MBA 8480 - Portfolio Theory and Asset Pricing

1. Portfolio Theory &
Asset Pricing
Professor Mike Pagano
michael.pagano@villanova.edu

2. Magnitude of cash flows expected by shareholders
Riskiness of the cash flows
Timing of the cash flow stream
2
Key Factors that Affect Stock Price
“M.R.T.”

3. 3
Value = + + +
FCF1 FCF2 FCF∞
(1 + WACC)1 (1 + WACC)∞(1 + WACC)2
Free cash flow
(FCF)
Market interest rates
Firm’s business riskMarket risk aversion
Firm’s debt/equity mixCost of debt
Cost of equity
Weighted average
cost of capital
(WACC)
Net operating
profit after taxes
Required investments
in operating capital
−
=
Determinants of Intrinsic Value: The Cost of Equity
...

4. 4
What is investment risk?
Investment risk is exposure to the chance of earning less than
expected.
The greater the chance of a return far below the expected
return, the greater the risk.

5. 5
Scenarios and Returns for a 10-Year Zero Coupon
T-bond Over the Next Year
Scenario Probability Return
Worst Case 0.10 −14%
Poor Case 0.20 −4%
Most Likely 0.40 6%
Good Case 0.20 16%
Best Case 0.10 26%
1.00

6. Discrete Probability Distribution for Scenarios
0.0
0.1
0.2
0.3
0.4
-14% -4% 6% 16% 26%
Probability
Returns
6

7. Example of a Continuous Probability Distribution
-30% -20% -10% 0% 10% 20% 30% 40%
Returns
7

8. Calculate the expected rate of return r on the bond
𝐫 = 𝐢=𝟏
𝐧
𝐩𝐢 𝐫𝐢
r = 0.10(-14%) + 0.20(-4%) + 0.40(6%)
+ 0.20(16%) + 0.10(26%)
𝐫 = 6%
8

9. Consider these probability distributions for two investments.
Which riskier? Why?
-30% -20% -10% 0% 10% 20% 30% 40%
Return
9

10. Portfolio Returns
The percentage of a portfolio’s value that is invested in Stock i
is denoted by the “weight” wi.
Notice that the sum of all the weights must equal 1.
With n stocks in the portfolio, its return each year will be:
rp,t =
i=1
n
wi ri,t
10

11. Example: 2-Stock Portfolio
Form a portfolio by selling 25% of the Blandy stock and
investing it in the higher-risk Gourmange stock.
The portfolio return each year will be:
– rP,t = wBlandy rBlandy,t + wGour. rGour.,t
– rP,t = 𝟎. 𝟕𝟓 rBlandy,t + 𝟎. 𝟐𝟓 rGour.,t
11

12. 12
Historical Data for Stocks and Portfolio Returns
Year Blandy Gourmange
Portfolio of Blandy and
Gourmange
1 26% 47% 31.3%
2 15 −54 −2.3
3 −14 15 −6.8
4 −15 7 −9.5
5 2 −28 −5.5
6 −18 40 −3.5
7 42 17 35.8
8 30 −23 16.8
9 −32 −4 −25.0
10 28 75 39.8

13. Portfolio Historical Average and Standard Deviation
The portfolio’s average return is the weighted
average of the stocks’ average returns.
The portfolio’s standard deviation is less than
either stock’s σ!
What explains this? Diversification works!
13
Blandy Gourmange Portfolio
Average return 6.4% 9.2% 7.1%
Standard deviation 25.2% 38.6% 22.2%

14. How closely do the returns follow one another?
Blandy
Gourmange
-75%
-50%
-25%
0%
25%
50%
75%
1 2 3 4 5 6 7 8 9 10
Return
Year
Notice that the returns
don’t move in perfect
lock-step: Sometimes
one is up and the
other is down.
14

15. 15
Correlation Coefficient (ρi,j)
Loosely speaking, the correlation (r) coefficient
measures the tendency of two variables to move
together.
Estimating ρi,j with historical data is tedious:
t=1
T
r𝑖,t − ri,Avg r𝑗,t − rj,Avg
t=1
T
r𝑖,t − ri,Avg
2
t=1
T
r𝑗,t − rj,Avg
2

16. Excel Functions to Estimate the Correlation Coefficient (ρi,j)
“Stocki” and “Stockj” are the cell ranges with historical returns
for Stocks i and j.
Can use the =Correl(x,y) command in Excel:
Est. ρi,j = Rij =Correl(Stocki,Stockj)
Correlation between Blandy (B) and Gourmange (G):
Est. ρB,G = 0.11
16

17. 17
2-Stock Portfolios
r = −1
– 2 stocks can be combined to form a riskless
portfolio: σp = 0.
r = +1
– Risk is not “reduced”
– σp is just the weighted average of the 2 stocks’
standard deviations.
−1 < r < −1
– Risk is reduced but not eliminated.

18. 18
Adding Stocks to a Portfolio
What would happen to the risk of an average 1-stock portfolio
as more randomly selected stocks were added?
sp would decrease because the added stocks would not be
perfectly correlated (the benefits of diversification).

19. 19
Risk vs. Number of Stocks in Portfolio
10 20 30 40 2,000 stocks
Company Specific
(Diversifiable) Risk
Market Risk
20%
0
Total Portfolio Risk, sp
sp
35%

20. 20
Total risk = Market risk + Diversifiable risk
Market risk is that part of a security’s stand-alone risk that
cannot be eliminated by diversification.
Firm-specific, or diversifiable, risk is that part of a security’s
stand-alone risk that can be eliminated by diversification.

21. 21
Insights from Portfolio Theory
As more stocks are added, each new stock has a
smaller risk-reducing impact on the portfolio.
sp falls very slowly after about 40 stocks are included.
The lower limit for sp is about 20% = sM
By forming well-diversified portfolios, investors can
eliminate about half the risk of owning a single
stock.

22. 22
Portfolio Theory
Suppose Asset A has an expected return of 10 percent
and a standard deviation of 20 percent.
Asset B has an expected return of 16 percent and a
standard deviation of 40 percent.
If the correlation between A and B is 0.35, what are
the expected return and standard deviation for a
portfolio comprised of 30 percent Asset A and 70
percent Asset B?

23. 23
Portfolio Expected Return
rp = wArA + (1 – wA) rB
^ ^ ^
= 0.3(0.10) + 0.7(0.16)
= 0.142 = 14.2%

24. 24
Portfolio Standard Deviation
σP = √w2
Aσ2
A + (1-wA)2σ2
B + 2wA(1-wA)ρABσAσB
= √0.32(0.22) + 0.72(0.42) + 2(0.3)(0.7)(0.35)(0.2)(0.4)
= 0.306

25. 25
Attainable Portfolios: rAB = 0.35
r AB = +0.35: Attainable Set of
Risk/Return Combinations
0%
5%
10%
15%
20%
0% 10% 20% 30% 40%
Risk, sp
Expectedreturn

26. 26
Attainable Portfolios: rAB = +1.0
r AB = +1.0: Attainable Set of Risk/Return
Combinations
0%
5%
10%
15%
20%
0% 10% 20% 30% 40%
Risk, p
Expectedreturn
s

27. 27
Attainable Portfolios: rAB = -1.0
r AB = -1.0: Attainable Set of Risk/Return
Combinations
0%
5%
10%
15%
20%
0% 10% 20% 30% 40%
Risk, s p
Expectedreturn

28. 28
Attainable Portfolios with a Risk-Free Asset
(Expected risk-free return = 5%)
Attainable Set of Risk/Return
Combinations with Risk-Free Asset
0%
5%
10%
15%
0% 5% 10% 15% 20%
Risk, sp
Expectedreturn

29. 29
Expected
Portfolio
Return, rp
Risk, sp
Efficient Set
Feasible Set
Feasible and Efficient Portfolios

30. 30
Feasible and Efficient Portfolios
The feasible set of portfolios represents all portfolios
that can be constructed from a given set of stocks.
An efficient portfolio is one that offers:
– the most return for a given amount of risk, or
– the least risk for a given amount of return.
The collection of efficient portfolios is called the
efficient set or efficient frontier.

31. 31
What is the CAPM?
The CAPM is an equilibrium model that specifies the
relationship between risk and required rate of return for
assets held in well-diversified portfolios.
It is based on the premise that only one factor affects risk.
What is that factor?

32. 32
What are the key assumptions of the CAPM?
Investors all think in terms of a single holding period.
All investors have identical expectations.
Investors can borrow or lend unlimited amounts at the risk-free
rate.
(More...)

33. 33
CAPM Assumptions (Cont.)
All assets are perfectly divisible.
There are no taxes and no transactions costs.
All investors are price takers, that is, investors’ buying and
selling won’t influence stock prices.
Quantities of all assets are given and fixed.

34. 34
What impact does rRF have on
the efficient frontier?
When a risk-free asset is added to the feasible set, investors
can create portfolios that combine this asset with a portfolio
of risky assets (and improves the risk-return trade-off).
The straight line connecting rRF with M, the tangency point
between the line and the old efficient set, becomes the new
efficient frontier.

35. 35
M
Z
.A
rRF
sM Risk, sp
The Capital Market
Line (CML):
New Efficient Set
.
.B
rM
^
Expected
Return, rp
Efficient Set with a Risk-Free Asset

36. 36
What is the Security Market Line (SML)?
The CML gives the risk/return relationship for efficient
portfolios.
The Security Market Line (SML), also part of the CAPM, gives
the risk/return relationship for individual stocks.

37. 37
The SML Equation
The measure of risk used in the SML is the beta coefficient of
company-i, bi.
The Security Market Line (SML) equation:
ri = rRF + (RPM) bi

38. 38
Using a Regression to Estimate Beta
Run a regression with returns on the stock plotted on the Y-axis
and returns on the market portfolio plotted on the X-axis.
The slope of the regression line is equal to the stock’s beta
coefficient.

39. 39
Method of Calculation
Analysts use a computer with statistical or Excel spreadsheet
software to perform the regression.
– At least 3 years of monthly returns or 1 year’s of
weekly (or daily) returns are used.
– Many analysts use 5 years of monthly returns.

40. Excel: Plot Trendline Right on Chart
y = 0.6027x + 0.0158
R² = 0.2316
-0.45
0
0.45
-0.45 0 0.45
Blandy Returns
Market
Returns
40

41. 41
Estimated Beta from Regression
The trendline is plotted on the previous slide, including the
regression equation.
– y = 0.6027x + 0.0158
– y = Blandy’s stock returns
– x = Broad Stock Market returns (e.g., S&P 500)
– b = Slope = 0.6027 (same as before)
Much Easier way—use the Excel SLOPE function.
– Beta = b =SLOPE(y_values,x_values)

42. 42
Interpreting Regression Results
If beta = 1.0, stock is average risk.
If beta > 1.0, stock is riskier than average.
If beta < 1.0, stock is less risky than average.
Most stocks have betas in the range of 0.5 to 1.5.
The R2 measures the percent of a stock’s variance that is
explained by the market. The typical R2 is:
– 0.30 for an individual stock
– over 0.90 for a well diversified portfolio

43. 43
Interpreting Regression Results (Cont.)
The 95% confidence interval shows the range in which we are
95% sure that the true value of beta lies. The typical range
is:
– from about 0.5 to 1.5 for an individual stock
– from about 0.9 to 1.1 for a well diversified
portfolio

44. 44
Web Sites for Beta
http://finance.yahoo.com
– Enter the ticker symbol for a “Stock Quote”, such
as IBM or Dell, then click GO.
– When the quote comes up, select Key Statistics
from panel on left.
www.valueline.com
– Enter a ticker symbol at the top of the page.
Can also access raw return data from sites such as WRDS’s CRSP
return database (1926-present).

45. Calculate the weights for a portfolio with $1.4 million in
Blandy and $0.6 million in Gourmange:
Find the weights based on total portfolio value of $2 million:
– wB = $1.4/($1.4+$0.6) = 70%
– wG = $0.6/($1.4+$0.6) = 30%
The portfolio beta is the weighted average of the stocks’ betas:
𝐛 𝐩 =
𝐢=𝟏
𝐧
𝐰𝐢 𝐛𝐢
45

46. 46
Calculate the portfolio beta:
bp = 0.7(bBlandy) + 0.3(bGour.)
= 0.7(0.60) + 0.3(1.30)
= 0.81

47. 47
What is the Required Return on the Portfolio?
(1) Can use SML with portfolio beta:
rp = rRF + bp (RPM)
= 4.0% + 0.81%(5%) = 8.05%
or,
(2) Can use fact that rp= 𝐢=𝟏
𝐧
𝐰𝐢 𝐫𝐢
rp= 0.7(7.0%) + 0.3(10.5%)
= 8.05%

48. 48
s2 = b2 s2 + se
2.
s2 = variance
= total risk of Stock j.
b2 s2 = market risk of Stock j.
se
2 = variance of error term
= diversifiable risk of Stock j.
j j M j
j
j
j M
What is the relationship between total, market, and
diversifiable risk?

49. 49
Market Risk vs. Diversifiable Risk
10 20 30 40 2,000 stocks
Company Specific
(Diversifiable) Risk, se
Market Risk, sM
20%
0
Total Portfolio Risk, sp
sp
35%

50. 50
What are two potential tests that can be conducted
to verify the CAPM?
Beta stability tests
Tests based on the slope of the SML

51. 51
Tests of the SML indicate:
A more-or-less linear relationship between realized returns and
market risk.
Slope is less than predicted.
Irrelevance of diversifiable risk specified in the CAPM model
can be questioned.

52. 52
Tests of the SML indicate:
“Historical” Estimated Betas of individual securities are not
good estimators of future risk.
Can use an “adjusted beta” as follows:
Adjusted Beta = 0.35 + 0.67 x Estimated Beta
Adjusted Beta is a better forward-looking estimate of risk.
Betas of portfolios of 10 or more randomly selected stocks are
reasonably stable.
Past portfolio betas are good estimates of future portfolio
volatility.

53. 53
Are there problems with the CAPM tests?
Yes.
– Richard Roll questioned whether it was even
conceptually possible to test the CAPM.
– Roll showed that it is virtually impossible to prove
investors behave in accordance with CAPM theory.

54. 54
What are our conclusions regarding the CAPM?
It is impossible to verify.
Recent studies have questioned its validity.
Investors seem to be concerned with both market risk and
total, stand-alone risk. Therefore, the SML may not produce
a correct estimate of ri.

55. 55
What are our conclusions
regarding the CAPM? (cont.)
CAPM/SML concepts are based on expectations, yet
betas are calculated using historical data. A
company’s historical data may not reflect investors’
expectations about future riskiness.
Other models are being developed that will one day
replace the CAPM, but it still provides a good
framework for thinking about risk and return.

56. 56
What is the difference between the CAPM and the
Arbitrage Pricing Theory (APT)?
The CAPM is a single factor model.
The APT proposes that the relationship between risk and return
is more complex and may be due to multiple factors such as
GDP growth, expected inflation, tax rate changes, and
dividend yield.
Other multi-factor models such as the 3-factor “Fama-French”
model can be used to create more precise estimates of
expected return (includes Rm – Rf; SMB, and HML).