MEASURING INVESTMENT RETURNS AND RISKS

COMPARING RETURN INFORMATION

Holding period return

Pt = Price of asset at time tRt = % return from time t-1 to tCFt = cash flow from time t-1 to t - e.g. dividend

R1 =

= (%capital gain) + (%dividend yield)

P P CF

P

P P

P

CF

P1 0 1

0

1 0

0

1

0

QUESTION: How can you measure the return you expect from an asset?

The best guess at the future return i.e., what one should expect is the mean return:

where Ri is an observation of the variable (return) and

is the arithmetic sample mean of the variable (return)

R R ni

n

i

1 /

R

EXAMPLE OF ARITHMETIC MEAN

P0 = 100 - price nowP1 = 200 - price at time 1 period from now

R1 = (200 - 100)/100

= 1.00 or 100%

Suppose P2 = 100 - period 2 price

R2 = ((100 - 200)/200= -.5 or -50%

Arithmetic mean return = (R1 + R2 )/2 = .25 or 25%R

QUESTION: Is an average return of 25% a good return? Is it accurate?

ANSWER: Not for past returns. Instead use Geometric return (also called compound return) especially when returns are volatile. Geometric mean ≤ arithmetic mean.

G =

G = is the geometric mean returnRi = the return in period i = the product operator

The geometric mean using the previous data is:

G = (1 + 1.0)1/2 x (1 + (-.5))1/2 - 1 = [(2)(.5)]1/2 - 1 = 0

i

n

inR

1

11 1( ) /

Question: Who knows what this symbol means?Answer: This is the Chinese ideograph for Risk (or crisis). It is a combination of the ideograph for Danger (first symbol) and the ideograph for Opportunity (second one). Why are they together?

Who likes taking risks? –playing card games – poker?

Who wants to make millions of dollars?

RISK AVERSE INVESTORS REQUIRE MORE RETURN TO HOLD ASSETS WITH MORE RISK

QUESTION: What is risk?

ANSWER: The likelihood that you will not receive what you expect - i.e. the mean risk free

you always get what you expect

Variance and Standard Deviation - ivolatility.com

sample variance = 2 /n sample standard deviation =

for = 10% and = 10%

-20% -10% 0 10% 20% 30% 40%

R

i

n

iR R

1 ( )

-3 -2 -1 Mean +1 +2 +3

For the normal distribution, one, two and three standard deviations about themean delineate where observations of a variable should fall 68, 95 and 99percent of the time, respectively.

COEFFICIENT OF VARIATION (CV)

CV = (Std. Dev. of return)/(mean return)

= risk/ arithmetic expected return

says, for each percent of mean return, how much volatility must you bear.

PROBLEM: Suppose asset 1 has a standard deviation of.08 and a mean return of .06 while asset 2 has astandard deviation of .05 and a mean return of .03.

Which is best? (Asset 1)

CV1 = .08/.06 = 1.33 still has more total riskCV2 = .05/.03 = 1.67

1

2

These two distributions have the same mean but 1’s variance is smaller than 2’s.

If these represent stock returns, a risk averse investor should choose stock 1.

1

2

When two stock return distributions have different means and variances, a risk averse investor choosing between them needs a method that compares mean return relative to risk, such as coefficient of variation or the capital asset pricing model.

Ibbotson Sinquifield Data – 1926-2009

Geometric Arithmetic StandardStocks Mean Mean DeviationLarge Cos. 10.4 12.4 20.3Small Cos. 12.7 17.5 33.1

BondsLong Corp. 5.9 6.2 8.6Long Treas. 5.4 5.8 9.3Med. Treas. 5.4 5.5 5.7Tbills 3.7 3.8 3.1

QUESTION: Compare the Risk/Return tradeoff of Bonds to stocks and T Bills to Bonds. -anything

unusual here?

ANSWER: Bonds have relatively low returns but large variation due to unexpected inflation

-87% -52% -17% 18% 53% 88% 123%

-48 -28 -8 12 32 52 72

-22 -13 -4 5 14 23 32

-5 -2 1 4 7 10 13

SMALL STOCKSmean=18%standard deviation=35%

LARGE STOCKSmean=12%standard deviation=20%

LONGTERM U.S. GOVERNMENT BONDSmean=5%standard deviation=9%

U.S. TREASURY BILLSmean=4%standard deviation=3%

There are considerable differences among return distributions for these common asset types.

Expected Return and Variance Using Probability Distributions

Probabilities are weights attached to scenarios or observation classes where i indexes scenarios.

Expected return = E(R1) = Probabilityi x Return1i

Return Variance = 2(R1) = Probabilityi x [Return1i - E(R1)]2

Sample mean and variance assumes that each observation has equal probability which is acceptable if the sample covers a full economic cycle.

Return standard deviation = (R1) = [2(R1)]1/2

How to Get Probabilities

You can get some probabilities from past data. If stocks earned 10% in 30 years over the last 100 years, then the probability of earning 10% is .30.

But this approach uses past data which may not reflect expected future events.

Another approach is to use data from websites like intrade.com that offer bets on events for prices that reflect the probability that the event will occur.

Price = probability1(payoff1) + probability2(payoff2)

= probability1(1) + probability2(0)

Price = probability1

EXPECTED RETURN, VARIANCE, AND COVARIANCE OF A PORTFOLIO USING JOINT

PROBABILITY DISTRIBUTION

Assume portfolio consisting of stock 1 and stock 2:

Expected portfolio return =

E(Rp) = Weight1 x E(R1) + Weight2 x E(R2)

Portfolio Variance = p2

=Weight12 x (R1) + Weight2

2 x (R2) + 2 x Weight1 x Weight2 x (R1, R2)

where (R1, R2) = covariance of stock 1 and stock 2 returns.

NO COST TO DIVERSIFY

Diversifiable risk can be eliminated easily so - no compensation. Only undiversified risk should receive compensation - Covij risk

where Covariance = (R1, R2) Probabilityi x [Return1i - E(R1)][Return2i - E(R2)]

In a portfolio, there is a covariance for each asset pairing – the many covariances account for most of a portfolio’s variance.

All else equal, covariance is large when the data points fall along the regression line instead of away from it because, on the line, the deviations from the means of each variable are equal – the products are squares - larger than otherwise.

Illustrate surprising probabilities with student birthdays. Question: What is the probability that two students in class have the same birthday? Individual pairings have small probability but there are many pairings.

It’s usually best to diversify, except in this case.

Stock 1Return

NegativeCorrelation

PositiveCorrelation

ZeroCorrelation

Stock 2 Return

Stock 1Return

Stock 1Return

Stock 2 Return Stock 2 Return

PortfolioRisk

Number of securities in the portfolio

Diversifiable risk drops as more securitiesare added to a portfolio.

Diversifiable Risk

Nondiversifiable Risk

Example: Bill Gates started with $100 billion in Microsoft - now sells $50 million each month and buys other stocks.

QUESTION: Consider two stocks with the following return distributions. Find the variance for a portfolio with 40% invested in stock 1 and 60% invested in stock 2.

Obama Liberal Conservative Moderate

Probability .50 .10 .40stock 1 return .10 .15 .20stock 2 return .25 .10 .05

Portfolio Return DistributionExample: Probability Distribution Results

E(R1) = .50(.10) + .40(.20) + .10(.15) = .145

E(R2) = .50(.25) + .40(.05) + .10(.10) = .155

(R1) = .50(.10 - .145)2 + .40(.20 - .145)2 + .10(.15 - .145)2 = .0022375

(R2) = .50(.25 - .155)2 + .40(.05 - .155)2 + .10(.10 - .155)2 = .009235

(R1, R2) = .50(.10 - .145)(.25 - .155) + .40(.20 - .145) (.05 - .155) + .10(.15 - .145)(.10 - .155) = -.004476

correlation = (R1, R2)/(R1) (R2) = -.004475/[(.0473022)(.0960989)] = -.98

E(Portfolio return) = .4(.145) + .6(.155) = .151

Portfolio Variance = (.4)2 (.0022375) + (.6)2 (.009235) + 2(.4)(.6)(-.004475) = .0015346

QUESTION: Suppose Obama chose Chris Dodd as Treasury Secretary instead of New York Fed President Timothy Geithner. How would you restructure the portfolio? - more of stock 2 and less of stock 1.

Illustrate correlation and optimal portfolio weights using

www.wolframalpha.com (click “Examples”, then Money and Finance” then under “compare several stocks” put in up to 4 stock ticker symbols)

Explain the optimal portfolio return and volatility.

For more complex portfolio optimization statistics and to input more than 4 stocks go to Macroaxis.com, register for free and create a portfolio – then try the “management” option and try “optimize” or “suggest”.

CORRELATION AND HEDGING - FIRE INSURANCE

Correlation - in general - hedging takes advantage ofnegative correlation but less than perfect correlation can be used to reduce risk.

EXAMPLE Suppose you own a $200,000 house.

QUESTION: If the probability of fire = .001 and the cost of a fire insurance policy is $700, what is your expected return on the insurance policy alone. Is it risky? Is insurance a good investment?

Probability Outcome % Return

Fire .001 200,000-700 199,300/700= 28471%

No fire .999 -700 -700 / 700 = -100%

Expected return and variance of policy on its own (like Walmart buys life insurance policies on its employees).

E(R) = .001(28471) + .999(-100) = -71.43%

2 = .001( 284.71 - (-0.7143))2 + .999( -1 - (-0.7143))2 = 81.46 + .0815 = 81.54 = 8154%

Expected return on house without insurance = [.999 (0) + .001 (-1.00)] = -.001 = -.1%

Variance of return on house without insurance= [.999 (0 - (-.001))2 + .001 (-1.00 - (-.001))2] = .0001

Expected return with insurance = [ .001 (-700/200,700) + .999 (- 700/200,700)]

= 1.00 (- 700/200,700) = -.0035 = -.35%

Variance of return with insurance

= [.001 [(- 700/200,700) - (-.0035)]2 + .999 [(-700/200,700) - (-.0035)]2

= 1.00 (0) = 0%

Insurance doesn't look like a good investment but return variance is zero with insurance, hence insurance is valuable for hedging or risk reduction purposes.

Question: Many stores offer insurance or service contracts on items such as DVD players to cover costs after warranties run out. Why is this coverage typically an even poorer investment than home or automobile coverage?

Beta is a Standardized Covariance

Beta is the slope of a regression line of an asset’s returns on the market portfolio’s return.

Beta1 = =

where m signifies market and 1 signifies stock 1.

All asset betas are measured against the market so all are being compared with the same gauge.

If B1 = 1 and B2 =.5, stock 1 is twice as risky as stock 2

Cov m

m

1

2

,

Corr m m

m

1 12

CAPM - Capital Asset Pricing ModelBeta is used as a measure of risk in a theoretical return equation called CAPM.

E(Ri) = Rf + Bi[E(Rm) - Rf]

where E(Ri) = expected return on stock i

Rf = the risk-free rate of return

Bi = stock i’s beta

E(Rm) = expected return on the market portfolio

Illustrate diversification with

stockcharts.com – carpets

Holding a diversified portfolio is best unless you have better estimates of payoff probabilities for a stock than the market.

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Homestake's Return

0.40.30.20.10-0.1

S&P 500 Return

The correlation between Homestake and the S&P 500 is 0.18 and its beta is 0.54

Annual return pairs for the S&P 500 and Homestake Mining's stock

Slope is 0.54

Year S&P Homestake1983 0.23 0.011984 0.06 -0.261985 0.32 0.11986 0.18 0.091987 0.05 0.391988 0.17 -0.271989 0.31 0.551990 -0.03 -0.091991 0.3 -0.161992 0.08 -0.251993 0.1 0.831994 0.01 -0.19

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0.6

0.5

0.4

0.3

0.2

0.1

0

-0.1

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Gasoline Return

0.40.30.20.10-0.1

S&P 500 Return

Gasoline's correlation with the S&P 500 is -0.47 and its beta is -2.11.

Annual return pairs for the S&P 500 and gasoline

Slope is -2.11

Year S&P Gas1983 0.23 0.081984 0.06 -0.11985 0.32 0.091986 0.18 -0.451987 0.05 0.191988 0.17 -0.041989 0.31 -0.081990 -0.03 0.731991 0.3 -0.331992 0.08 -0.071993 0.1 -0.291994 0.01 0.2

Year S&P Gold1983 0.23 -0.11984 0.06 -0.161985 0.32 01986 0.18 0.251987 0.05 0.21988 0.17 -0.171989 0.31 01990 -0.03 -0.051991 0.3 -0.051992 0.08 -0.061993 0.1 0.121994 0.01 0.02

0.4

0.2

0

-0.2

0.40.30.20.10-0.1

S&P 500 Return

The correlation between gold and the S&P 500 and its beta is approximately zero.

Annual return pairs for the S&P 500 and Gold

Slope is zero

Go ld

Return

High Beta

Low Beta

Market

StockReturn

During this time period the market rises, falls, and then rises again. A high (low) beta stock varies more (less) than the market.

Positive Beta

Negative Beta

StockReturn

Positive and negative beta stock returns move opposite one another.

Illustrate beta and return distributions using

www.wolframalpha.com (click “Examples”, then Money and Finance” then under Stock Data put in a company)