or 25% 2.56% 8.00%
30% 1.44% 8.20%
minimum
14.00%
standard dev portfolio? 35% 0.39% 8.40%
40% 0.86% 8.60%
45% 1.97% 8.80%
12.00%
50% 3.08% 9.00%
55% 4.20% 9.20%
10.00%
60% 5.32% 9.40%
Portfolio Return

65% 6.45% 9.60%

8.00%
100% stock fund 70% 7.57% 9.80%
75% 8.69% 10.00%
6.00% 80% 9.81% 10.20%
85% 10.94% 10.40%
100% bond fund 90% 12.06% 10.60%
4.00%
95% 13.18% 10.80%
100% 14.31% 11.00%
2.00% 105% 15.43% 11.20%
110% 16.55% 11.40%
0.00% 115% 17.68% 11.60%
0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 14.00% 16.00% 18.00% 20.00% 22.00% 120% 18.80% 11.80%
Portfolio Risk (standard deviation σ) 50
125% 19.92% 12.00%
Derivation of formula
 Workings

 B 2   B S 66.67  (1)(14.31)(8.16) 66.67  116.77 183.44

xs  2     0.3631
 S   B 2  2  B S 205  66.67  2(1)(14.31)(8.16) 205  66.67  233.54 505.21
 The Efficient Set for Two Securities
% in stocks Risk (σ) Return
Portfolio Risk and Return 0% 8.16% 7.00%
14.00% 5% 7.04% 7.20%
10% 5.92% 7.40%
12.00% 15% 4.80% 7.60%
20% 3.68% 7.80%
10.00% 25% 2.56% 8.00%
100% stock fund
Portfolio Return

30% 1.44% 8.20%

8.00% 35% 0.39% 8.40%
40% 0.86% 8.60%
6.00%
100% bond fund 45% 1.97% 8.80%
50% 3.08% 9.00%
4.00% 55% 4.20% 9.20%
60% 5.32% 9.40%
2.00% 65% 6.45% 9.60%
70% 7.57% 9.80%
0.00% 75% 8.69% 10.00%
0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 14.00% 16.00% 18.00% 20.00% 22.00%
80% 9.81% 10.20%
Portfolio Risk (standard deviation σ) 85% 10.94% 10.40%
90% 12.06% 10.60%
95% 13.18% 10.80%
• Some portfolios are “better” than others, as they 100% 14.31% 11.00%
have higher returns for the same level of risk. 105% 15.43% 11.20%
110% 16.55% 11.40%
• These portfolios comprise the efficient frontier. 115% 17.68% 11.60%
120% 18.80% 11.80%
53
125% 19.92% 12.00%
 Two-Security Portfolios with Various Correlations (ρ)
• The risk-return relation of
a two-security portfolio
depends on the
return

100%
correlation coefficient:
 = -1.0 stocks -1.0 <  < +1.0

• If  = +1.0, no risk
 = 1.0 reduction is possible.

100%
bonds

54
 Portfolios – Correlation Coefficient
 The correlation coefficient between two stocks (X and Y) , denoted by ρXY
measures the extent to which two securities X and Y move together

Rho – pronounced “roe”

ρXY = -1 ρXY = 0

In general
1  XY  -1

Perfectly Perfectly
positively negatively
ρXY = 0 correlated correlated
ρXY = +1

 The variance of a portfolio depends on the correlation between the assets

included in the portfolio.
55
 Workings

σ P2  (wB σ B )2  (wS σ S )2  2(wB σ B )(wS σ S )ρBS

 wB σ B  wS σ S  2wB σ B wS σ S ρBS
2 2 2 2

 wB σ B  wS σ S  2wB wS σ B σ S ρBS
2 2 2 2

when   1,
σ P2  wB σ B  wS σ S  2wB wS σ B σ S  ( wB σ B  wS σ S ) 2
2 2 2 2

σ p  ( wB σ B  wS σ S )

𝑟𝑃 = 𝑤𝐵 𝑟𝐵 + 𝑤𝑆 𝑟𝑆
56
 Workings

σ P2  (wB σ B )2  (wS σ S )2  2(wB σ B )(wS σ S )ρ BS

 wB 2 σ B 2  wS 2 σ S 2  2wB σ B wS σ S ρ BS
 wB 2 σ B 2  wS 2 σ S 2  2wB wS σ B σ S ρ BS

when   1,
σ P2  wB 2 σ B 2  wS 2 σ S 2  2wB wS σ B σ S  ( wB σ B  wS σ S ) 2
σ p  ( wB σ B  wS σ S )

when   1,
σ P2  wB 2 σ B 2  wS 2 σ S 2  2wB wS σ B σ S  ( wB σ B  wS σ S ) 2
σ p  ( wB σ B  wS σ S )

57
 Two-Security Portfolios with Various Correlations (ρ)
• The risk-return relation of
a two-security portfolio
depends on the
return

100%
correlation coefficient:
 = -1.0 stocks -1.0 <  < +1.0

• If  = +1.0, no risk
 = 1.0 reduction is possible.

100%
 = 0.2
bonds
• The smaller the
correlation, the greater
 the risk reduction
potential.

58
Learning Outcomes:
1. Know how to compute the expected return and variance of
individual securities, as well as the covariance and
correlation between 2 securities.
2. Know how to compute portfolio risk and return of 2
securities
3. Appreciate how portfolios can reduce risk and be able to
identify the efficient set of a 2 security portfolio.
4. Apply the same reasoning to a portfolio of many securities
5. Understand the rationale behind the CAPM

59
Portfolio Risk as a Function of the Number of Securities in the Portfolio
In a large portfolio, the nonsystematic risk are effectively diversified

away, but the systematic risk are not.

Portfolio risk

Nondiversifiable risk;
Systematic Risk; Market Risk
n

60
Portfolio Risk as a Function of the Number of Securities in the Portfolio
In a large portfolio, the nonsystematic risk are effectively diversified

away, but the systematic risk are not.

Diversifiable Risk;
Nonsystematic Risk; Firm
Specific Risk; Unique Risk

Portfolio risk

Nondiversifiable risk;
Systematic Risk; Market Risk
n

Diversification can eliminate firm specific risk but not market wide risk.
Investors are only rewarded for bearing systematic risk.
61
 Question

• Which one of the following is an example of a nondiversifiable risk?

– a well-respected president of a firm suddenly resigns
– a well-respected chairman of the Federal Reserve suddenly resigns
– a key employee suddenly resigns and accepts employment with a key
competitor
– a well-managed firm reduces its work force and automates several
jobs
– a poorly managed firm suddenly goes out of business due to lack of
sales

62
 Question

• Which one of the following is an example of a nondiversifiable risk?

– a well-respected president of a firm suddenly resigns
– a well-respected chairman of the Federal Reserve suddenly resigns
– a key employee suddenly resigns and accepts employment with a key
competitor
– a well-managed firm reduces its work force and automates several
jobs
– a poorly managed firm suddenly goes out of business due to lack of
sales

63
 The Efficient Set for Many Securities

return

Individual
Assets

P
Consider a world with many risky assets; we can still identify
the opportunity set of risk-return combinations of various
64
portfolios.
 The Efficient Set for Many Securities

return

Individual
Assets

P
Consider a world with many risky assets; we can still identify
the opportunity set of risk-return combinations of various
65
portfolios.
 The Efficient Set for Many Securities

return
minimum
variance
portfolio

Individual Assets

P
Given the opportunity set, we can identify the minimum
variance or minimum standard deviation portfolio.
 The Efficient Set for Many Securities

return
minimum
variance
portfolio

Individual Securities

P
The section of the opportunity set from the minimum
variance portfolio and up is the efficient frontier.
 Riskless Borrowing and Lending

return
rf


In addition to risky securities, consider a world that also has a
risk-free security.
In practice, we use the yield on the 10 year government bond as
the risk-free rate.
 Riskless Borrowing and Lending

return

rf

P
Now an investor can allocate his money between the risk free
security and a portfolio of risky securities on the
opportunity set.
 Riskless Borrowing and Lending

return
M

rf

P
With a risk free security and the efficient frontier identified,
the investor will choose the capital allocation line with the
steepest slope, i.e., combinations of 𝑟𝑓 and M.
Learning Outcomes:
1. Know how to compute the expected return and variance of
individual securities, as well as the covariance and
correlation between 2 securities.
2. Know how to compute portfolio risk and return of 2
securities
3. Appreciate how portfolios can reduce risk and be able to
identify the efficient set of a 2 security portfolio.
4. Apply the same reasoning to a portfolio of many securities
5. Understand the rationale behind the CAPM

71
 Key Assumptions

• If all investors have the same perception regarding the

probability distribution of risky securities (homogeneous
expectations)
and
• if they can borrow and lend at the same rate (same 𝑟𝑓 )

then
• they will agree on the same risky portfolio M, i.e., all investors
have the same capital allocation line (going through 𝑟𝑓 and M).

72
 Market Equilibrium
All investors have the same capital
allocation line. This line is the Capital
return

Market Line (CML).

Where a specific investor chooses to
invest along the CML depends on his
Optimal risk tolerance.
Risky
Portfolio M

rf


 Market Equilibrium
All investors have the same capital
allocation line. This line is the Capital
return

Market Line (CML).

Where a specific investor chooses to
invest along the CML depends on his
Optimal risk tolerance.
Risky
Portfolio M
The Separation Property states that the
rf
market portfolio, M, is the same for all
investors— that is, the choice of the
market portfolio is separate from the
 investors’ risk tolerance.
 Equation of CML
CML is a straight line y = mx + c
return

• 2 points: risk free asset and M

M – Risk free asset (0, 𝑟𝑓 )
E(rm)
– M (𝜎𝑚 , E(𝑟𝑚 ))
E(ri) 𝐸 𝑟𝑚 −𝑟𝑓
rf • Slope =
𝜎𝑚 −0
• Intercept =𝑟𝑓
𝜎𝑖 𝜎𝑚  • Y = mx + c
𝐸 𝑟𝑚 −𝑟𝑓
• 𝐸 𝑟𝑖 = (σi ) + 𝑟𝑓
𝜎𝑚 −0
𝐸 𝑟𝑚 −𝑟𝑓
• 𝐸 𝑟𝑖 = 𝑟𝑓 + 𝜎𝑖
𝜎𝑚 75
 Equation of CML

𝐸 𝑟𝑚 −𝑟𝑓
CML: 𝐸 𝑟𝑖 = 𝑟𝑓 + 𝜎𝑖
return

𝜎𝑚

M
E(rm) • CML applies only to 2
E(ri) benchmark securities: the
rf market portfolio (M) and the
risk free security.

𝜎𝑖 𝜎𝑚  • It does not apply to individual

securities or just any portfolio!!

76
 Capital Asset Pricing Model (CAPM)

• The CML does not say anything about the expected returns on
individual securities or other portfolios.
• We will now turn to a relationship that does so --- CAPM.
• The CAPM is mathematically derived from the CML.

From our earlier discussion,

• The market rewards investors only for holding systematic risk.
• Beta measures the systematic risk.

77
 Risk in a Diversified Portfolio

• The best measure of the risk of a security in a large portfolio is

the beta (b) of the security.
• Beta measures the responsiveness of a security to movement
in the market portfolio.

𝐶𝑜𝑣 𝑅𝑖 , 𝑅𝑀
𝛽𝑖 =
𝜎 2 R𝑀

– Clearly, the estimate of beta will depend on the choice of a

proxy for the market portfolio.

78
Beta of the market portfolio

cov(rm , rm )  2r
b m arket   2 1 m

 2r m  r m

Beta of the risk free asset

cov(rf , rm ) 0
b riskfree   0
 2
rm  2
rm

79
 CAPM
• In equilibrium, all securities and portfolios must have the
same reward-to-risk ratio and that must equal the reward-to-
risk ratio for the market portfolio.

𝐸 𝑅𝑖 − 𝑅𝑓 𝐸 𝑅𝑀 − 𝑅𝑓
=
𝛽𝑖 𝛽𝑀

• As βM = 1 by definition, rearranging the above gives the CAPM

equation
𝐸 𝑅𝑖 −𝑅𝑓 𝐸 𝑅𝑀 −𝑅𝑓
𝛽𝑖
= 1

𝐸 𝑅𝑖 = 𝑅𝑓 + 𝛽𝑖 𝐸 𝑅𝑀 − 𝑅𝑓 80
 Expected Return on any Security
• This equation is called the Capital Asset Pricing Model
or Security Market Line
𝐸 𝑅𝑖 = 𝑅𝑓 + 𝛽𝑖 𝐸 𝑅𝑀 − 𝑅𝑓
Expected
Riskfree Beta of Market risk
return on = + ×
rate security premium
any
security

• If bi = 0, then the expected return is Rf.

• If bi = 1, then the expected return is 𝐸 𝑅𝑀 .
CAPM or Security Market Line (SML)

SML: 𝐸 𝑅𝑖 = 𝑅𝑓 + 𝛽𝑖 𝐸 𝑅𝑀 − 𝑅𝑓

82
Estimating b with regression
Model: (Ri -Rf)= a i + bi (Rm -Rf )+ ei

Excess Security Returns

𝐸 𝑅𝑖 − 𝑅𝑓 Slope = bi
Excess Return on
market %
𝐸 𝑅𝑀 − 𝑅𝑓
Impact of Inflation Change on SML

84
Impact of a Risk Aversion Change

What happens when investors become more risk averse?

Slope steepens as risk

premium increases

85
 More on β (Systematic Risk Measure)
• In summary, how do we measure systematic risk?
– We use the beta coefficient to measure systematic risk
• What does beta tell us?
 A beta = 1 implies the asset has the same systematic risk as
the overall market
 A beta < 1 implies the asset has less systematic risk than the
overall market
 A beta > 1 implies the asset has more systematic risk than
the overall market

86
 Example: Total Risk versus Systematic Risk
• Consider the following information:
Standard Deviation Beta
Security C 20% 1.25
Security K 30% 0.95

• Which security has more total risk? K

• Which security has more systematic risk? C
• Which security should have the higher expected return? C

87
Calculate the expected returns and std deviation
based on possible returns and probabilities

Calculate the expected returns and standard deviations of the

above investment alternatives.
88
 Expected Return of Alta

n
rˆAlta   ri Pi
i 1

=0.1 (-22%) +0.2 (-2%) +

0.4 (20%) +0.2 (35%) +
0.1 (50%)

= 17.4%
 Standard Deviation of Alta

n 2

 Alta   P r  rˆ 
i 1
i i

σ 2 Alta  0.1 (-22% -17.4%)2 +

0.2 (-2% -17.4 %)2 +
0.4 (20% -17.4%)2 +
0.2 (35% -17.4%)2 +
0.1 (50% -17.4%)2
= 401.44%2

s Alta = 20%
Expected Returns & Standard Deviations
given possible returns and probabilities

Investment r̂ 

Alta 17.4% 20%

Market 15.0 15.3

Am. F. 13.8 18.8

T-bonds 8.0 0

Repo Men 1.7 13.4

 Calculate the required return given beta

Investment r̂ beta
Alta 17.4% 1.29
Market 15.0 1.00
Am. F. 13.8 0.68
T-bonds 8.0 0.00
Repo Men 1.7 -0.86

Given a risk free rate of 8% and market risk premium of 7%, what are the
required returns of the various alternatives, given the betas?
 Apply CAPM

• Given a risk free rate of 8% and market risk premium of

7% and a beta of 1.29 for Alta

𝐸 𝑅𝐴𝑙𝑡𝑎 = 𝑅𝑓 + 𝛽𝐴𝑙𝑡𝑎 𝐸 𝑅𝑀 − 𝑅𝑓
= 8% + 1.29 (7%)
= 8% + 9.03
= 17.03%
≈ 17.0%
 Expected Returns & Required Returns

Required
Investment r̂ return Attractive?

Alta 17.4% 17.0% Underpriced

Market 15.0 15.0
Am. F. 13.8 12.8
T-bonds 8.0 8.0
Repo Men 1.7 2.0

►For a fairly priced asset, the expected return is on the SML.

►For an underpriced asset, the expected return is above the SML.
►For an overpriced asset, the expected return is below the SML.
Alta has an expected
. return of 17.4% which is higher than what is required based on its beta
of 1.29 (which gives a required return of 17%).

This is an underpriced asset. Its expected return is above the SML.

Everybody wants to hold Alta. That will bid up the price P0. That means P0 is larger
causing (P1-P0) to be smaller.

The return will therefore be lower: (P1-P0)/P0 = smaller/larger = a smaller number.

That is, Alta's expected return will move down towards the SML. In equilibrium, all investments
95
will be on SML.
 Expected Returns & Required Returns

Required
Investment r̂ return Attractive?

Alta 17.4% 17.0% Underpriced

Market 15.0 15.0 Fairly Valued
Am. F. 13.8 12.8 Underpriced
T-bonds 8.0 8.0 Fairly Valued
Repo Men 1.7 2.0 Overpriced

►For a fairly priced asset, the expected return is on the SML.

►For an underpriced asset, it would be above the SML.
►For an overpriced asset, it would be below the SML.
Portfolio βs – Portfolio Systematic Risk Measure
• Given a large number (m) of assets in a portfolio, we would
multiply each asset’s beta by its portfolio weight and then
sum up the results to get the portfolio beta:

m
b p   wj b j
j 1
• Consider the following four securities in a portfolio:

Security Weight Beta

DCLK 0.133 3.69
KO 0.2 0.64
INTC 0.267 1.64
KEI 0.4 1.79

What is the portfolio beta (βP)?

98
Example: Portfolio Betas

• Consider the following four securities in a portfolio:

Security Weight Beta w i βi
DCLK 0.133 3.69 0.491
KO 0.2 0.64 0.128
INTC 0.267 1.64 0.438
KEI 0.4 1.79 0.716
portfolio beta (βP) = 1.77

99
 Finding Betas

• One source of betas is Yahoo Finance

• Try it out: Visit http://finance.yahoo.com/
– Enter a ticker symbol and get a basic quote
– Click on Key Statistics
• Compare the betas of
• Pepsi (PEP), Coca-Cola (KO)
• General Motors (GM), Honda (HMC), Toyota Motor (TM)
• Apple (AAPL), Samsung Electronics (SSNLF)
• Alibaba (BABA), Amazon (AMZN)

100
PepsiCo, Inc (PEP)

101
Coca-Cola Company (KO)

102
General Motors Company (GM)

103
Honda (HMC)

104
 Application to Capital Budgeting: Cost of Capital
• Learning Outcomes:
1. Understand why the risk and return concept (so far applied only
to financial securities) can be applied to capital budgeting
2. Know how to compute the cost of capital if project is an
expansion of the firm (has same risk) and the firm uses only
equity
3. Know how to compute the cost of capital if project is an
expansion of the firm (has same risk) and the firm uses both debt
and equity
4. Know how to compute the cost of capital if project has different
risk from the firm

105
 Application to Capital Budgeting
Firm with Shareholder
Pay cash dividend
excess cash getting dividends

A firm with excess cash can either make a

capital investment or pay a dividend

Invest in project Shareholder’s Invest in

Terminal Value financial asset

Because stockholders can reinvest the dividend in risky financial

securities, the expected return on a capital-budgeting project
should be at least as great as the expected return on a financial
security of comparable risk.
106
Learning Outcomes:
1. Understand why the risk and return concept (so far applied only
to financial securities) can be applied to capital budgeting
2. Know how to compute the cost of capital if project is an
expansion of the firm (has same risk) and the firm uses only
equity
3. Know how to compute the cost of capital if project is an
expansion of the firm (has same risk) and the firm uses both debt
and equity
4. Know how to compute the cost of capital if project has different
risk from the firm

107
 Cost of Capital for an expansion project
of an all-equity firm

• Suppose the stock of Stansfield Enterprises, a publisher of

PowerPoint presentations, has a beta of 1.5. The firm is
100% equity financed.

• Assume a risk-free rate of 3% and a market risk premium

of 6%.

• What is the appropriate discount rate for an expansion of

this firm?

108
• Appropriate discount rate for the expansion is
the Cost of Equity Capital (CAPM):

R i  R F  β i (R M  R F )

R i  3 %  1 .5  6 %  12 %
Learning Outcomes:
1. Understand why the risk and return concept (so far applied only
to financial securities) can be applied to capital budgeting
2. Know how to compute the cost of capital if project is an
expansion of the firm (has same risk) and the firm uses only
equity
3. Know how to compute the cost of capital if project is an
expansion of the firm (has same risk) and the firm uses both debt
and equity
4. Know how to compute the cost of capital if project has different
risk from the firm

110
Cost of Capital when the project is an expansion
and the firm has debt
• Firstly, estimate cost of equity and cost of debt.
– need equity beta to estimate cost of equity (CAPM)
– use yield to maturity (YTM) to estimate cost of debt
• Secondly, compute WACC by weighting these two
costs appropriately:
𝑆 𝐵
𝑟𝑊𝐴𝐶𝐶 = ∗ 𝑟𝑆 + ∗ 𝑟𝐵 ∗ (1 − 𝑇𝑐 )
𝑆+𝐵 𝑆+𝐵
– Where Tc is the corporate tax rate. We multiply the
last term by (1-Tc) because interest is tax deductible.
 Example
• International Paper has equity beta of 1.2, risk free rate of 5%
and market risk premium of 5%.
• The cost of equity capital is ri  RF  βi ( R M  RF )
 5 %  1 .2  5 %
 11%

• Yield on debt = 6%, tax rate = 17%, debt to value ratio = 32%
𝑆 𝐵
𝑟𝑊𝐴𝐶𝐶 = ∗ 𝑟𝑆 + ∗ 𝑟𝐵 ∗ (1 − 𝑇𝑐 )
𝑆+𝐵 𝑆+𝐵

= 0.68 x 11% + 0.32 x 6% x (1-.17)

= 9.0736%

• If project’s risk and leverage are the same as the firm, then its 112

cost of capital=9.0736%.
Learning Outcomes:
1. Understand why the risk and return concept (so far applied only
to financial securities) can be applied to capital budgeting
2. Know how to compute the cost of capital if project is an
expansion of the firm (has same risk) and the firm uses only
equity
3. Know how to compute the cost of capital if project is an
expansion of the firm (has same risk) and the firm uses both debt
and equity
4. Know how to compute the cost of capital if project has different
risk from the firm

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 Extension of the Basic Model:
project has different risk from the firm
• What discount rate to use?
– The cost of capital depends on the use to which the capital
is being put (project) —not the source (firm).
– It depends on the risk of the project .

• If a firm’s cost of capital is 15%, should it only accept projects

with return ≥ 15%?
– No
– Accepting projects with return ≥ cost of capital is known as
the cost of capital rule.

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 Project
return SML
Incorrectly accepted
negative NPV projects
Hurdle
rate
Incorrectly rejected
rf positive NPV projects

Risk of project

Thus, a firm that uses one discount rate for all projects will
have a lower value over time.

Note: In capital budgeting, hurdle rate is the minimum rate that a company expects from a
115
project. Under this COST OF CAPITAL RULE, a project will only be accepted if its return is higher
than the hurdle rate.
 Example: Assume an all equity firm
• Conglomerate Company has risk-free rate of 5%, market
risk premium of 8% and equity beta of 1.3. According to
CAPM, the cost of equity capital is:
5% + 1.3 (8%) =15.4%

Breakdown of the company’s investment projects:

1/3 Automotive retailer b = 2.0
1/3 Computer Hard Drive Mfr. b = 1.3
1/3 Electric Utility b = 0.6
Average b of assets = 1.3
When evaluating a new electrical utility investment, which discount rate
should be used? Use b = 0.6 116
Project IRR SM L
21%
Investments in hard
drives and auto retailing
15.4%
should have higher
9.8% discount rates, according
to CAPM.

Risk (beta)
0.6 1.3 2.0
E.U. H.D. Auto.
9.8% reflects the opportunity cost of capital on an investment in
electrical utility.

5% + 0.6 (8%) =9.8% 117