fm10e ch06

Chapter 6 - Risk and Rates Chapter 6 - Risk and Rates of Returnof Return

2005, Pearson Prentice Hall

Chapter 6: ObjectivesChapter 6: Objectives

Inflation and rates of returnInflation and rates of return How to How to measuremeasure risk risk

(variance, standard deviation, beta)(variance, standard deviation, beta) How to How to reducereduce risk risk

(diversification)(diversification) How to How to priceprice riskrisk

(security market line, Capital Asset (security market line, Capital Asset Pricing Model)Pricing Model)

Inflation, Rates of Return, Inflation, Rates of Return, and the Fisher Effectand the Fisher Effect

InterestRates

Interest RatesInterest RatesConceptually:

Nominalrisk-freeInterest

Realrisk-freeInterest

Inflation-risk

premium

Conceptually:

Inflation-risk

premium

Mathematically:

Interest RatesInterest Rates

Conceptually:

Inflation-risk

premium

Mathematically:

(1 + krf) = (1 + k*) (1 + IRP)

Conceptually:

Inflation-risk

premium

Mathematically:

(1 + krf) = (1 + k*) (1 + IRP)

This is known as the “Fisher Effect”

Suppose the real rate is 3%, and the nominal Suppose the real rate is 3%, and the nominal rate is 8%. What is the inflation rate rate is 8%. What is the inflation rate premium?premium?

(1 + k(1 + krfrf) = (1 + k*) (1 + IRP)) = (1 + k*) (1 + IRP)

(1.08) = (1.03) (1 + IRP)(1.08) = (1.03) (1 + IRP)

(1 + IRP) = (1.0485),(1 + IRP) = (1.0485), so so

IRP = 4.85%IRP = 4.85%

Term Structure of Interest RatesTerm Structure of Interest Rates

The pattern of rates of return for debt The pattern of rates of return for debt securities that differ only in the length of securities that differ only in the length of time to maturity.time to maturity.

yieldto

maturity

time to maturity (years)

yieldto

maturity

yieldto

maturity

The yield curve may be downward The yield curve may be downward sloping or “inverted” if rates are sloping or “inverted” if rates are expected to fall.expected to fall.

yieldto

maturity

The yield curve may be downward The yield curve may be downward sloping or “inverted” if rates are sloping or “inverted” if rates are expected to fall.expected to fall.

For a Treasury security, what is For a Treasury security, what is the required rate of return?the required rate of return?

RequiredRequired

rate of rate of

returnreturn==

For a Treasury security, what is For a Treasury security, what is the required rate of return?the required rate of return?

Since Treasuries are essentially Since Treasuries are essentially free of free of default riskdefault risk, the rate of return on a , the rate of return on a Treasury security is considered the Treasury security is considered the

““risk-freerisk-free”” rate of return. rate of return.

RequiredRequired

rate of rate of

returnreturn==

Risk-freeRisk-free

rate of rate of

returnreturn

For a For a corporate stock or bondcorporate stock or bond, , what is the required rate of return?what is the required rate of return?

RequiredRequired

rate of rate of

returnreturn==

RequiredRequired

rate of rate of

returnreturn==

Risk-freeRisk-free

rate of rate of

returnreturn

How large of a How large of a risk premiumrisk premium should we should we require to buy a corporate security? require to buy a corporate security?

RequiredRequired

rate of rate of

returnreturn== + +

Risk-freeRisk-free

rate of rate of

returnreturn

RiskRisk

premiumpremium

ReturnsReturns

Expected ReturnExpected Return - the return that an - the return that an investor expects to earn on an asset, investor expects to earn on an asset, given its price, growth potential, etc.given its price, growth potential, etc.

Required ReturnRequired Return - the return that an - the return that an investor requires on an asset given investor requires on an asset given itsits riskrisk and market interest rates.and market interest rates.

Expected ReturnExpected Return

State of Probability ReturnState of Probability Return

Economy (P) Economy (P) Orl. Utility Orl. TechOrl. Utility Orl. Tech

Recession .20 4% -10%Recession .20 4% -10%

Normal .50 10% 14%Normal .50 10% 14%

Boom .30 14% 30%Boom .30 14% 30%

For each firm, the expected return on the For each firm, the expected return on the stock is just a stock is just a weighted averageweighted average::

Normal .50 10% 14%Normal .50 10% 14%

Boom .30 14% 30%Boom .30 14% 30%

For each firm, the expected return on the For each firm, the expected return on the stock is just a stock is just a weighted averageweighted average::

k = P(kk = P(k11)*k)*k11 + P(k + P(k22)*k)*k22 + ...+ P(k + ...+ P(knn)*kn)*kn

Normal .50 10% 14%Normal .50 10% 14%

Boom .30 14% 30%Boom .30 14% 30%

k k (OU) (OU) = .2 (4%) + .5 (10%) + .3 (14%) = 10%= .2 (4%) + .5 (10%) + .3 (14%) = 10%

Normal .50 10% 14%Normal .50 10% 14%

Boom .30 14% 30%Boom .30 14% 30%

k k (OI) (OI) = .2 (-10%)+ .5 (14%) + .3 (30%) = 14%= .2 (-10%)+ .5 (14%) + .3 (30%) = 14%

Based only on your Based only on your expected returnexpected return

calculations, which calculations, which stock would you stock would you

prefer?prefer?

RISK?Have you considered

What is Risk?What is Risk?

The possibility that an The possibility that an actualactual return return will differ from our will differ from our expectedexpected return. return.

Uncertainty in the distribution of Uncertainty in the distribution of possible outcomes.possible outcomes.

What is Risk?What is Risk? Uncertainty in the distribution of Uncertainty in the distribution of

possible outcomes.possible outcomes.

4 8 12

Company A

returnreturn

-10 -5 0 5 10 15 20 25 30

Company B

4 8 12

Company A

returnreturn

How do We Measure Risk?How do We Measure Risk?

To get a general idea of a stock’s To get a general idea of a stock’s price variability, we could look at price variability, we could look at the the stock’s price rangestock’s price range over the over the past year.past year.

52 weeks Yld Vol NetHi Lo Sym Div % PE 100s Hi Lo Close Chg134 80 IBM .52 .5 21 143402 98 95 9549 -3

115 40 MSFT … 29 558918 55 52 5194 -475

How do We Measure Risk?How do We Measure Risk?

A more scientific approach is to A more scientific approach is to examine the stock’s examine the stock’s standard standard deviationdeviation of returns. of returns.

Standard deviation is a measure of Standard deviation is a measure of the the dispersion of possible outcomesdispersion of possible outcomes. .

The greater the standard deviation, The greater the standard deviation, the greater the uncertainty, and, the greater the uncertainty, and, therefore, the greater the risk.therefore, the greater the risk.

Standard DeviationStandard Deviation

= (k= (kii - k) - k)22 P(k P(kii)) n

Orlando Utility, Inc. Orlando Utility, Inc. Orlando Utility, Inc. Orlando Utility, Inc.

= (ki - k)2 P(ki) n

Orlando Utility, Inc. Orlando Utility, Inc.

( 4% - 10%)( 4% - 10%)22 (.2) = 7.2 (.2) = 7.2

= (ki - k)2 P(ki) n

( 4% - 10%)( 4% - 10%)22 (.2) = 7.2 (.2) = 7.2

(10% - 10%)(10% - 10%)22 (.5) = 0 (.5) = 0

( 4% - 10%)( 4% - 10%)22 (.2) = 7.2 (.2) = 7.2

(10% - 10%)(10% - 10%)22 (.5) = 0 (.5) = 0

= (ki - k)2 P(ki) n

( 4% - 10%)( 4% - 10%)22 (.2) = 7.2 (.2) = 7.2

(10% - 10%)(10% - 10%)22 (.5) = 0 (.5) = 0

(14% - 10%)(14% - 10%)22 (.3) (.3) = = 4.84.8

( 4% - 10%)( 4% - 10%)22 (.2) = 7.2 (.2) = 7.2

(10% - 10%)(10% - 10%)22 (.5) = 0 (.5) = 0

(14% - 10%)(14% - 10%)22 (.3) (.3) = = 4.84.8

= (ki - k)2 P(ki) n

( 4% - 10%)( 4% - 10%)22 (.2) = 7.2 (.2) = 7.2

(10% - 10%)(10% - 10%)22 (.5) = 0 (.5) = 0

(14% - 10%)(14% - 10%)22 (.3) (.3) = = 4.84.8Variance = 12Variance = 12

( 4% - 10%)( 4% - 10%)22 (.2) = 7.2 (.2) = 7.2

(10% - 10%)(10% - 10%)22 (.5) = 0 (.5) = 0

= (ki - k)2 P(ki) n

( 4% - 10%)( 4% - 10%)22 (.2) = 7.2 (.2) = 7.2

(10% - 10%)(10% - 10%)22 (.5) = 0 (.5) = 0

Stand. dev. = 12 =Stand. dev. = 12 =

( 4% - 10%)( 4% - 10%)22 (.2) = 7.2 (.2) = 7.2

(10% - 10%)(10% - 10%)22 (.5) = 0 (.5) = 0

Stand. dev. = 12 =Stand. dev. = 12 =

= (ki - k)2 P(ki) n

( 4% - 10%)( 4% - 10%)22 (.2) = 7.2 (.2) = 7.2

(10% - 10%)(10% - 10%)22 (.5) = 0 (.5) = 0

Stand. dev. = 12 = Stand. dev. = 12 = 3.46%3.46%

( 4% - 10%)( 4% - 10%)22 (.2) = 7.2 (.2) = 7.2

(10% - 10%)(10% - 10%)22 (.5) = 0 (.5) = 0

= (ki - k)2 P(ki) n

Orlando Technology, Inc. Orlando Technology, Inc.

= (ki - k)2 P(ki) n

(-10% - 14%)(-10% - 14%)22 (.2) = 115.2 (.2) = 115.2

= (ki - k)2 P(ki) n

(-10% - 14%)(-10% - 14%)22 (.2) = 115.2 (.2) = 115.2

(14% - 14%)(14% - 14%)22 (.5) = 0 (.5) = 0

= (ki - k)2 P(ki) n

(-10% - 14%)(-10% - 14%)22 (.2) = 115.2 (.2) = 115.2

(14% - 14%)(14% - 14%)22 (.5) = 0 (.5) = 0

(30% - 14%)(30% - 14%)22 (.3) (.3) = = 76.8 76.8

= (ki - k)2 P(ki) n

(-10% - 14%)(-10% - 14%)22 (.2) = 115.2 (.2) = 115.2

(14% - 14%)(14% - 14%)22 (.5) = 0 (.5) = 0

(30% - 14%)(30% - 14%)22 (.3) (.3) = = 76.8 76.8Variance = 192Variance = 192

= (ki - k)2 P(ki) n

(-10% - 14%)(-10% - 14%)22 (.2) = 115.2 (.2) = 115.2

(14% - 14%)(14% - 14%)22 (.5) = 0 (.5) = 0

Stand. dev. = 192 = Stand. dev. = 192 =

= (ki - k)2 P(ki) n

(-10% - 14%)(-10% - 14%)22 (.2) = 115.2 (.2) = 115.2

(14% - 14%)(14% - 14%)22 (.5) = 0 (.5) = 0

= (ki - k)2 P(ki) n

Which stock would you prefer?Which stock would you prefer?

How would you decide?How would you decide?

Which stock would you prefer?Which stock would you prefer?

How would you decide?How would you decide?

Orlando OrlandoOrlando Orlando

UtilityUtilityTechnologyTechnology

Expected ReturnExpected Return 10% 14%10% 14%

Standard DeviationStandard Deviation 3.46% 13.86%3.46% 13.86%

SummarySummary

It depends on your tolerance for risk! It depends on your tolerance for risk!

Remember, there’s a tradeoff between Remember, there’s a tradeoff between risk and return.risk and return.

Return

PortfoliosPortfolios

Combining several securities Combining several securities in a in a portfolioportfolio can actually can actually reduce overall riskreduce overall risk..

How does this work?How does this work?

Suppose we have stock A and stock B. The returns on these stocks do not tend to move together over time (they are not perfectly correlated).

rateof

return

rateof

return

rateof

return

What has happened to the variability of returns for the

portfolio?

rateof

return

rateof

return

What has happened to the variability of returns for the

portfolio?

DiversificationDiversification

Investing in Investing in more than onemore than one security security to to reduce riskreduce risk..

If two stocks are If two stocks are perfectly perfectly positivelypositively correlatedcorrelated, diversification has , diversification has no no effecteffect on risk. on risk.

If two stocks are If two stocks are perfectly perfectly negativelynegatively correlatedcorrelated, the portfolio is , the portfolio is perfectlyperfectly diversified.diversified.

If you owned a share of every stock If you owned a share of every stock traded on the NYSE and NASDAQ, traded on the NYSE and NASDAQ, would you be diversified?would you be diversified?

YES!YES! Would you have eliminated all of Would you have eliminated all of

your risk?your risk?

NO!NO! Common stock portfolios still Common stock portfolios still have risk. have risk.

Some risk can be diversified Some risk can be diversified away and some cannot.away and some cannot.

Market riskMarket risk ( (systematic risk)systematic risk) is is nondiversifiable. nondiversifiable. This type of risk This type of risk cannot be diversified away.cannot be diversified away.

Company-unique riskCompany-unique risk (unsystematic (unsystematic risk)risk) is is diversifiablediversifiable. This type of risk . This type of risk can be reduced through can be reduced through diversification.diversification.

Market RiskMarket Risk

Unexpected changes in interest Unexpected changes in interest rates.rates.

Unexpected changes in cash flows Unexpected changes in cash flows due to tax rate changes, foreign due to tax rate changes, foreign competition, and the overall competition, and the overall business cycle.business cycle.

Company-unique RiskCompany-unique Risk

A company’s labor force goes on A company’s labor force goes on strike.strike.

A company’s top management dies A company’s top management dies in a plane crash.in a plane crash.

A huge oil tank bursts and floods a A huge oil tank bursts and floods a company’s production area.company’s production area.

As you add stocks to your portfolio, As you add stocks to your portfolio, company-unique risk is reduced.company-unique risk is reduced.

portfolioportfolioriskrisk

number of stocksnumber of stocks

portfoliorisk

number of stocks

Market risk

portfoliorisk

number of stocks

Market risk

company-unique

Do some firms have more Do some firms have more market risk than others?market risk than others?

YesYes.. For example: For example:

Interest rate changes affect all firms, but Interest rate changes affect all firms, but which would be which would be moremore affected: affected:

a) Retail food chaina) Retail food chain

b) Commercial bankb) Commercial bank

YesYes.. For example: For example:

Interest rate changes affect all firms, but Interest rate changes affect all firms, but which would be which would be moremore affected: affected:

a) Retail food chaina) Retail food chain

b) b) Commercial bankCommercial bank

Do some firms have more Do some firms have more market risk than others?market risk than others?

NoteNoteAs we know, the market compensates As we know, the market compensates

investors for accepting risk - but investors for accepting risk - but only for only for market riskmarket risk.. Company- Company-unique risk can and should be unique risk can and should be diversified away.diversified away.

So - we need to be able to So - we need to be able to measuremeasure market risk.market risk.

This is why we have This is why we have Beta.Beta.

Beta: a measure of market risk.Beta: a measure of market risk. Specifically, beta is a measure of how Specifically, beta is a measure of how

an individual stock’s returns vary an individual stock’s returns vary with market returns.with market returns.

It’s a measure of the It’s a measure of the “sensitivity”“sensitivity” of of an individual stock’s returns to an individual stock’s returns to changes in the market.changes in the market.

A firm that has a A firm that has a beta = 1beta = 1 has has average average market riskmarket risk. The stock is no more or less . The stock is no more or less volatile than the market.volatile than the market.

A firm with a A firm with a beta > 1beta > 1 is is more volatilemore volatile than than the market. the market.

The market’s beta is The market’s beta is 11

A firm with a A firm with a beta > 1beta > 1 is is more volatilemore volatile than than the market. the market. (ex: technology firms)(ex: technology firms)

A firm with a A firm with a beta < 1beta < 1 is is less volatileless volatile than than the market.the market.

A firm with a A firm with a beta < 1beta < 1 is is less volatileless volatile than than the market.the market. (ex: utilities)(ex: utilities)

Calculating BetaCalculating Beta

-5-15 5 10 15

XYZ Co. returns

S&P 500returns

-5-15 5 10 15

XYZ Co. returns

S&P 500returns

. . . .

. . . .. . . .

. . . .

-5-15 5 10 15

XYZ Co. returns

S&P 500returns

. . . .

. . . .. . . .

. . . .

-5-15 5 10 15

XYZ Co. returns

S&P 500returns

. . . .

. . . .. . . .

. . . .

Beta = slope = 1.20

Summary:Summary:

We know how toWe know how to measuremeasure risk, using risk, using standard deviationstandard deviation for overall risk for overall risk and and betabeta for market risk. for market risk.

We know how to We know how to reducereduce overall risk overall risk to only market risk through to only market risk through diversificationdiversification..

We need to know how to We need to know how to priceprice risk so risk so we will know how much extra return we will know how much extra return we should require for accepting extra we should require for accepting extra risk.risk.

What is the Required Rate of What is the Required Rate of Return?Return?

The return on an investment The return on an investment requiredrequired by an investor given by an investor given market interest rates and the market interest rates and the investment’s investment’s riskrisk..

Required

rate of

return=

Required

rate of

return= +

Risk-free

rate of

return

Required

rate of

return= +

Risk-free

rate of

return

premium

marketrisk

Required

rate of

return= +

Risk-free

rate of

return

premium

marketrisk

company-unique risk

Required

rate of

return= +

Risk-free

rate of

return

premium

marketrisk

company-unique risk

can be diversifiedaway

Required

rate of

return= +

Risk-free

rate of

return

premium

RequiredRequired

rate of rate of

returnreturn

Let’s try to graph thisrelationship!

RequiredRequired

rate of rate of

returnreturn

Risk-freerate ofreturn(6%)

RequiredRequired

rate of rate of

returnreturn

securitymarket

line (SML)

This linear relationship between This linear relationship between risk and required return is risk and required return is known as the known as the Capital Asset Capital Asset

Pricing ModelPricing Model (CAPM). (CAPM).

RequiredRequired

rate of rate of

returnreturn

RequiredRequired

rate of rate of

returnreturn

Is there a riskless(zero beta) security?

RequiredRequired

rate of rate of

returnreturn

Is there a riskless(zero beta) security?

Treasurysecurities are

as close to risklessas possible. Risk-free

rate ofreturn(6%)

RequiredRequired

rate of rate of

returnreturn

SMLWhere does the S&P 500fall on the SML?

RequiredRequired

rate of rate of

returnreturn

SMLWhere does the S&P 500fall on the SML?

The S&P 500 isa good

approximationfor the market

RequiredRequired

rate of rate of

returnreturn

UtilityStocks

RequiredRequired

rate of rate of

returnreturn

SMLHigh-techstocks

The CAPM equation:The CAPM equation:

kkjj = k = krfrf + + jj (k (kmm - k - krf rf ))

where:where:

kkjj = the required return on security j, = the required return on security j,

kkrfrf = the risk-free rate of interest, = the risk-free rate of interest,

jj = the beta of security j, and = the beta of security j, and

kkmm = the return on the market index. = the return on the market index.

Example:Example:

Suppose the Treasury bond rate is Suppose the Treasury bond rate is 6%6%,, the average return on the the average return on the S&P 500 index is S&P 500 index is 12%12%,, and Walt and Walt Disney has a beta of Disney has a beta of 1.21.2..

According to the According to the CAPMCAPM, what , what should be the should be the required rate of required rate of returnreturn on Disney stock? on Disney stock?

kkjj = k = krfrf + (k + (kmm - k - krf rf ))

kkjj = .06 + 1.2 (.12 - .06) = .06 + 1.2 (.12 - .06)

kkjj = .132 = = .132 = 13.2%13.2%

According to the CAPM, Disney According to the CAPM, Disney stock should be priced to give a stock should be priced to give a 13.2%13.2% return. return.

RequiredRequired

rate of rate of

returnreturn

RequiredRequired

rate of rate of

returnreturn

Theoretically, every security should lie on the SML

RequiredRequired

rate of rate of

returnreturn

Theoretically, every security should lie on the SML

If every stock is on the SML,

investors are being fully compensated for risk.Risk-free

rate ofreturn(6%)

RequiredRequired

rate of rate of

returnreturn

If a security is abovethe SML, it isunderpriced.

RequiredRequired

rate of rate of

returnreturn

If a security is abovethe SML, it isunderpriced.

If a security is below the SML, it

is overpriced.Risk-freerate ofreturn(6%)

Simple Return CalculationsSimple Return Calculations

$50 $60

= = = 20% = 20%PPt+1t+1 - P - Pt t 60 - 50 60 - 50

PPtt 50 50

$50 $60

PPt+1t+1 60 60

PPtt 50 50

= = = 20% = 20%PPt+1t+1 - P - Pt t 60 - 50 60 - 50

PPtt 50 50

- 1- 1 = = -1-1 = 20% = 20%

$50 $60

(a) (b)monthly expected

month price return return (a - b)2

Dec $50.00Jan $58.00Feb $63.80Mar $59.00Apr $62.00May $64.50Jun $69.00Jul $69.00Aug $75.00Sep $82.50Oct $73.00Nov $80.00Dec $86.00

Dec $50.00Jan $58.00 0.160Feb $63.80Mar $59.00Apr $62.00May $64.50Jun $69.00Jul $69.00Aug $75.00Sep $82.50Oct $73.00Nov $80.00Dec $86.00

Dec $50.00Jan $58.00 0.160Feb $63.80 0.100Mar $59.00Apr $62.00May $64.50Jun $69.00Jul $69.00Aug $75.00Sep $82.50Oct $73.00Nov $80.00Dec $86.00

Dec $50.00Jan $58.00 0.160Feb $63.80 0.100Mar $59.00 -0.075Apr $62.00May $64.50Jun $69.00Jul $69.00Aug $75.00Sep $82.50Oct $73.00Nov $80.00Dec $86.00

Dec $50.00Jan $58.00 0.160Feb $63.80 0.100Mar $59.00 -0.075Apr $62.00 0.051May $64.50Jun $69.00Jul $69.00Aug $75.00Sep $82.50Oct $73.00Nov $80.00Dec $86.00

Dec $50.00Jan $58.00 0.160Feb $63.80 0.100Mar $59.00 -0.075Apr $62.00 0.051May $64.50 0.040Jun $69.00Jul $69.00Aug $75.00Sep $82.50Oct $73.00Nov $80.00Dec $86.00

Dec $50.00Jan $58.00 0.160Feb $63.80 0.100Mar $59.00 -0.075Apr $62.00 0.051May $64.50 0.040Jun $69.00 0.070Jul $69.00Aug $75.00Sep $82.50Oct $73.00Nov $80.00Dec $86.00

Dec $50.00Jan $58.00 0.160Feb $63.80 0.100Mar $59.00 -0.075Apr $62.00 0.051May $64.50 0.040Jun $69.00 0.070Jul $69.00 0.000Aug $75.00Sep $82.50Oct $73.00Nov $80.00Dec $86.00

Dec $50.00Jan $58.00 0.160Feb $63.80 0.100Mar $59.00 -0.075Apr $62.00 0.051May $64.50 0.040Jun $69.00 0.070Jul $69.00 0.000Aug $75.00 0.087Sep $82.50Oct $73.00Nov $80.00Dec $86.00

Dec $50.00Jan $58.00 0.160Feb $63.80 0.100Mar $59.00 -0.075Apr $62.00 0.051May $64.50 0.040Jun $69.00 0.070Jul $69.00 0.000Aug $75.00 0.087Sep $82.50 0.100Oct $73.00Nov $80.00Dec $86.00

Dec $50.00Jan $58.00 0.160Feb $63.80 0.100Mar $59.00 -0.075Apr $62.00 0.051May $64.50 0.040Jun $69.00 0.070Jul $69.00 0.000Aug $75.00 0.087Sep $82.50 0.100Oct $73.00 -0.115Nov $80.00Dec $86.00

Dec $50.00Jan $58.00 0.160Feb $63.80 0.100Mar $59.00 -0.075Apr $62.00 0.051May $64.50 0.040Jun $69.00 0.070Jul $69.00 0.000Aug $75.00 0.087Sep $82.50 0.100Oct $73.00 -0.115Nov $80.00 0.096Dec $86.00

Dec $50.00Jan $58.00 0.160Feb $63.80 0.100Mar $59.00 -0.075Apr $62.00 0.051May $64.50 0.040Jun $69.00 0.070Jul $69.00 0.000Aug $75.00 0.087Sep $82.50 0.100Oct $73.00 -0.115Nov $80.00 0.096Dec $86.00 0.075

Dec $50.00Jan $58.00 0.160 0.049Feb $63.80 0.100 0.049Mar $59.00 -0.075 0.049Apr $62.00 0.051 0.049May $64.50 0.040 0.049Jun $69.00 0.070 0.049Jul $69.00 0.000 0.049Aug $75.00 0.087 0.049Sep $82.50 0.100 0.049Oct $73.00 -0.115 0.049Nov $80.00 0.096 0.049Dec $86.00 0.075 0.049

Dec $50.00Jan $58.00 0.160 0.049 0.012321Feb $63.80 0.100 0.049 0.002601Mar $59.00 -0.075 0.049 0.015376Apr $62.00 0.051 0.049 0.000004May $64.50 0.040 0.049 0.000081Jun $69.00 0.070 0.049 0.000441Jul $69.00 0.000 0.049 0.002401Aug $75.00 0.087 0.049 0.001444Sep $82.50 0.100 0.049 0.002601Oct $73.00 -0.115 0.049 0.028960Nov $80.00 0.096 0.049 0.002090Dec $86.00 0.075 0.049 0.000676

0.0781St. Dev: sum, divide by (n-1), and take sq root:

Calculator solution using HP 10B:Calculator solution using HP 10B:

Enter monthly return on 10B calculator, Enter monthly return on 10B calculator, followed by followed by sigmasigma key (top right corner). key (top right corner).

Shift 7Shift 7 gives you the expected return. gives you the expected return. Shift 8Shift 8 gives you the standard deviation. gives you the standard deviation.

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