This document discusses key concepts in portfolio theory, including how to calculate investment returns over single and multiple periods. It defines holding period return (HPR) to measure single period returns and arithmetic average, geometric average, and dollar-weighted return to measure returns over many periods. It also explains how to calculate the expected return, variance, and standard deviation of investments to quantify the expected reward and risk.

1. 1
FINC4101
Investment Analysis
Instructor: Dr. Leng Ling
Topic: Portfolio Theory I

2. 2
Learning objectives
1. Compute different measures of investment
performance:
 Holding-period return (HPR)
 Arithmetic average
 Geometric average
 Dollar-weighted return
2. Compute the expected return, variance
and standard deviation of a risky
investment.

3. 3
Concept Map
FI4101
Portfolio
Theory
Asset
Pricing
Equity
Fixed
Income
Market
Efficiency
Derivatives
Foreign
Exchange

4. 4
Portfolio Theory I: Concept Map
Portfolio
Theory I
HPR
Arithmetic,
Geometric,
$-weighted
Expected
return
Variance

5. 5
Investment return over 1 period:
Holding period return (HPR)
 Rate of return over a given investment (holding)
period. Has two components:
 Price change = ending price – beginning price
 Cash income
Price change + Cash income
Beginning price
Ending price - Beginning price + Cash income
Beginning price
HPR
=
=

6. 6
Holding period return (HPR)
 Assumes that cash income is paid at the
end of the holding period.
 If cash income is received earlier,
reinvestment income is ignored.
 HPR can be used for different types of
investments: stock, bond, mutual fund etc.
 For stock, cash income = dividend
 For bond, cash income = coupon

7. 7
Holding period return (HPR)
 Stock
Price change + Cash dividend
Beginning price
Price change Cash dividend
+
Beginning price Beginning price
Capital gains yield + Dividend yield
HPR =
=
=

8. 8
Simple HPR example
 You are thinking of investing in ABC Inc’s stock. You
intend to hold the stock for 1 year. ABC’s stock is
currently selling at $50 and is expected to rise to $56 by
the end of the year. The company is expected to pay a
per share dividend of $0.60 during the year.
Compute:
 HPR
 Capital gains yield
 Dividend yield.
 Sum up the capital gains yield and the dividend yield. Is
that the same as the HPR?

9. 9
Investment return
over many periods
 Three alternative ways of measuring
average returns over multiple periods:
 Arithmetic average (arithmetic mean)
 Geometric average (geometric mean)
 Dollar-weighted return
 Use the following example to illustrate each
return measure.

10. 10
Example: Table 5.1
 Quarterly cash flows and HPRs of a mutual fund.
1st Qtr 2nd Qtr 3rd Qtr 4th Qtr
Beg. Assets ($m) 1 1.2 2 0.8
HPR (%) 10 25 (20) 25
Assets before flows ($m) 1.1 1.5 1.6 1
Net inflow ($m) 0.1 0.5 (0.8) 0.0
End. Assets ($m) 1.2 2 0.8 1
What is the arithmetic average, geometric average
and dollar-weighted return over the four quarters?

11. 11
Arithmetic average
 Suppose we hold an asset over N periods:
1, 2,…, and N.
 And we label the HPR in each period as:
r1, r2, …, rN
 The arithmetic average is the sum of returns in
each period divided by number of periods.
Arithmetic average = (r1 + r2 + r3 + ... rN) /N
 In the example,
arithmetic average = (10+25-20+25)/4 = 10%

12. 12
Geometric Average (time-weighted)
 Single per period return that gives the same
cumulative performance as the sequence of
actual returns.
Geometric average
= [(1+r1) x (1+r2) ... x (1+rN)]1/N - 1
 In the example, geometric average
= [(1.1) x (1.25) x (.8) x (1.25)] 1/4 - 1
= (1.375) 1/4 -1 = .0829 = 8.29%

13. 13
Dollar-weighted return (IRR)
 This is simply the internal rate of return (IRR) on
an investment !
 IRR: the interest rate that will make the PV of
cash inflows equal to the PV of cash outflows.
 In other words, IRR is the discount rate such that the
NPV is 0.
   
0
1
1 0
0





 

N
t
t
t
N
t
t
t
IRR
COF
IRR
CIF

14. 14
 In the example, think in terms of capital
budgeting. So, the mutual fund is a
“project” from investor’s perspective.
• Initial Investment is an outflow
• Ending value is an inflow
• Additional investment is an outflow
• Reduced investment (withdraw money) is an
inflow
Dollar-weighted return (IRR)

15. 15
Dollar-Weighted Return
Time
0 1 2 3 4
Net cash flow ($m) -1 -0.1 -0.5 0.8 1
Using the definition of the IRR,
2 3 4
2 3 4
0.1 0.5 0.8 1
1
1 (1 ) (1 ) (1 )
0.1 0.5 0.8 1
1
1 (1 ) (1 ) (1 )
IRR IRR IRR IRR
IRR IRR IRR IRR
+ + = +
+ + + +
- -
= + + +
+ + + +

16. 16
Quoting Rates of Return
Annual percentage rate, APR
= rate per period X n
Where n = no. of compounding periods per year
Effective annual interest rate,
EAR 1
1 








m
m
APR

17. 17
Quoting Rates of Return
 With continuous compounding, the relationship
between EAR and APR becomes
EAR = eAPR – 1
‘e’ is the exponential function (that appears on your
financial calculator as [ex])
Equivalently,
APR = Ln(1 + EAR)
Ln is the natural log function.

18. 18
HPR, APR, EAR problem
 Suppose you buy a bond of General Electric at a
price of $990. The bond pays coupons semi-
annually, has an annual coupon rate of 6%, a
face value of $1,000 and will mature in six
months’ time. You intend to hold the bond till it
matures.
 What is the 6-month HPR?
 What is the APR of this investment?
 What is the EAR of this investment?

19. Another example of HPR
 Suppose you bought a bond of General Electric
at a price of $990 6 months ago. The bond pays
coupons semi-annually, has an annual coupon
rate of 6%, a face value of $1,000 and will
mature in 12 months from today. Today it just
paid the coupon and you intend to sell it
immediately at current market price. The current
YTM is 15%.
 What is the current market price?
 What will be your HPR?
19

20. 20
Describing investment uncertainty:
Scenario analysis
 Investment is risky simply because we don’t know what
will happen in the future for certain. One way of
quantifying risk is through scenario analysis.
 Scenario analysis: The process of devising a list of
possible economic scenarios and specifying: -
 The likelihood (probability) of each scenario.
 The HPR that will be realized in each scenario.
 The list of possible HPRs with associated probabilities is
called the probability distribution of HPRs. This is critical
in helping us to evaluate risky investments.

21. 21
Probability distribution of HPR
 The probability distribution provides
information for us to measure the reward
and risk of an investment.
 Reward of the investment: Expected return
 Also known as ‘mean return’, ‘mean of the
distribution of HPRs’.
 Risk of the investment: Variance
Let’s start with a simple scenario analysis.

22. 22
Say you want to buy Google’s stock
and hold it for a year.
 During this coming year, you think there are 3
possible economic scenarios: boom, normal
growth, recession.
State of the
Economy
Scenario Probability, p(s) HPR (%)
Boom 1 0.25 44
Normal 2 0.50 14
Recession 3 0.25 -16

23. 23
Expected Return, E(r)
 The weighted average of returns in all
possible scenarios, s = 1,2,…S, with
weights equal to the probability of that
particular scenario.
p(s): probability of scenario s
r(s): HPR in scenario s







S
s
s
r
s
p
s
r
s
p
r
p
r
p
r
E
1
)
(
)
(
)
(
)
(
...
)
2
(
)
2
(
...
)
1
(
)
1
(
)
(

24. 24
Expected Return, E(r)
 With the formula, Google’s expected
return is:
E(r) = (0.25 x 44) + (0.5 x 14) + (0.25 x -16)
= 14%
Probability in each scenario
HPR in each scenario

25. 25
Variance, Var(r)
 When we talk about risk, we often think
of surprises or deviations from what we
expect. Variance captures this idea.
 Variance: The expected value of squared
deviation from the mean.
 Also known as σ2 (read as ‘sigma squared’).





S
s
r
E
s
r
s
p
r
Var
1
2
2
)]
(
)
(
)[
(
)
( 

26. 26
Standard deviation, SD(r)
Standard deviation: Square root of variance
Returning to the Google example,
Var(r)
= 0.25(44 – 14)2 + 0.5(14 – 14)2 + 0.25(-16 -14)2
= 450
SD(r)= (450)1/2 = 21.21%
)
(
)
( r
Var
r
SD 


27. 27
How to interpret
E(r), Var(r) and SD(r)
 The bigger the expected return, the bigger
the potential reward from the investment,
vice versa.
 The bigger the variance, the bigger the
risk of the investment, vice versa.
 The bigger the standard deviation, the
bigger the risk of the investment, vice
versa.

28. 28
Describing investment performance
in the past
 If we are interested in the rewards from
investing in the past (using historical data), we
can use (1) arithmetic average, (2) geometric
average.
 To quantify risk, use ‘historical’ or ‘sample’
variance: Arithmetic average
No. of periods





n
i
i r
r
n 1
2
2
)
(
1
1


29. 29
Example: S&P500 index,
1988-1992
Year HPR(%)
1988 16.9
1989 31.3
1990 -3.2
1991 30.7
1992 7.7
Compute the arithmetic average, geometric average, and variance.

30. 30
Example: S&P500 index,
1988-1992
Year (1)
HPR(%)
(2)
1+HPR
(3)
Deviation from
arithmetic average
(4)
Squared
deviation
1988 16.9 1.169 16.9 – 16.7 = 0.2 0.04
1989 31.3 1.313 31.3 – 16.7 = 14.6 213.16
1990 -3.2 0.968 -3.2 – 16.7 = -19.9 396.01
1991 30.7 1.307 30.7 – 16.7 = 14 196
1992 7.7 1.077 7.7 – 16.7 = -9 81
Total 83.4 886.21

31. 31
Example: S&P500 index,
1988-1992, cont’d
Arithmetic average = 83.4/5 =16.7%
Geometric average =
[(1.169) x (1.313) x (0.968) x (1.307) x (1.077)]1/5 – 1
= 0.15902 or 15.9%
Variance = 886.21/(5 – 1) = 221.6
Standard deviation = (221.6)1/2 =14.9%

32. 32
Risk premiums & risk aversion
 If you don’t want to invest in a risky asset like a
stock, what is the alternative?
 Risk-free assets like treasury bills.
The return you get is the risk-free rate (rf).
 Risk-free rate = rate of return that can be earned with
certainty.
 Risk premium: expected return in excess of the
risk-free rate.
Risk premium = E(r) – rf

33. 33
Risk premium depends on risk
aversion and variance
 Risk aversion: reluctance to accept risk.
 Risk premium of a portfolio, E(rp) – rf
E(rp) – rf = A x var(rp)
A = measures degree of investor’s risk aversion,
Var(rp) = variance (risk) of the portfolio
Risk premium increases if:
 Portfolio variance increases OR
 Risk-aversion, A, increases

34. Inferred Risk Aversion
(price of risk)
34
p
f
p r
r
E
A 2
)
(




35. Average A in market
35
M
f
M r
r
E
A 2
)
(




36. Sharpe Ratio
 Measure the risk-return tradeoff
is the standard deviation
* It works for portfolio only, not individual
security.
36
p
f
p
p
r
r
E
S



)
(


37. Problem sets after Chapter 5,
# 11.
 The expected cash flow is: (0.5 x $50,000) + (0.5 x $150,000) = $100,000
With a risk premium of 10%, the required rate of return is 15%. Therefore, if the value of the
portfolio is X, then, in order to earn a 15% expected return:
X(1.15) = $100,000  X = $86,957
 If the portfolio is purchased at $86,957, and the expected payoff is $100,000, then the
expected rate of return, E(r), is:
= 0.15 = 15.0%
The portfolio price is set to equate the expected return with the required rate of return.
 If the risk premium over T-bills is now 15%, then the required return is:
5% + 15% = 20%
The value of the portfolio (X) must satisfy:
X(1.20) = $100, 000  X = $83,333
 For a given expected cash flow, portfolios that command greater risk premium must sell at
lower prices. The extra discount from expected value is a penalty for risk. 37
957
,
86
$
957
,
86
$
000
,
100
$ 

38. 38
Summary
 Single period: holding-period return (HPR)
 Many periods: arithmetic average,
geometric average, and dollar-weighted
return.
 Expected return measures the ‘reward’
from an investment.
 Variance (standard deviation) measures
the ‘risk’ from an investment.

39. Practice 1
1.Chapter 5 problem sets : 5,6,7,8.
2. Suppose you bought a bond of BT Co. at a price of $990 6 months ago.
The bond pays coupons semi-annually, has an annual coupon rate of 6%, a
face value of $1,000 and will mature in 24 months from today. Today it just paid
the coupon. The current YTM is 15%.
 What is the current market price? What is your HPR for the last 6 months?
 If the interest rate on the market (YTM) does not change for the next 2
years, what will be your HPR for one year if you intend to sell the bond after
6 months receiving the 2nd coupon payment? (assume all coupons do not
earn any investment returns)
39

40. Homework 1
1. Suppose an investor’s risk aversion A=2 and the variance of the
return of the portfolio he chose is 0.0260. The risk-free rate is 2%. The
value of the portfolio has 0.3 probability to go to $2000 in one year, 0.5
probability to $1500 and 0.2 probability to $1000. What is the maximum
price he would pay for this portfolio?
2. Suppose you bought a bond at a price of $990 12 months ago. The
bond pays coupons annually, has an annual coupon rate of 6%, a face
value of $1,000 and will mature in 36 months. Today it just paid the
coupon. The current YTM is 15%. Suppose the YTM will decrease to
10% after 12 months and then remains the same till expiration. What is
the arithmetic average annual return for the next two years?
40