Ch5 Portfolio Theory -Risk and Return Liang (Kevin) Guo Learning Objectives Be able to calculate ex post and ex ante risk and return statistical measures, such as holding period return, average returns, expected returns, and standard deviation. Understand the difference between time-weighted and dollar-weighted returns, geometric and arithmetic averages. Be able to construct portfolios of different risk levels, given information about risk free rates and returns on risky assets. Be able to explain the CML theory. Table of Contents 5.1 Rates of Return 5.2 Risk and Risk Premiums 5.3 Inflation and Real Rates of Return 5.4 Asset Allocation Across Risky and Risk Free Portfolios 5.5 Passive Strategies and The Capital Market Line (CML) 5.1 Rates of Return Considering one-single period investment: regardless of the length of the period. Holding period return (HPR): measuring Ex-Post (Past) Returns over one-single period. HPR = [PS - PB + CF] / PB where PS = Sale price (or P1) PB = Buy price ($ you put up) (or P0) CF = Cash flow during holding period ( Such as dividend, interest) Example: You put up $50 at the beginning of the year for an investment. The value of the investment grows 4% and you earn a dividend of $3.50. What is your HPR? Annualizing HPRs Annualize a holding period return: translate it into percentage per year. (1) Without compounding (Simple or APR): HPRann = HPR/n (2) With compounding: EAR HPRann = [(1+HPR)1/n ]-1 where n = number of years held Annualizing HPRs for holding periods of greater than one year Example: Suppose you buy one share of a stock today for $45 and you hold it for two years and sell it for $52. You also received $8 in dividends at the end of the two years. What is the annual rate of return with and without compounding? HPR = (1) Annualized w/out compounding (2) The annualized HPR assuming annual compounding is (n =2 ): ( Annualizing HPRs for holding periods of less than one year Example: Suppose you buy one share of a stock today for $45 and you hold it for 3 months and sell it for $52. You also received $8 in dividends at the end of the two years. What is the annual rate of return with and without compounding? HPR = (1) Annualized w/out compounding (2) The annualized HPR assuming annual compounding is (n =0.25 ): Investment Returns over multiple periods The holding period return (HPR) is a simple measure of investment return over a single period. But how to measure the performance of a mutual fund over the last ten-year period? Several measures to find the average investment return for a time series of returns . (a) Arithmetic average return (simple Time-weighted average) (b) Geometric average return (Geometric time-weighted average) (c) Dollar-weighted return (a) Arithmetic Average Return (AAR) (a) Arithmetic average (simple Time-weighted average) Arithmetic means are the sum of ...

1. Ch5 Portfolio Theory -Risk and Return
Liang (Kevin) Guo
Learning Objectives
Be able to calculate ex post and ex ante risk and return
statistical measures, such as holding period return, average
returns, expected returns, and standard deviation.
Understand the difference between time-weighted and dollar-
weighted returns, geometric and arithmetic averages.
Be able to construct portfolios of different risk levels, given
information about risk free rates and returns on risky assets.
Be able to explain the CML theory.
Table of Contents
5.1 Rates of Return
5.2 Risk and Risk Premiums

2. 5.3 Inflation and Real Rates of Return
5.4 Asset Allocation Across Risky and Risk Free Portfolios
5.5 Passive Strategies and The Capital Market Line (CML)
5.1 Rates of Return
Considering one-single period investment: regardless of the
length of the period.
Holding period return (HPR): measuring Ex-Post (Past) Returns
over one-single period.
HPR = [PS - PB + CF] / PB
where
PS = Sale price (or P1)
PB = Buy price ($ you put up) (or P0)
CF = Cash flow during holding period ( Such as
dividend, interest)
Example: You put up $50 at the beginning of the year for an
investment. The value of the investment grows 4% and you earn
a dividend of $3.50. What is your HPR?
Annualizing HPRs
Annualize a holding period return: translate it into percentage

3. per year.
(1) Without compounding (Simple or APR):
HPRann = HPR/n
(2) With compounding: EAR
HPRann = [(1+HPR)1/n ]-1
where n = number of years held
Annualizing HPRs for holding periods of greater than one year
Example: Suppose you buy one share of a stock today for $45
and you hold it for two years and sell it for $52. You also
received $8 in dividends at the end of the two years. What is the
annual rate of return with and without compounding?
HPR =
(1) Annualized w/out compounding
(2) The annualized HPR assuming annual compounding is (n =2
):

4. (
Annualizing HPRs for holding periods of less than one year
Example: Suppose you buy one share of a stock today for $45
and you hold it for 3 months and sell it for $52. You also
received $8 in dividends at the end of the two years. What is the
annual rate of return with and without compounding?
HPR =
(1) Annualized w/out compounding
(2) The annualized HPR assuming annual compounding is (n
=0.25 ):
Investment Returns over multiple periods
The holding period return (HPR) is a simple measure of
investment return over a single period.
But how to measure the performance of a mutual fund over the
last ten-year period?
Several measures to find the average investment return for a
time series of returns .
(a) Arithmetic average return (simple Time-weighted average)
(b) Geometric average return (Geometric time-weighted
average)

5. (c) Dollar-weighted return
(a) Arithmetic Average Return (AAR)
(a) Arithmetic average (simple Time-weighted average)
Arithmetic means are the sum of returns in each period divided
by the number of periods.
Ignore compounding (Ignore reinvestment)
Used to forecast next-period return
n = number of time periods
(b) Geometric Average Return (GAR)
(b) Geometric average (Geometric time-weighted average
return)
Consider reinvestment (compounding)

6. n = number of time periods
Example: Continuing previous example, what is geometric
average return?
( c) Dollar-Weighted Return (DWR)
(c )Dollar-weighted return
It is the internal rate of return on an investment.
IRR method: (i.e. find the discount rate that makes the NPV of
the net cash flows equal zero.)
This measure of return considers both security performance and
changes in investment (accounting for cash flow).
If different amounts of money were managed in the portfolio for
each period it may be useful to see the Dollar weighted returns.
The DWR gives you an average return based on the stock’s
performance and dollar amount invested each period.
Tips on Calculating Dollar Weighted Returns
Initial Investment is an outflow
Ending value is considered as an inflow
Additional investment is an outflow
Security sales are an inflow

7. Dollar-Weighted Return (Example)
Example: You initially buy one share of mutual fund AAA at
$50, in one year collect a $2 dividend, and you buy another
share at $53. In two years you sell the stock for $54, after
collecting another $2 dividend per share. What is dollar-
weithted return?
NPV = $0 = -$50 - $51/(1+IRR) + $112/(1+IRR)2
Solve for IRR:
IRR is average annual dollar weighted return.
Dollar-weighted return vs Time-weighted return
Example: You initially buy one share of mutual fund AAA at
$50, in one year collect a $2 dividend, and you buy another
share at $53. In two years you sell the stock for $54, after
collecting another $2 dividend per share. What are the time-
weighted average return (Arithmetic Average Return and
Geometric Average Return )? What makes it different from
dollar-weighted return?
Time-weighted return (TWR) assumes you buy ONE share of

8. the stock at the beginning of each period and sell ONE share at
the end of each period. TWRs are thus independent of the
dollar amount invested in a given period.
DWR vs TWR (cont.)
TWR cash flows:
HPR for year 1 = HPR for year 2 =
To calculate TWRs:
(1) Arithmetic Average Return =
(2) Geometric Average Return =
Q: When should you use the DWR and when should you use the
TWR?
- If you want to measure the performance of your investment
in a fund, including the timing of your purchases and
redemptions you should calculate the DWR instead of TWR.
Year 0-1Year 1-20112-$50$ 2-$53$ 2+$53+$54
Arithmetic Average Return (AAR) vs Geometric Average
Return (GAR)

9. Q: When should you use the GAR and when should you use the
AAR?
A1: When you are evaluating PAST RESULTS (ex-post):
Use the AAR (average without compounding) if you ARE NOT
reinvesting any cash flows received before the end of the
period.
Use the GAR (average with compounding) if you
ARE reinvesting any cash flows received before the end of the
period.
A2: When you are trying to estimate an expected return (ex-ante
return):
Use the Arithmetic Average Return (AAR)
Class Discussion
(1) If you want to measure the performance of your investment
in a fund, including the timing of your purchases and
redemptions you should calculate the __________.
(2) If you desire to forecast performance for next period, the
best forecast will be given by the ________.
(3) If you always reinvest your dividends and interest earned
on the portfolio. Which method provides the best measure of the
actual average historical performance of the investments you
have chosen?

10. A. Dollar weighted return
B. Geometric average return
C. Arithmetic average return
D. Index return
5.2 Risk and Risk Premiums
Risk = Uncertainty or potential variability in future cash flows
How likely/closely will the realized return be to the expected
return?
To quantify risk, we can begin with the question: What holding
period return are possible? And how likely are they?
To determine the variability, we calculate the standard deviation
of the distribution of realized returns.
Indicates the dispersion around the expected (historical average)
return
Two approaches to estimate the standard deviation.
Scenario Analysis
- Requires analysts’ estimates for probability & outcomes
(b) Using historical data
Assume the past data will extend into the future
Scenario Analysis

11. (a) Scenario Analysis: Describes a probability distribution of
future returns
List all possible economic outcomes (scenarios)
Specify both the probability (likelihood) of each scenario and
the HPR the security will realized in that scenario.
Example: The stock of Business Adventures sells for $40 a
share. Its likely dividend payout and end-of-year price depend
on the state of the economy by the end of the year as follows.
Economic statesProbabilityDividendStock priceHPRBoom
1/3250Normal conomy 1/3143Recession 1/30.534
The list of possible HPRs
with associated prob is
called the probability
Distribution of HPRs.
Scenario Analysis (Formula)
Scenario analysis
Requires analysts’ estimates for probability & outcomes
Subjective Expected Return (mean) is the weighted average of
all the possible returns, weighted by the probability that each
return will occur.
Subjective Variance is the expected value of the squared
deviation from the mean.
Standard deviation is square root of variance, describing the
expected value of deviation from the mean.
E(R) = Expected Return

12. VAR(R) = Variance
Pi = probability of a state
Ri = return if a state occurs
Example (Scenario Analysis)
The stock of Business Adventures sells for $40 a share. Its
likely dividend payout and end-of-year price depend on the state
of the economy by the end of the year as follows. Calculate the
expected HPR and standard deviation of the HPR.
Economic state ProbDividendStock priceHPRColumn B *
Column EDeviation from Mean ReturnColumn B * squared
deviationBoom 1/3250Normal Economy 1/3143Recession
1/30.534Column sums
Expected return =Variance =Square root of variance = Standard
deviation =
Using historical data
(b) Using historical data
Assume the past data will extend into the future
Estimating Expected HPR (E[r]) from ex-post data. Use the
arithmetic average of past returns as a forecast of expected
future returns
Expected return is arithmetic mean of historical realized return
Variance is the expected value of the squared deviation from the
mean.
Standard deviation is square root of variance, describing the

13. expected value of deviation from the mean.
Example (Using historical data)
Compute the Facebook’s expected daily return and the standard
deviation using its realized return from May 18, 2014 to June
30,2017.
How to interpret Standard Deviation?
Standard deviations are useful for ranking the investments from
riskiest to least risky.
High standard deviation indicates the high risk
About two-thirds (68.26%) of all possible outcomes fall within
one standard deviation above or below the average
About 95% of all possible outcomes fall within two standard
deviations above and below the average.
Example: How to interpret the standard deviation of precious
example? (Expected return =10.8% and S.D.=16.37%)
23

14. Normal Distribution
Risk is the possibility of getting returns different from
expected.
mean.
Returns > E[r] may not be considered as risk, but with
Frequency distributions of annual HPRs
Which is the most risky and which is the least risky?
Risk Premium
Risk Premium is additional return we must expect to receive for
assuming risk.
The risk premium is the difference between the expected return
of a risky asset and the risk-free rate.
Excess Return or Risk Premium= E[r] – rf
The risk free rate is the rate of return that can be earned with
certainty.
Risk premium depends on level of risk associated with the
assets. As the level of risk associated with asset increases, we
will demand additional expected returns.

15. Risk Aversion
Risk aversion is an investor’s reluctance to accept risk.
The degree to which investors are willing to invest risky asset
depends on their risk aversion.
Risk premium on risky assets is to induce risk-averse investor
to hold these assets.
Risk premium an investor demands of a risky portfolio also
depends on their risk aversion.
Quantifying risk aversion:
E(rp) = Expected return on portfolio p
rf = the risk free rate
0.5 = Scale factor
A = risk aversion, between 2 to 4
The larger (lower) A is, the more risk averse (tolerant) the
investors are, the larger (smaller) will be the investor’s added
return required to bear risk.

16. Sharpe (reward-to-volatility) Ratio
Sharpe (reward-to-volatility) Ratio: Risk-return trade-off,
measure risk-adjusted performance
The Sharpe ratio tells us whether a portfolio's returns are due to
smart investment decisions or a result of excess risk.
Class discussion: Considering two portfolios. Portfolio A
generated a return of 15% and a 25% standard deviation last
year while Portfolio B generated a return of 18% and a 32%
standard deviation last year. T-bills were paying 4% last year.
Which portfolio do you prefer?
Higher Sharpe measure indicates a more efficient portfolio
5.3 Inflation and Real Rates of Return
The average inflation rate for the last 40 years was about 4%.

17. For the last 40 years, this relatively small inflation rate reduces
the terminal value of $1 invested in T-bills from a nominal
value of $10.08 to a real value of $1.63.
Nominal rate of interest determines how much more money you
will have while real rate of interest represents the rate of
increase in your actual purchasing power, after adjusting
inflation.
inflation rate
We can express their precise relationship as follows (exact
Fisher effect): :
( 1+ nominal interest rate) = (1+ real rate of interest) (1+ rate
of inflation)
Class discussion (Interest rate)
Example: what is the nominal rate of interest if real interest
rate is 10% and inflation rate is 4%?
Example: what is the real rate of interest if nominal interest
rate is 10% and inflation rate is 4%?
Historical Real Returns & Sharpe Ratios

18. Real returns have been much higher for stocks than for bonds
Sharpe ratios measure the excess return to standard deviation.
The higher the Sharpe ratio the better.
Stocks have had much higher Sharpe ratios than bonds.
Real Returns%Sharpe RatioSeriesWorld Stock6.000.37US Large
Stock6.130.37Small Stock8.170.36World Bond2.460.24Long
term Bond2.220.24
Motivation (Portfolio Theory)
What is a Portfolio and Why is it useful?
A portfolio is simply a specific combination of securities,
usually defined by portfolio weights that sum to 1:
Weights can be positive (long positions) or negative (short
positions).

19. Example
Your investment account of $100,000 consists of three stocks:
200 shares of stock A, 1,000 shares of stock B, and 750 shares
of stock C. Your portfolio is summarized by the following
weights:
AssetSharesPrice/ShareDollar
InvestmentPortfolio
WeightA200$50 $10,000 10%B1000$60 $60,000 60%C750$40
$30,000 30%Total$100,000 100%
Motivation (Portfolio Theory)
Why not Pick the Best Stock instead of forming a portfolio?
We don’t know which stock is best!
Portfolios provide diversification, reducing unnecessary risks.
Portfolio can enhance performance by focusing bets.
Portfolios can customize and manage risk/reward trade-offs.
How do we construct a “Good” portfolio?
What does “good” mean?
What characteristics do we care about for a given portfolio?
Risk and return trade-offs
Investors like higher expected returns
Investors dislike risk
Question: How can we choose portfolio weights to optimize the
risk/reward characteristics of the overall portfolio?
Mean Variance Analysis
Objective：

20. Assume investors focus only on the expected return and
variance (or standard deviation) of their portfolios: higher
expected return is good, higher variance is bad
Develop a method for constructing optimal portfolios
Portfolio Returns and Risk
The expected return on a portfolio is the weighted average of
the expected returns of the individual assets in the portfolio.
The portfolio’s risk is measured by its return variance (Variance
is more complicated:
36
Portfolio Return Variance
Portfolio variance is the weighted sum of all the variances and

21. covariances:
There arenvariances, and n2 −n covariances�Covariances
dominate portfolio variance�Positive covariances increase
portfolio variance; negative covariances decrease portfolio
variance (diversification)
5.4 Asset Allocation Across Risky and Risk Free Portfolios
Possible to split investment funds between safe and risky assets
Risk free asset rf : proxy; T-bills or money market fund
Risky asset portfolio rp: risky portfolio
Example. Your total wealth is $10,000. You put $2,500 in risk
free T-Bills and $7,500 in a stock portfolio invested as follows:
Stock A you put 2,500
Stock B you put 3,000
Stock C you put 2,000
$7,500
Allocating Capital Between Risky & Risk-Free Assets
Weights in risky portfolio rp :

22. WA =
WB =
WC =
The complete portfolio includes the riskless investment and rp.
Wrf = 25% Wrp = 75%
Weights in the complete portfolio
WA =
WB =
WC =
Allocating Capital Between Risky & Risk-Free Assets
How much should be invested in the risky asset and risk free
asset respectively?
Examine risk & return tradeoff
Demonstrate how different degrees of risk aversion will affect
allocations between risky and risk free assets
Depending on your level of risk you must merely choose
between your weights of the risk free and the risky portfolio

23. Combined Portfolio Expected Return and Risk
Example: The information about T-bill and risky portfolio runs
as follows:
Expected Return rate for T-bill, rf= 5%
Standard deviation for T-
Expected Return rate for risky portfolio, rp= 14%
Suppose you invest y of your total wealth in the risky portfolio
What is the expected return and standard deviation for the
complete or combined portfolio?
E(rc) =y E(rp) + (1 - y) rf
E(r)

24. E(rp) = 14%
rf = 5%
22%
0
P
F
Possible Combinations of asset allocation choices
s
E(rp) = 11.75%
16.5%
y =.75
y = 1
y = 0
5-42
Risk-return Trade Offs
42
Using Leverage with Capital Allocation Line
Borrow at the Risk-Free Rate and invest in stock
Using 50% Leverage, which means y = 1.5
E(rc) =

25. E(rC) =18.5%
33%
y = 1.5
E(r)
E(rp) = 14%
rf = 5%
= 22%
0
P
F
) Slope = 9/22
E(rp) -
rf = 9%
CAL
(Capital
Allocation
Line)
s
Complete portfolio offers a return per unit of risk of 9/22.
5-44

26. Capital Allocation Line (CAL) and its Slope
Capital Allocation Line plots all risk-return combinations
available by varying asset allocation between a risk free asset
and a risky portfolio
44
Risk Aversion and Allocation
How much should be invested in the risky portfolio and risk
free asset respectively? – depending on risk aversion and trade-
off between risk and return.
Greater levels of risk aversion lead investors to choose larger
proportions of the risk-free assets (risk free rate)
Lower levels of risk aversion (more risk tolerance) lead
investors to choose larger proportions of the portfolio of risky
assets
Willingness to accept high levels of risk for high levels of
returns would result in leveraged combination
If the reward-to-volatility ratio increases, investors might well
decide to take a greater position in the risky portfolio.
5.5 Passive Strategies and The Capital Market Line (CML)
How can investor choose the assets included in the risky
portfolio?
Using either passive or active strategies

27. Passive strategy
The investor makes no attempt to actively find undervalued
strategies nor actively switch their asset allocations.
Investment policy that avoids security analysis: securities are
fairly priced
Two advantages compared to active strategy
Avoids the costs involved in the undertaking security analysis
(active trading strategies may not guarantee higher returns )
free ride benefit: the activity of knowledge investors force
prices to reflect currently available information
Capital Market Line (CML)
A simple passive strategy (Indexing strategy): Investing in a
broad stock index (like S&P 500 index) and a risk free
investment.
Indexing has become an extremely popular strategy for passive
investors.
Capital Market Line: the Capital allocation line (CAL) provided
by combinations of one month T-bills and a broad index of
common stocks (or an index that mimics overall market
performance).
What does Portfolio theory suggest?

28. Investors should only invest two passive portfolios
Short-term T-bills
Fund of common stocks that mimics a broad market index
Vary combinations according to investor’s risk aversion.
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Total Cash Flows Each Year
Year
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-$50 $ 2 $ 4
-$53 $108
Net -$50 -$51 $112
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Stock A $2,500
Stock A $2,500
Stock B $3,000
Stock B $3,000
Stock C $2,000
Stock C $2,000
Your total wealth is $10,000. You put $2,500 in risk free
T-Bills and $7,500 in a stock portfolio invested as follows
r
f
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r
f
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rf
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rf
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p
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rp

36. = 22%
rp
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37. y = % in r
p
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E(r)
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r
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f
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= 5%
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Possible Combinations
Possible Combinations
E(r
E(r
p
p
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) = 11.75%
16.5%

38. 16.5%
E(r
E(r
p
p
) = 11.75%
) = 11.75%
16.5%
16.5%
y =.75
y =.75
y = 1
y = 1
E(r)
E(r)
E(r
E(r
p
p
) = 14%
) = 14%
r
r
f
f
= 5%
= 5%
22%
22%
0
0
P
P
F
F
Possible Combinations

39. Possible Combinations
E(r
E(r
p
p
) = 11.75%
) = 11.75%
16.5%
16.5%
E(r
E(r
p
p
) = 11.75%
) = 11.75%
16.5%
16.5%
y =.75
y =.75
y = 1
y = 1
E(r
E(r
C
C
) =18.5%
) =18.5%
33%
33%
E(r
E(r
C
C
) =18.5%
) =18.5%

40. 33%
33%