The Trade-off betweenRisk and Return

Professor ThomsonFin 3013

2

Risk and Return

The return earned on investments represents the marginal benefit of investing.

Risk is one of the marginal costs of investing (the other is the pure time value of money).

A trade-off always arises between expected risk and expected return.

3

Risk and Return

Valuing risky assets - a task fundamental to financial management

Three-step procedure for valuing a risky asset

1. Determine the asset’s expected cash flows2. Choose discount rate that reflects asset’s risk3. Calculate present value (PV cash inflows - PV

outflows)

The three-step procedure is called discounted cash flow (DCF) analysis.

4

Financial Return

Total return: the total gain or loss experienced on an investment over a given

period of time

Component

s of the total

return

Income stream from the investment

Capital gain or loss due to changes in asset prices

Total return can be expressed either in dollar terms or in percentage terms.

5

Cash Flow Time Line

Pt-1 cash payments Pt

+-----------------------------------------------+Time t-1 Time t

Pt = Price at time t (today)

Capital Gain = Pt – Pt-1

= Price today – Price last period

Dollar Return = Cash Payments + Capital Gain

= Cash Payments + Pt – Pt-1

6

Example 6.1

• You purchased a stock last year for $25. It has paid $1 in dividends and is not worth $21. What is your Dollar Return?

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Example 6.2

• You bought an 11% coupon bond one year ago for $1125. You can sell that bond today for $1100. What is your Dollar Return?

8

Holding Period Return (hpr) or

Percentage Return

• This is the most common way to express the gains or losses over a period

• It is the $Return relative to the amount invested

1

1%Return

t

tt

P

PPCashPmthpr

1+hpr is often called the wealth relative

9

Example 6.1 (Revised)

• You purchased a stock last year for $25. It has paid $1 in dividends and is not worth $21. What is your Dollar Return?

• What is your hpr?

10

Example 6.2 Revised

• You bought an 11% coupon bond one year ago for $1125. You can sell that bond today for $1100. What is your Dollar Return?

• What is your hpr?

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Measuring Wealth over TimeYear

hpr $1 Investment $1 Investment each year

1 15% 1*(1.15) = 1.15Vt-1(1+it) = Vt

1*(1.15)=1.15Vt-

Vt- Vt-1 (1+it) = Vt

2 -10% 1.15*(0.90) =1.035

(1+1.15)*(0.90) = 1.935

3 13% 1.035*1.13 = 1.1696

(1+1.935)*(1.13) = 3.3166

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Arithmetic Average Return

• Add the individual hpr’s and divide by the number of years

%00.63

%13%10%15

x

13

Geometric Rate of Return

• Multiply by the wealth relatives, raise to the 1/N power and subtract 1

• Is the constant rate of wealth building over time that results in the observed future value

%36.50536.01)]13.1)(90.0)(15.1[(

1)]1)...(1)(1)(1[(

31

1

321

NNrrrrg

By Financial Calculator: P/YR=1

I/YR(FV=1.1696, PV=-1, N=3) = 5.36%

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IRR from a Constant Investment

t CF

0 -1

1 -1

2 -1

3 3.3166

P/YR=1

•Press IRR = 5.10%

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Value of $1 Invested in Equities, Treasury Bonds and Bills, 1900 - 2003

Year

$15,579

$148

$61

$22

10,000

100,000

1,000

100

10

1

Equities Bonds

Bills Inflation

1900 1920 1940 1960 1980 2000 2003

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Geometric Return Calculation

• A $1 investment in Large Stocks (with dividends reinvested) was worth 15579 after 103 years. The geometric mean return can be computed as (P/Yr = 1)

I/YR(FV=15579, PV=-1, N=103)=9.83% StocksI/YR(FV=148, PV=-1, N=76)=4.97% Long US

BondI/YR(FV=61, PV=-1, N=76)=4.07% US

Treasury BillsI/YR(FV=22, PV=-1, N=76)=3.05% Inflation

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Geometric Real Rates of Return• To compute the long run real rate of return

one can divide the ending value of the investment by the ending value of the inflation figure to determine the purchasing power of the investment. Then compute the return using “Real Dollars”

• Real value of the Large Stocks at end of period is 15579/22 = 708.14

• I/YR(FV=708.14, PV=-1, N=103)=6.58%• I/YR(FV=6.73, PV=-1, N=103)=1.87% Long US

Bond• I/YR(FV=2.73, PV=-1, N=103)=1.00% TBill

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Arithmetic versus Geometric Returns (1900-2003)

Stocks Bonds Bills

Nominal Returns Arithmetic Avg

11.7 5.2 4.1

Nominal Returns Geometric

9.8 5.0 4.1

Real ReturnsArithmetic Avg

8.5 2.3 1.1

Real Returns Geometric

6.6 1.9 1.0

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Annual Returns for Securties

-30%

-20%

-10%

0%

10%

20%

30%

40%

50%

1980 1985 1990 1995 2000

Year

Ret

urn

Stocks

Bonds

T-Bills

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IRR From a Constant Annual Investment (Jan 1, each year) through Oct 2002

-25%

-20%

-15%

-10%

-5%

0%

5%

10%

15%

20%

19

80

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81

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82

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00

20

01

20

02

Year of First Investment

Stocks

LongGovBond

Tbills

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IRR From a Constant Annual Investment (Jan 1, each year) through Dec 2003

-5%

0%

5%

10%

15%

20%

25%

30%

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80

19

81

19

82

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83

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84

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01

20

02

20

03

Year of First Investment

Stocks

LongGovBond

Tbills

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IRR From a Constant Annual Investment (Jan 1, each year) through Dec 2004

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

19

80

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81

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82

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00

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01

20

02

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03

20

04

Year of First Investment

Stocks

LongGovBond

Tbills

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Dollar Gain per Dollar Invested From a Constant Annual Investment (Jan 1, each year) through Dec 2004

0.00

1.00

2.00

3.00

4.00

5.00

6.00

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80

19

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99

20

00

20

01

20

02

20

03

20

04

Year of First Investment

Stocks

LongGovBond

Tbills

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Take home message

• If you have a short holding period, stocks are very risky, but from a longer term perspective they have provided the best returns both recently and historically

• Investment is not about saving money for the future, its about earning money from the money you invested so that most of your portfolio is from the earning of that portfolio and not from your deposits into that fund

25

Percentage Returns on Bills, Bonds, and Stocks, 1900 - 2003

Difference between average return of stocks and bills = 7.6%

Difference between average return of stocks and bonds = 6.5%

Risk premium: the difference in returns offered by a risky asset relative to the risk-free return

available

Nominal (%) Real (%)Asset Class Average Best Year Worst Year Average Best Year Worst Year

Bills 4.1 14.7 0.0 1.1 19.7 -15.1Bonds 5.2 40.4 -9.2 2.3 35.1 -19.4Stocks 11.7 57.6 -43.9 8.5 56.8 -38

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Why are Treasury Bills considered risk free?

• If the government default on Treasury Bills, your last concern will be the money you might have earned on the TB

• When you buy a Treasury Bill, you purchase it at a discount and redeem it at par, so you know when you buy it, what your return will be

• If you buy a stock, you don’t know what you will sell it for, or what dividends it will pay; thus, it is risky

• The yield on Treasury Bills, is generally taken to be the risk free return

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Distribution of Historical Stock Returns, 1900 - 2003

Histogram of Nominal Returns on Equities 1900-2003

<-30 -30 to -20 to -10 to 0 to 10 to 20 to 30 to 40 to >50 -20 -10 0 10 20 30 40 50

Percent return in a given year

Probability distribution for future stock returns is unknown. We can approximate the unknown distribution by assuming a normal

distribution.

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Variability of Stock ReturnsNormal distribution can be described by

its mean and its variance. A Normal Distribution is symmetric around the

mean• Variance (2) - the expected value of

squared deviations from the mean

1

)(1

2

2

N

RRVariance

N

tt

Units of variance (%-squared) - hard to interpret, so calculate standard deviation, a

measure of volatility equal to square root of 2

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The Normal Distribution

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Volatility of Asset Returns

AssetClass Average(%) Std. Dev. (%) Average(%) Std. Dev. (%)

Equities 11.7 20.1 8.5 20.4Bonds 5.2 8.2 2.3 10Bills 4.1 2.8 1.1 4.7

Nominal Returns Real Returns

Asset classes with greater volatility pay higher average returns.

• Average return on stocks is more than double the average return on bonds, but stocks are 2.5 times more volatile.

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Average Returns and St. Dev. for Asset Classes, 1900-2003

1. Investors who want higher returns have to take more risk

2. The incremental reward from accepting more risk seems constant

Bills Bonds

Stocks

Average Return (%)

Standard Deviation (%)

32

Average Return and St. Dev. for Individual Securities, 1994-2003

Average risk for all stocks in this period was 60%

For various asset classes, a trade-off arises between risk and return. Does the trade-off appear to hold for all

individual securities?

Anheuser-Busch 19.2 16.1Coca Cola 12.1 22.6Wendy's International 11.8 23.3Archer Daniels Midland 7.6 23.5General Motors 8.3 26.0General Electric 20.3 32.1Merck 17.8 32.7Nordstrom 14.3 38.1Wal-Mart 22.7 44.7American Airlines (AMR) 10.0 47.8Advanced Micro Devices (AMD) 17.6 56.4Average of individual 11 stocks 14.7 33.0Average for portfolio of U.S stocks 12.5 21.0

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Average Return and St. Dev. for Individual Securities, 1994-2003

Average Return (%)

Standard Deviation (%)

Wal-MartAnheuser-Busch

Archer Daniels Midland

American Airlines

No obvious pattern here

34

Diversification

Most individual stock prices show higher volatility than the price volatility of portfolio

of all common stocks.

How can the standard deviation for individual stocks be higher than the standard deviation of the portfolio?

Diversification: investing in many different assets reduces the volatility of the portfolio.

The ups and downs of individual stocks partially cancel each other out.

35

Annual Return: Coke, Wendy's, 50/50 Portfolio

-20

-10

0

10

20

30

40

50

Year

Retu

rn%

Coke

Wendy's

Ptf

The standard deviation of the portfolio is lower than the standard deviation of either Coke or Wendy’s

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The Impact of Additional Assets on the Risk of a Portfolio

Po

rtfo

lio

Sta

nd

ard

Dev

iati

on

Number of StocksNumber of Stocks

Systematic RiskSystematic Risk

1 2 3 111 2 3 11

Portfolio of 11 stocks

AMD

Unsystematic RiskUnsystematic Risk

AMD + American Airlines

AMD + American Airlines + Wal-Mart

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Diversification reduces portfolio volatility, but only up to a point. Portfolio of all stocks still

has a volatility of 21%.

Systematic risk: the volatility of the portfolio that cannot be eliminated through

diversification.Unsystematic risk: the proportion of risk of

individual assets that can be eliminated through diversification, for example, by buying

mutual funds. Because this risk can be eliminated, there is no reward for holding

unsystematic risk

What really matters is systematic risk….how a group of assets move together.

Systematic and Unsystematic Risk

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Anheuser Busch stock had higher average returns than Archer-Daniels-Midland stock,

with smaller volatility.

Systematic and Unsystematic Risk

American Airlines had much smaller average returns than Wal-Mart, with similar volatility.

The tradeoff between standard deviation and average returns that holds for asset classes

does not hold for individual stocks.

Because investors can eliminate unsystematic risk through diversification, market rewards

only systematic risk.

Standard deviation contains both systematic and unsystematic risk.

Investment performance is measured by total return.

Trade-off between risk and return for assets: historically, stocks had higher returns and

volatility than bonds and bills.

One measure of risk: standard deviation (volatility)

Unsystematic and (systematic) risk: risk that can (cannot) be eliminated through

diversification, respectively

Risk and Return

40

41

Dollar Returns

Total dollar return = income + capital gain / loss

Terrell bought 100 shares of Micro-Orb stock for $25

A year later:Dividend = $1/shareSold for $30/share

Dollar return = (100 shares) x ($1 + $5) = $600

Owen bought 50 shares of Garcia Inc. stock for $15

A year later:No dividends paidSold for $25/share

Dollar return = (150 shares) x ($15)

= $500

42

Percentage Returns

Terrell’s dollar return exceeded Owen’s by $100. Can we say that Terrell was better off?

No, because Terrell and Owen’s initial investments were different: Terrell spent $2,500 in initial

investment, while Owen spent $750.

Percentage return: total dollar return divided by the initial investment

investment initialreturndollar total

return percentage Total

43

Percentage Returns

%2424.0

500,2$

5$1$100 '

returnpercentagesTerrell

%6767.0

750$

10$50 '

returnpercentagesOwen

In percentage terms, Owen’s investment performed better than Terrell’s did.