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1Fin 2802, Spring 10 - TangChapter 5: Risk and Return
Fina2802: Investments and Portfolio Analysis
Spring, 2010Dragon Tang
Fina2802: Investments and Portfolio Analysis
Spring, 2010Dragon Tang
Lecture 8
Risk and Return: Past and Prologue
February 4, 2010
Readings: Chapter 5
Practice Problem Sets: 3,4,5,14,15-17
2Chapter 5: Risk and Return
Fin 2802, Spring 10 - Tang
3Chapter 5: Risk and Return
Risk and Return in ChineseRisk and Return in Chinese
Risk: 危机
Danger | Opportunity
Return: 回报
Come back | Gratitude
Fin 2802, Spring 08 - TangFin 2802, Spring 10 - Tang
4Fin 2802, Spring 10 - TangChapter 5: Risk and Return
Risk and ReturnRisk and Return
Objectives:
1. Characterize the risk and return on stocks (risky) and bonds (risk-free).
2. Historical risk and return of various securities
5Fin 2802, Spring 10 - TangChapter 5: Risk and Return
Return over One Period: Holding Period Return (HPR)
Return over One Period: Holding Period Return (HPR)
HPR: Rate of return over a given investment period
Price Beginning
DividendsPrice Beginning-Price EndingHPR
6Fin 2802, Spring 10 - TangChapter 5: Risk and Return
Rates of Return: Single Period ExampleRates of Return: Single Period Example
Beginning Price = 100
Ending Price = 110
Dividend = 4
HPR = ( 110 - 100 + 4 )/ ( 100) = 14%
7Fin 2802, Spring 10 - Tang
Real vs. Nominal RatesReal vs. Nominal Rates
• Notation:–R=nominal return
– i =inflation rate
– r =real return
• Exact relationship
• Approximate relationship
• Example R = 9%, i = 6%: what is r?
Rir 111
iRr
Chapter 5: Risk and ReturnChapter 5: Risk and Return
8Fin 2802, Spring 10 - TangChapter 5: Risk and Return
Quoting ConventionsQuoting Conventions
APR = annual percentage rate
(periods in year) X (rate for period)
EAR = effective annual rate
( 1+ rate for period)Periods per yr - 1
Example: monthly return of 1%
APR = 1% X 12 = 12%
EAR = (1.01)12 - 1 = 12.68%
9Fin 2802, Spring 10 - TangChapter 5: Risk and Return
Return over Multiple PeriodsReturn over Multiple Periods
• Dollar-weighting: Internal Rate of Return (IRR)
• Time-weighting:
– Arithmetic Average: rA = (r1+r2)/2
– Geometric Average: rG = [(1+r1)(1+r2)]1/2 – 1
– rA ? rG always
t = 0 1 2
$X $Y $Z
r1 r2
$X, $Y, $Z: Cash Flows; r1, r2: one-period HPR
0
1
$
1
$$ 21
IRR
Z
IRR
YX
10Fin 2802, Spring 10 - TangChapter 5: Risk and Return
ExampleExample
1 2 3 4
Assets(Beg.) 1.0 1.2 2.0 .8
HPR .10 .25 (.20) .25
TA (Before
Net Flows) 1.1 1.5 1.6 1.0
Net Flows 0.1 0.5 (0.8) 0.0
End Assets 1.2 2.0 .8 1.0
11Fin 2802, Spring 10 - TangChapter 5: Risk and Return
Returns Using Arithmetic and Geometric Averaging
Returns Using Arithmetic and Geometric Averaging
Arithmetic
ra = (r1 + r2 + r3 + ... rn) / n
ra = (.10 + .25 - .20 + .25) / 4
= .10 or 10%
Geometric
rg = {[(1+r1) (1+r2) .... (1+rn)]} 1/n - 1
rg = {[(1.1) (1.25) (.8) (1.25)]} 1/4 - 1
= (1.5150) 1/4 -1 = .0829 = 8.29%
12Fin 2802, Spring 10 - TangChapter 5: Risk and Return
Dollar Weighted Average ExampleDollar Weighted Average Example
Net CFs 0 1 2 3 4
$ (mil) -1.0 - 0.1 - 0.5 0.8 1.0
Solving for IRR1.0 = -.1/(1+r)1 + -.5/(1+r)2 + .8/(1+r)3 +1.0/(1+r)4
r = .0417 or 4.17%
13Fin 2802, Spring 10 - TangChapter 5: Risk and Return
Which One to Use?Which One to Use?
Dollar-weighted return(IRR):
• Use if focus is total amount of money at some terminal date (wealth)
Time-weighted return:
- Arithmetic Average: , ignore compounding
- Geometric Average: , compounding over time.
• Use if there is no control over timing• Used most by money management industry
T
ttt
T
tt
t
IRR
Outflow
IRR
Inflow
11 11
n
rrrr n
A
21
1111 /121 n
nG rrrr
14Fin 2802, Spring 10 - TangChapter 5: Risk and Return
HPR - Expected ReturnHPR - Expected Return
s
srsprE
p s
r s
s
the probability of each scenario
the HPR in each scenario
indexation variable for scenarios
15Fin 2802, Spring 10 - Tang
Normal distributionNormal distribution
Chapter 5: Risk and ReturnChapter 5: Risk and Return
16Fin 2802, Spring 10 - TangChapter 5: Risk and Return
HPR - Risk MeasureHPR - Risk Measure
Variance or standard deviation:
22 s
rEsrsp
2
17Fin 2802, Spring 10 - Tang
Why do we need the variance?Why do we need the variance?
-5 -4 -3 -2 -1 0 1 2 3 4 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Outcomes
Probability
•Two variables with the same mean.
•What do we know about their dispersion?
Chapter 5: Risk and ReturnChapter 5: Risk and Return
18Fin 2802, Spring 10 - TangChapter 5: Risk and Return
ExampleExample
Suppose your expectations regarding the stock market are as
follows:
State of the economy Scenario(s) Probability(p(s)) HPR
Boom 1 0.3 44%
Normal Growth 2 0.4 14%
Recession 3 0.3 -16%
Compute the mean and standard deviation of the HPR on stocks.
E( r ) = 0.3*44% + 0.4*14%+0.3*(-16%)=14%
Sigma^2=0.3*(44%-14%)^2+0.4*(14%-14%)^2
+0.3*(-16%-14%)^2=0.54
Sigma=0.7348=73.48%
19Fin 2802, Spring 08 - TangChapter 5: Risk and Return
Historical Mean and VarianceHistorical Mean and Variance
Data in the n-point time series are treated as realization of a particular scenario each with equal probability 1/n
n
t
t
n
rr
1
n
t
t
n
rr
n
n
1
22
1
Fin 2802, Spring 10 - TangChapter 5: Risk and Return
20Fin 2802, Spring 08 - TangChapter 5: Risk and Return
Annual Holding Period ReturnsAnnual Holding Period Returns
Geom. Arith. Stan.
Series Mean% Mean% Dev.%
World Stk 9.41 11.17 18.38
US Lg Stk 10.23 12.25 20.50
US Sm Stk 11.80 18.43 38.11
Wor Bonds 5.34 6.13 9.14
LT Treas 5.10 5.64 8.19
T-Bills 3.71 3.79 3.18
Inflation 2.98 3.12 4.35
Historical Returns: 1926-2003
Fin 2802, Spring 10 - TangChapter 5: Risk and Return
21Fin 2802, Spring 10 - Tang
rrNegativeNegative PositivePositive
Skewed Distribution: Large Negative Returns PossibleSkewed Distribution: Large Negative Returns Possible
Median
Chapter 5: Risk and ReturnChapter 5: Risk and Return
22Fin 2802, Spring 10 - Tang
rrNegativeNegative PositivePositive
Skewed Distribution: Large Positive Returns PossibleSkewed Distribution: Large Positive Returns Possible
Median
Chapter 5: Risk and ReturnChapter 5: Risk and Return
23
Table 5.5 Risk Measures for Non-Normal DistributionsTable 5.5 Risk Measures for Non-Normal Distributions
Fin 2802, Spring 10 - TangChapter 5: Risk and Return
24Fin 2802, Spring 10 - TangChapter 5: Risk and Return
SummarySummary
Definition of Returns: HPR, APR and AER.
Risk and expected return
Next: Asset Allocation