1 fin 2802, spring 10 - tang chapter 5: risk and return fina2802: investments and portfolio analysis...

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


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


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


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


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%


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


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


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%


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%


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


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


Normal distributionNormal distribution



HPR - Risk MeasureHPR - Risk Measure

Variance or standard deviation:

22 s

rEsrsp

2


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?



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%


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


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



rrNegativeNegative PositivePositive

Skewed Distribution: Large Negative Returns PossibleSkewed Distribution: Large Negative Returns Possible

Median



rrNegativeNegative PositivePositive

Skewed Distribution: Large Positive Returns PossibleSkewed Distribution: Large Positive Returns Possible

Median


23

Table 5.5 Risk Measures for Non-Normal DistributionsTable 5.5 Risk Measures for Non-Normal Distributions



SummarySummary

Definition of Returns: HPR, APR and AER.

Risk and expected return

Next: Asset Allocation

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