Week 2
GD31403
Basic Performance Analysis
Concepts
What Is An Investment?
• A current commitment of $ for a period of time
in order to derive future payments that will
compensate for:
– The time the funds are committed (time value of
money)
– The expected rate of inflation
– Uncertainty of future flow of funds (risk premium)
Risk and return are 2 fundamental concepts in
investment.
1-2
Required Rate of Return
The minimum rate of return to
compensate investors for the time, the
expected rate of inflation and the
uncertainty of the return.
1-3
Historical Rates of Return
(1) Return over a Holding Period
Holding Period Return (HPR)
Ending Value of Investment
HPR 
Beginning Value of Investment
Holding Period Yield (HPY)
HPY=  HPR-1 *100
1-4
Historical Rates of Return
Assume that you invest $200 at the beginning of the
year and get back $220 at the end of the year. What
are the HPR and the HPY for your investment?
HPR = $220/200
= 1.1
HPY = (HPR – 1) * 100
= (1.1 -1) * 100
= 10%
1-5
Historical Rates of Return
Annual Holding Period Return (Annual HPR)
Annual HPR  HPR
1n
Annual Holding Period Yield (Annual HPY)
Annual HPY=  Annual HPR-1 *100
1-6
Historical Rates of Return
Your investment of $250 in Stock A is worth $350 in
two years, while the investment of $1000 in Stock B is
worth $750 over the same holding period. What are the
annual HPRs and the annual HPYs on these two
stocks?
Stock A
Annual HPR = HPR1/n = ($350/$250)1/2 = 1.1832
Annual HPY = (Annual HPR – 1)*100 = 18.32%
Stock B
Annual HPR = HPR1/n = ($750/$1000)1/2 = 0.866
Annual HPY = (Annual HPR – 1)*100 = -13.4%
1-7
Historical Rates of Return
When converting HPY to an annual basis, we
assume a constant annual yield for each year.
Stock A
HPY = [($350/$250)–1]*100 = 40% (over 2 years)
Annual HPY = [(1+0.40)1/2 – 1]*100 = 18.32%
Ending Value = $1 * (1.1832)(1.1832)
= $1.40
The annual yield assumes investor earns the same
18.32% return for the second year (compounded).
1-8
Historical Rates of Return
Your investment of $100 in Stock C is worth $112 over
a six-month holding period. What is the annual HPY on
this stock? What is the implicit assumption in the
calculation?
HPY = [($112/$100)–1]*100 = 12% (over 6 months)
Annual HPY = [(1+0.12)1/0.5 – 1]*100 = 25.44%
Ending Value = $1 * (1.12)(1.12)
= $1.2544
The annual yield assumes investor earns the same 12%
for the second six-month (compounded).
1-9
Historical Rates of Return
(2) Average/Mean Historical Returns
The assumption of constant annual yield for each
year is not realistic.
Over the holding period of several years,
investors will likely get different annual rates of
return for an investment.
How do you measure the mean/average annual
return?
1-10
Historical Rates of Return
Single Asset
Arithmetic Mean Return (AM)
n
AM =
å HPY
i=1
n
Geometric Mean Return (GM)
(
)(
) (
)
1n
GM= éë 1+ HPY1 1+ HPY2 ... 1+ HPYn ùû -1
1-11
Historical Rates of Return
Suppose you invested $100 three years ago and it is
worth $110.40 today. The information below shows
the annual ending values and HPR and HPY. This
example illustrates the computation of the AM and
the GM over a three-year period for an investment.
Year
1
2
3
1-12
Beginning
Value
100
115
138
Ending
Value
115.0
138.0
110.4
HPR
HPY
1.15
1.20
0.80
15%
20%
-20%
Historical Rates of Return
n
AM =
å HPY
i=1
n
0.15 + 0.20 - 0.20
=
= 0.05 = 5%
3
GM= éë(1+ HPY1 ) (1+ HPY2 ) ...(1+ HPYn ) ùû -1
1n
= éë(1+ 0.15) (1+ 0.20 ) (1- 0.20 ) ùû -1
13
= 1.03353 -1 = 0.03353 = 3.353%
AM versus GM: Which one should we use?
1-13
Historical Rates of Return
Suppose you invested $50 in a stock, and the price
increases to $100 at the end of year 1. However, the
stock is worth $50 at the end of year 2. Compute the
AM and the GM over a 2-year period for the stock.
Year
1
2
1-14
Beginning
Value
50
100
Ending
Value
100
50
HPR
HPY
2.00
0.50
100%
-50%
Historical Rates of Return
n
AM =
å HPY
i=1
n
1- 0.50
=
= 0.25 = 25%
2
(
)(
= éë(1+1) (1- 0.50)ùû
) (
)
1n
GM= éë 1+ HPY1 1+ HPY2 ... 1+ HPYn ùû
12
-1
-1
= 1-1 = 0%
AM versus GM: which one is more sensible?
1-15
Historical Rates of Return
AM versus GM
 When rates of return are the same for all years,
the AM and the GM will be equal.
 When rates of return are not the same for all
years, the AM will always be higher than the
GM.
 The AM is best used as an “expected value”
for an individual year
 The GM is the best measure of an asset’s
long-term performance as it measures the
compound annual rate of return.
1-16
Historical Rates of Return
Portfolio
1-17
The HPY is referred to as the value-weighted mean rate
of return
Expected Rates of Return
• In previous examples, we discussed realized
historical rates of return.
• In contrast, an investor would be more
interested in the expected return on a future
risky investment.
• An investor determines how certain the
expected rate of return on an investment by
analyzing estimates of possible returns (by
assigning probability values).
1-18
Expected Rates of Return
• These probabilities are typically subjective
estimates
• From Business Statistics, you learned that
there are 3 approaches to assign probability:
– Classical approach
– Relative frequency
– Subjective approach
1-19
Expected Rates of Return
n
E(R i )   (Probabilit y of Return)  (Possible Return)
i 1
 [(P1 )(R 1 )  (P2 )(R 2 )  ....  (Pn R n )]
n
  ( Pi )( Ri )
i 1
1-20
Expected Rates of Return
Scenario 1: The investor is absolutely certain of a
return of 5% (risk-free investment)
1.00
0.80
E ( Ri ) = 0.05 = 5%
0.60
0.40
0.20
0.00
-5%
1-21
0%
5% 10% 15%
Expected Rates of Return
Scenario 2: Investors expect 3 possible returns
based on different economic conditions
(risky investment)
1.00
0.80
0.60
0.40
0.20
0.00
(10%,0.70)
( -20%,0.15)
-30%
1-22
-10%
( )
E Ri = 0.07 = 7%
( 20%,0.15)
10%
30%
Expected Rates of Return
Scenario 3: Risky investment with ten possible
returns
1.00
0.80
0.60
E ( Ri ) = 0.05 = 5%
0.40
0.20
0.00
-40% -20% 0%
1-23
20% 40%
Expected Rates of Return
• Since Scenario 1 and Scenario 3 provide the
same rate of return (5%), which one would you
invest?
• It is generally assume that most investors are
risk averse- ceteris paribus, they will select
the investment that offers greater certainty
(less risk).
1-24
Risk of Expected Return
• Risk refers to the uncertainty of an investment;
therefore the measure of risk should reflect the
degree of the uncertainty.
• The risk of expected return reflect the degree
of uncertainty that actual return will be different
from the expected return.
• The common measures of risk are based on
the variance and the standard deviation of the
estimated distribution of expected returns.
1-25
Risk of Expected Return
Variance  2     Probability  *  Possible Return-Expected Return 
n
i 1
n
=   Pi   Ri  E  Ri  
2
i 1
• Variance measures the dispersion of possible returns
around the expected returns
• The larger the variance, the greater the dispersion of
expected returns, and the greater the risk.
1-26
2
Risk of Expected Return
Standard Deviation    Variance  2 
• Standard deviation is the square root of the variance
• Why standard deviation is the preferred measure of
total risk?
• For variance, the unit is the square of the unit attached
to the original observation. In the case of returns, the
variance will be “% squared”.
• The unit associated with the standard deviation is the
unit of the original data. For returns, the standard
deviation will be “%”.
1-27
Risk of Expected Return
• For your practice, you can try this website
1-28
Risk of Expected Return
1-29
Risk of Expected Return
• For comparing alternative investments, the
coefficient of variation (CV) is more suitable.
• CV, a measure of relative variability, indicates
the risk per unit of expected return.
Coefficient of Variation 
CV =
1-30
Standard Deviation of Expected Returns
Expected Rate of Return
i
E  R
Historical Rates of Return
Which one of the following 2 stocks is riskier?
Expected Return
Standard Deviation
Stock A
0.07
0.05
Stock B
0.12
0.07
• By comparing the standard deviation (absolute
variability), Stock B appears to be riskier.
• But Stock B has higher returns
1-31
Risk of Expected Return
CVA
0.05
=
 0.714
0.07
CVB
0.07
=
 0.583
0.12
The CV shows that Stock B has less relative
variability or lower risk per unit of expected
return.
1-32
Risk of Historical Rates of Return
To measure the risk for a series of historical
rates of returns, the same variance and standard
deviation are used.
The only difference is that we consider the
historical holding period yields (HPY).
For illustration, see Textbook (Brown and Reilly,
2009), pp.28-29.
1-33
Determinants of Required Returns
• Three Components of Required Return:
– The time value of money
– The expected rate of inflation
– The risk premium
• Complications of Estimating Required Return
– A wide range of rates is available for alternative
investments at any time.
– The rates of return on specific assets change
dramatically over time.
– The difference between the rates available on
different assets change over time.
1-34
Determinants of Required Returns
(1) The time value of money
– An investor would demand the real risk-free rate
(RRFR) as a compensation for deferring the use of
money in current period.
– RRFR assumes no inflation.
– RRFR assumes no uncertainty about future cash
flows.
– RRFR is influenced by the long-run real growth
rate of the economy.
1-35
Determinants of Required Returns
(2) The expected rate of inflation
– An investor would demand the rate of return to
include compensation for the expected rate of
inflation.
– The nominal risk-free rate (NRFR) that prevails in
the market are determined by RRFR, the expected
rate of inflation and conditions in the capital market
(supply and demand).
NRFR  1  RRFR  * 1  Expected Rate of Inflation    1
1-36
Determinants of Required Returns
(3) Risk Premium
– Most investors require higher rates of return on
investments if they perceive that there is any
uncertainty about the expected rate of return.
– The increase in the required rate of return over the
NRFR is the risk premium (RP).
The risk premium can be related to fundamental
factors (fundamental risk) OR systematic market risk
(beta).
1-37
Determinants of Required Returns
– Fundamental risk comprises business risk,
financial risk, liquidity risk, exchange rate risk, and
country risk.
– Systematic risk refers to the portion of an individual
asset’s total variance attributable to the variability
of the total market portfolio.
RP =
f
(Business Risk, Financial Risk, Liquidity Risk,
Exchange Rate Risk, Country Risk)
OR
RP = f (Systematic Risk)
1-38
Determinants of Required Returns
Risk Premium and Portfolio Theory
– From a portfolio theory perspective, the relevant risk
measure for an individual asset is its co-movement
with the market portfolio (systematic risk or beta).
– Individual assets have variance that is unrelated to
the market portfolio, known as unsystematic risk.
– Unsystematic risk is unimportant because it is
eliminated in a large diversified portfolio.
1-39
Relationship Between Risk & Return
The Security Market Line (SML)
– It shows the relationship between risk and return for
all risky assets in the capital market at a given time.
– Investors select investments that are consistent
with their risk preferences.
Expected Return
Security
Market Line
NRFR
The slope indicates the
required return per unit of risk
Risk
1-40
(fundamental risk or systematic risk)
Relationship Between Risk & Return
• Movement along the SML
– When the risk of an investment changes due to a
change in one of its risk sources, the expected
return will also change, moving along the SML.
Expected
Return
SML
NRFR
1-41
Movements along the curve
that reflect changes in the
risk of the asset
Risk
(fundamental risk or systematic risk)
Relationship Between Risk & Return
• Changes in the Slope of the SML
– When there is a change in the attitude of investors
toward risk, the slope of the SML will also change.
– If investors become more risk averse, then the SML
will have a steeper slope, indicating a higher risk
premium for the same risk level.
Expected Return
R m’
New SML
Original SML
Rm
NRFR
Risk
1-42
(fundamental risk or systematic risk)
Relationship Between Risk and Return
• Changes in Market Condition or Inflation
– A change in the RRFR or the expected rate of
inflation will cause a parallel shift in the SML.
– When nominal risk-free rate increases, the SML will
shift up, implying a higher rate of return while still
having the same risk premium.
Expected Return
New SML
Original SML
NRFR'
NRFR
1-43
(fundamental risk or systematic risk) Risk