Investment Analysis and Portfolio Management

Instructor: Attila Odabasi

Introduction to Risk, Return, and the Historical Record on T-bills

and US equityRef Ch 5, BKM

Learning Objectives

• Time Value of Money – Review

• How to calculate the return on an investment using different methods.

• The historical returns on various important types of investments.

• The historical risks of various important types of investments.

• The relationship between risk and return.

1-2

The Time Value of Money

• Future Value:• $V invested for n years at simple interest

rate r per year• Compounding of interest occurs at the end

of year

FVt = $V * (1+r)t

where FVt is future value after t years

5-3

The Time Value of Money• Example: Consider putting $1000 in an

interest checking account that pays a simple annual percentage rate of 3%. The future value after t = 1, 5, 10 years is, respectively,

FV1 = $1000 * (1.03)1 = $1030,

FV5 = $1000 * (1.03)5 = $1159.27,

FV10 = $1000 * (1.03)10 = $1343.92

5-4

TVM

• FV function is a relationship between four variables: FVn, V, r, t. Given three variables, you can solve for the fourth:

• Present Value

• Compound annual return

• Investment horizon

5-5

tt

r

FVV

)1(

1/1

t

t

V

FVr

)1ln(

)/ln(

r

VFVt t

Compounding occurs m times per year

return periodic

,)1($

m

rm

rVFV tmm

t

5-6

Continuous compounding:

71828.2e where

$)1($lim

1

trtm

mt Ve

m

rVFV

Example

• If the simple annual percentage rate is 10% then the value of $1000 at the end of one year (t =1) for different values of m is given in the table below.

5-7

Compounding Frequency

Value of $1000 at end of 1 year (r = 10%)

Annually (m = 1) 1100.00

Quarterly (m = 4) 1103.81

Weekly (m = 52) 1105.06

Daily (m = 365) 1105.16

Continuously (m = ∞) 1105.17

Quoting Rates of Return

• Annual Percentage Return (or Simple Annual Return) can be translated to an effective annual rate (EAR) by:

• Note that APR or a given EAR:

5-8

11

11

1$)1($

m

m

m

m

APREAR

m

APREAR

m

APRVEARV

]1)1[(

)1(1

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m

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EARm

APR

Continuous Compounding

5-9

)1ln(

lyequivalentor

1

)1(

EARAPR

eEAR

eEARAPR

APR

Example: The effective annual rate associated with continuously compounded (m=∞) APR= 10% is determined by:

%517.10

10517.0

110517.1110.0

EAR

EAR

eEAR

EAR of APR with continuous compounding:

10

Measuring Past Returns

One period investment: • Holding Period Percentage Return (HPR): • HPR= r1 = P1 – P0 + D1

P0

• P0 = Beginning price (or, PV)

• P1 = Ending price (or, FV)

• D1 = Dividend (cash flow) during period

• Q: Why use % returns at all?• Q: What are we assuming about the cash flows in

the HPR calculation?

Measuring Past Returns

• HPR defined as:• r1 = P1 – P0 + D1 = (P1+D1) - 1

P0 P0

• Then we can define ‘gross return’: (1 + r1) = (P1+D1) / P0

11

Annualizing HPRs (n > 1 year)Q: Why would you want to annualize returns?1. Annualizing HPRs for holding periods of greater than one year:

• Without compounding (Simple or APR): HPRann = HPR / n

• With compounding: EAR• HPRann = [(1+HPR)1/n]-1

where n = number of years held

Ex:Annualizing Ex-Post (Past) Returns

• Example: Suppose that the price of Microsoft stock 24 months ago is Pt-24 = $50 and the price today is Pt = $90.

• The two –year gross return is:

1+rt(2) = 90/50 = 1.80

• The two-year net return is: rt(2) = 0.80 = 80%.

• The annual return for this investment:HPRann = 0.80/2 = 0.40 or 40% Annualized w/out comp

• The annual HPR assuming annual compounding is (n=2):HPRann = (1+0.80)1/2 - 1 = 1.3416 – 1= 0.3416 or 34.16%

5-13

Annualizing Past Returns (n<1y)

Annualizing HPRs for holding periods of less than one year:

• Simple annual return: HPRann = HPR x n

• Compounded annual return: HPRann = [(1+HPR)n]-1

where n = number of compounding periods per year

Ex: Annualizing past returns (n < 1y)

• You buy 200 shares of Lowe’s Companies, Inc. at $48 per share. Three months later, you sell these shares for $51 per share. You received no dividends. What is your return? What is your annualized return?

• Return: (Pt+1 – Pt) / Pt = ($51 - $48) / $48 = 0.0625 or 6.25% for 3 months

• Without: n=12/3= 4 so HPann=HPRxn= 0.0625x4=0.25

• With: HPRann = (1.06254) - 1 = 0.2744

Q: Why is the compound return greater than the simple return?

1-15

Multi-period vs One-period Returns

5-16

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returns grossmonth -one twoofproduct thereturn grossmonth -two

)2(

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dividends) (ignoringreturn month - twoSimple

11

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Ex: Multi-period vs One-period Returns

• Suppose the price of Msoft stock in month t-2 is $80, $85 in t-1, and $90 at t. No dividends paid between t-2 and t.

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: toequalnot isreturn siplemonth - twoThe

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8590

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8085

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Averages of Multi-period Returns

18

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Arithmetic vs Geometric Average Returns

• Arithmetic and geometric average returns will give different values for the returns over some evaluation period.

• Arithmetic average return: the amount invested is assumed to be maintained at the initial market value

• The geometric average return: it is a return on an investment that varies in size because of the assumption that all proceeds are reinvested.

19

Measuring Ex-Post (Past) ReturnsQ: When should you use the GAR and when should

you use the AAR? A1: When you are evaluating PAST RESULTS (ex-

post):

A2: When you are trying to estimate an expected return (ex-ante return):

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.

Use the AAR

Continuously Compounded (cc) Returns

• Let rt = simple monthly (or daily) return on an investment

• Corresponding continuously compounded monthly return (rcc) :

5-21

tr

tr

t

tt

cc

tr

re

re

P

Prr

re

cc

cc

cc

1

: calculatecan wegiven also

)ln()1ln(

1

1

Example: Compute cc return

• Let Pt-1 = 85, Pt = 90 then simple return is rt= 0.0588.

• The cc monthly return can be computed as:• rcc = ln(1.0588) = 0.0571• rcc = ln(90) – ln(85) = 4.4998 – 4.4427 =

0.0571

• Notice that cc return is slightly smaller than simple return.

• Notice also: rt = ercc – 1 = e0.0571 – 1 = 0.05885-22

Multi-period cc Returns

5-23

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This is the main difference between simple and cc returns.

Ex: Multi and one-period cc returns

• Suppose Pt-2= 80, Pt-1= 85, Pt= 90. The cc two-month return can be computed in two equivalent ways:

1) Take difference in log prices:

2) Sum the two cc one-month returns:

5-24

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1178.00607.00571.0)2(

0607.0)80ln()85ln(

0571.0)85ln()90ln(

1

cct

cct

cct

r

r

r

Adjusting for Inflation• The computation of real returns on an asset:

• Deflate the nominal price Pt of the asset by an index of the general price level CPIt

• Compute returns in the usual way using the deflated prices

5-25

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Ex: Adjusting for Inflation• Consider a one-month investment in Msoft

stock. Suppose the CPI in months t-1 and t is 1 and 1.01, respectively. The real prices of the stock are:

5-26

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5-27

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28

Ex: Adjusting for Inflation

• Suppose you buy a 0-coupon T-Bond maturing in 20 years, priced to yield 12%

Price = $1000/(1.12)20 = $103.67

• If the CPI is 1.00 today and 2.65 in 20 years, what is your Real rate of return?

0667.01639997.31

65.2

1

67.103

10001 2020Re

k

kt

t

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Ex-Ante Return Estimation

• Some asset classes are called risky. They offer a risk premium over risk-free assets.

• These assets involve some degree of uncertainty about future holding-period returns.

• We have to estimate these future holding period returns and related uncertainty.

• How?29

Expected Return & Std; Scenario Analysis

Purchase Price 100

T-bill Rate 0.04

State of the market Prob Y-E Price

Cash Dividend HPR

Deviations from the mean

Squared Deviations from Mean

Excess Returns

Sqrd Deviations from Mean

Excellent 0.25 126.50 4.50 0.3100 0.2124 0.0451 0.2700 0.0729

Good 0.45 110.00 4.00 0.1400 0.0424 0.0018 0.1000 0.0100

Poor 0.25 89.75 3.50 -0.0675 -0.1651 0.0273 -0.1075 0.0116

Crash 0.05 46.00 2.00 -0.5200 -0.6176 0.3815 -0.5600 0.3136

Expected Value 0.0976<-- {=SUMPRODUCT(B5:B8,E5:E8)}

Variance of HPR 0.0380<-- {=SUMPRODUCT(B5:B8,G5:G8)}

Std of HPR 0.1949<-- =SQRT(E11)

Risk Premium 0.0576<-- {=SUMPRODUCT(B5:B8,H5:H8)}

Std of Excess Returns 0.2032 <-- {=SQRT(SUMPRODUCT(B5:B8,I5:I8))}

5-30

Time Series Analysis of Past Returns

5-31

Period

Implicitly Assumed Prob

= 1/5 HPR (decimal)Squared

DeviationGross HPR=

1 + HPR Wealth Index

2000 100.000 100.000

2001 0.2 88.110 -0.1189 0.0196 0.8811 88.110

2002 0.2 68.638 -0.2210 0.0586 0.7790 68.638

2003 0.2 88.330 0.2869 0.0707 1.2869 88.330

2004 0.2 97.940 0.1088 0.0077 1.1088 97.940

2005 0.2 102.749 0.0491 0.0008 1.0491 102.749

Arithmetic Average 0.0210<-- =AVERAGE(D4:D8)

Expected Value 0.0210<-- =SUMPRODUCT(B4:B8,D4:D8)

Standard Deviation 0.1774<-- =SUMPRODUCT(B4:B8,E4:E8)^0.5

Standard Deviation 0.1774<-- =STDEV.P(D4:D8)

Geometric Average Return 0.0054<-- =GEOMEAN(F4:F8)-1

Geometric Average Return 0.0054<-- =((G8/G3)^0.2)-1

Using Ex-Post Returns to estimate Expected HPR

Estimating Expected HPR (E[r]) from ex-post data.

Use the arithmetic average of past returns as a forecast of expected future returns as we did and,

Perhaps apply some (usually ad-hoc) adjustment to past returns

Problems?• Which historical time period?

• Have to adjust for current economic situation

1-33

The Normal Distribution• We assume that (past) returns are

distributed normally!• Normal distribution: A symmetric, bell-shaped

frequency distribution that can be described with only an average and a standard deviation.

• Mean: Average return• Variance is a common measure of return dispersion. • Standard deviation is the square root of the

variance. • Standard Deviation is handy because it is in the

same "units" as the average.• Standard Deviation is a good measure of risk

when returns are symmetric around the mean.• If security returns are symmetric, portfolio returns

will be, too.

5-34

Figure 5.4 The Normal Distribution

Characteristics of Probability Distributions

1. Mean: __________________________________ _

2. Median: _________________

3. Variance or standard deviation:

4. Skewness:_______________________________

5. Leptokurtosis: ______________________________

If a distribution is approximately normal, the distribution is fully described by _____________________

Arithmetic average & usually most likely

Dispersion of returns about the mean

Long tailed distribution, either side

Too many observations in the tails

Characteristics 1 and 3

Middle observation

1-36

Given Historical Returns

• Expected Return:

n

returnyearly Return AverageHistorical

n

1i

1-37

Return Variability: The Statistical Tools

• The formula for return variance is ("n" is the number of returns):

• Sometimes, it is useful to use the standard deviation, which is related to variance like this:

1N

rr σ VAR(r)

N

1i

2i

2

VAR(r) σ SD(r)

5-38

Skew and Kurtosis

Skew

Equation 5.19

Kurtosis

• Equation 5.20

3

^3

_

rr

averageskew 3

4

^4

_

rr

averagekurtosis

rrNegative Positive

Skewed Distribution: Large Negative Returns (Left Skewed)

Median

rr = averageImplication?

is an incomplete risk measure

Negative Positive

Skewed Distribution: Large Positive Returns (Right Skewed)

Median

rr = average

rr

LeptokurtosisImplication?

is an incomplete risk measure

Value at Risk (VaR)

Value at Risk attempts to answer the following question:

• How many dollars can I expect to lose on my portfolio in a given time period at a given level of probability?

• The typical probability used is 5%. • We need to know what HPR corresponds to a 5%

probability. • If returns are normally distributed then we can use a

standard normal table or Excel to determine how many standard deviations below the mean represents a 5% probability:• From Excel: =Norminv (0.05,0,1) = -1.64485

standard deviations

Value at Risk (VaR)From the standard deviation we can find the

correspondinglevel of the portfolio return:

VaR = E[r] + -1.64485

For Example:A $500,000 stock portfolio has an annual expected

return of 12% and a standard deviation of 35%. What is the portfolio VaR at a 5% probability level?

VaR = 0.12 + (-1.64485 * 0.35)VaR = -45.57% (rounded slightly)VaR$ = $500,000 x -.4557 = -$227,850

What does this number mean?

Value at Risk (VaR)

VaR versus standard deviation:• For normally distributed returns VaR is equivalent to

standard deviation (although VaR is typically reported in dollars rather than in % returns)

• VaR adds value as a risk measure when return distributions are not normally distributed. • Actual 5% probability level will differ from

1.68445 standard deviations from the mean due to kurtosis and skewness.

Value at Risk (VaR)

• Simple approach:

• Assume we have 100 HPR observations not necessrily normally distributed.

• To obtain an estimate of VaR of this sample of 100 observations, rank the returns from highest to lowest.

• Find the 5th percentile:• ........... -25% / -26% -30% -33% -35% -40%

45

46

Expected Shortfall (ES)

• Also called conditional tail expectation (CTE)

• More conservative measure of downside risk than VaR• VaR takes the highest return from

the worst cases• ES takes an average return of the

worst cases• (26% + 30% + 33% +35%

+40%)/5= 32.8%

Risk Premium & Risk Aversion• The risk free rate is the rate of return that can be

earned with certainty.• The risk premium is the difference between the

expected return of a risky asset and the risk-free rate.Excess Return or Risk Premiumasset =

Risk aversion is an investor’s reluctance to accept risk.

How is the aversion to accept risk overcome? By offering investors a higher risk premium.

E[rasset] – rf

Frequency distributions of annual HPRs,

1926-2008, Historical Records

Rates of return on stocks, bonds and bills, 1926-2008

Annual Holding Period Returns Statistics

1926-2008

• Geometric mean: Best measure of compound historical return over the period

• Arithmetic Mean:Expected return, best estimate for next year’s single-period return

• Deviations from normality?

Geom. Arith. Excess Sharpe

Series Mean% Mean% Return% SD Kurt. Skew. Ratio

World Stk 9.20 11.00 7.25 18.28 1.03 -0.16 0.396

US Lg. Stk 9.34 11.43 7.68 20.67 -0.10 -0.26 0.371

Sm. Stk 11.43 17.26 13.51 37.26 1.60 0.81 0.362

World Bnd 5.56 5.92 2.17 9.05 1.10 0.77 0.239

LT Bond 5.31 5.60 1.85 8.01 0.80 0.51 0.231

T-Bill 3.75 3.08

Deviations from Normality: Another Measure

Portfolio

World Stock

US Small Stock US Large Stock

Arithmetic Average .1100 .1726 .1143

Geometric Average .0920 .1143 .0934

Difference .0180 .0483 .0209

½ Historical Variance

.0186 .0694 .0214

If returns are normally distributed then the following relationship among geometric and arithmetic averages holds:Arithmetic Average – Geometric Average = ½ 2

• The comparisons above indicate that US Small Stocks may have deviations from normality and therefore VaR may be an important risk measure for this class.

Sharpe Ratio (Reward-to-volatility)

• Risk aversion implies that investors will accept a higher return in exchange of a higher risk as measured by the std of returns.

• A statistic commonly used to rank assets in terms of risk-return trade-off is the Sharpe Measure:

• The higher the Sharpe ratio the better. 5-52

fi rr

fi rrESR

][

return excess of SD

PremiumRisk Ratio Sharpe

Historic Returns on Risky Portfolios

• Observations:• Returns appear normally distributed• E(Geomean)=E(arith Mean) – ½ var• Except small stocks, equation is nearly satisfied.• Overall, no serious deviations from normality

observed.

1-54

Lesson: Risk and Return

• The First Lesson: There is a reward, on average, for bearing risk.

• That is if we are willing to bear risk, then we can expect to earn a risk premium, at least on average.

• Second Lesson: Further, the more risk we are willing to bear, the greater the expected risk premium.