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CHAPTER 8 Risk and Rates of Return. Stand-alone risk Portfolio risk Risk & return: CAPM / SML. Investment returns. The rate of return on an investment can be calculated as follows: (Amount received – Amount invested) Return = ________________________ Amount invested

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Chapter 8 risk and rates of return

CHAPTER 8Risk and Rates of Return

Stand-alone risk

Portfolio risk

Risk & return: CAPM / SML


Investment returns
Investment returns

The rate of return on an investment can be calculated as follows:

(Amount received – Amount invested)

Return =________________________

Amount invested

For example, if $1,000 is invested and $1,100 is returned after one year, the rate of return for this investment is:

($1,100 - $1,000) / $1,000 = 10%.


What is investment risk
What is investment risk?

  • Investment risk is related to the probability of earning a low or negative actual return.

  • The greater the chance of lower than expected or negative returns, the riskier the investment.


Selected realized returns 1990 2007
Selected Realized Returns, 1990 – 2007

Average Standard

ReturnDeviation

Small-company stocks 17.5% 33.1%

Large-company stocks 12.4 20.3

L-T corporate bonds 6.2 8.6

Source: Based on Stocks, Bonds, Bills, and Inflation: (Valuation Edition) 2008 Yearbook (Bursa Malaysia, 2008)



Investment alternatives rate of returns1
Investment alternatives & Rate of Returns( %)

  • T-bills will return the promised 5.5%, regardless of the economy- risk-free return

  • Do T-bills promise a completely risk-free return?

  • T-bills do not provide a completely risk-free return, as they are still exposed to inflation..

  • T-bills are also risky in terms of reinvestment rate risk.

  • However, T-bills are risk-free in the default sense of the word.


Reinvestment risk
“Reinvestment Risk”

  • CF

  • CF

  • CF

  • CF

  • CF

5.5%

5.5%

5.5%

5.5%

5.5%

  • “reinvest cash inflow at going market rates”

  • Thus, if market rates , may experience income reduction


Inflation risk
“inflation risk”

If inflation rate = 8%

- funds deposited loses the purchasing power

- “inflation risk”


How do the returns of ht and coll behave in relation to the market
How do the returns of HT and Coll. behave in relation to the market?

  • HT – Moves with the economy, and has a positive correlation. This is typical.

  • Coll. – Is countercyclical with the economy, and has a negative correlation. This is unusual.



Summary of expected returns
Summary of expected returns market?

Expected return

HT 12.4%

Market 10.5%

USR 9.8%

T-bill 5.5%

Coll. 1.0%

HT has the highest expected return, and appears to be the best investment alternative, but is it really? Have we failed to account for risk?


Expected returns standard deviation
Expected returns & Standard deviation market?

12.4

HT’s expected return



(-27-12.4) market?2(0.1) + (-7-12.4)2(0.2)

(15-12.4)2(0.4) + (30-12.4)2(0.2)

(45-12.4)2(0.1)

1/2

6HT

=

6HT

20%

=


Expected returns standard deviation1
Expected returns & Standard deviation market?

-7.6%

12.4%

32.4%



Comparing standard deviations

Prob. market?

T - bill

USR

HT

0 5.5 9.8 12.4 Rate of Return (%)

Comparing standard deviations


Comments on standard deviation as a measure of risk
Comments on standard deviation as a measure of risk market?

  • Standard deviation (σi) measures total, or stand-alone, risk.

  • The larger σi is, the lower the probability that actual returns will be closer to expected returns.

  • Larger σi is associated with a wider probability distribution of returns.



Investor attitude towards risk
Investor attitude towards risk market?

  • Risk aversion – assumes investors dislike risk and require higher rates of return to encourage them to hold riskier securities.

  • Risk premium – the difference between the return on a risky asset and a riskless asset, which serves as compensation for investors to hold riskier securities.


Portfolio construction risk and return
Portfolio construction: market?Risk and return

  • Assume a two-stock portfolio is created with $50,000 invested in both HT and Collections.

  • A portfolio’s expected return is a weighted average of the returns of the portfolio’s component assets.


Calculating portfolio expected return
Calculating portfolio expected return market?

= expected return

= 12.4%

= 1.0%


An alternative method for determining portfolio expected return
An alternative method for determining portfolio expected return

*A: 0.5(-27%) +

0.5(27)

= 0%

*B: 0.5(-7%) +

0.5(13%)

= 3%



Comments on portfolio risk measures
Comments on portfolio risk measures return

  • σp = 3.4% is much lower than the σi of either stock (σHT = 20.0%; σColl. = 13.2%).

  • Therefore, the portfolio provides the average return of component stocks, but lower than the average risk.

  • Why? Negative correlation between stocks.

  • Combining stocks in a portfolio generally lowers risk.


Returns distribution for two perfectly negatively correlated stocks 1 0

Stock W return

Stock M

Portfolio WM

25

15

0

0

0

-10

-10

Returns distribution for two perfectly negatively correlated stocks (ρ = -1.0)

25

25

15

15

-10


Returns distribution for two perfectly positively correlated stocks 1 0

Stock M’ return

Portfolio MM’

Stock M

25

25

25

15

15

15

0

0

0

-10

-10

-10

Returns distribution for two perfectly positively correlated stocks (ρ = 1.0)


Illustrating diversification effects of a stock portfolio

s returnp (%)

Diversifiable Risk

35

Stand-Alone Risk, sp

20

0

Market Risk

10 20 30 40 2,000+

# Stocks in Portfolio

Illustrating diversification effects of a stock portfolio


Creating a portfolio beginning with one stock and adding randomly selected stocks to portfolio
Creating a portfolio: returnBeginning with one stock and adding randomly selected stocks to portfolio

  • σp decreases as stocks added, because they would not be perfectly correlated with the existing portfolio.

  • Expected return of the portfolio would remain relatively constant.

  • Eventually the diversification benefits of adding more stocks dissipates (after about 10 stocks), and for large stock portfolios, σp tends to converge to  20%.


Breaking down sources of risk
Breaking down sources of risk return

  • Diversifiable risk – portion of a security’s stand-alone risk that can be eliminated through proper diversification.

  • Market risk – portion of a security’s stand-alone risk that cannot be eliminated through diversification. Measured by beta.


Beta return

  • Measures a stock’s market risk, and shows a stock’s volatility relative to the market.

  • If beta = 1.0, the security is just as risky as the average stock.

  • If beta > 1.0, the security is riskier than average.

  • If beta < 1.0, the security is less risky than average.

  • Most stocks have betas in the range of 0.5 to 1.5.


Calculating betas
Calculating betas return

  • Well-diversified investors are primarily concerned with how a stock is expected to move relative to the market in the future.

  • A typical approach to estimate beta is to run a regression of the security’s past returns against the past returns of the market.

  • The slope of the regression line is defined as the beta coefficient for the security.


Illustrating the calculation of beta

_ return

ri

.

20

15

10

5

.

Year rM ri

1 15% 18%

2 -5 -10

3 12 16

_

-5 0 5 10 15 20

rM

Regression line:

ri = -2.59 + 1.44 rM

.

-5

-10

^

^

Illustrating the calculation of beta


Comparing expected returns and beta coefficients
Comparing expected returns and beta coefficients return

SecurityExpected Return Beta

HT 12.4% 1.32

Market 10.5 1.00

USR 9.8 0.88

T-Bills 5.5 0.00

Coll. 1.0 -0.87

Riskier securities have higher returns, so the rank order is OK.


Can the beta of a security be negative
Can the beta of a security be negative? return

  • Yes, if the correlation between Stock i and the market is negative (i.e., ρi,m < 0).

  • If the correlation is negative, the regression line would slope downward, and the beta would be negative.

  • However, a negative beta is highly unlikely.


Beta coefficients for ht coll and t bills

_ return

ri

HT: b = 1.30

40

20

T-bills: b = 0

_

kM

-20 0 20 40

Coll: b = -0.87

-20

Beta coefficients for HT, Coll, and T-Bills


Capital asset pricing model capm
Capital Asset Pricing Model (CAPM) return

  • Model linking risk and required returns. CAPM suggests that there is a Security Market Line (SML) that states that a stock’s required return equals the risk-free return plus a risk premium.

    ri = rRF + (rM – rRF) bi


The security market line sml calculating required rates of return
The Security Market Line (SML): returnCalculating required rates of return

SML: ri = rRF + (rM – rRF) bi

ri = rRF + (RPM) bi

  • Assume the rRF = 5.5% and RPM = 5.0%.


What is the market risk premium rp m
What is the market risk premium (RP returnM)?

  • Additional return over the risk-free rate needed to compensate investors for assuming an average amount of risk.

  • Varies from year to year, but most estimates suggest that it ranges between 4% and 8% per year.


Calculating required rates of return
Calculating required rates of return return

  • rHT = 5.5% + (5.0%)(1.32)

    = 5.5% + 6.6% = 12.10%

  • rM = 5.5% + (5.0%)(1.00) = 10.50%

  • rUSR = 5.5% + (5.0%)(0.88) = 9.90%

  • rT-bill = 5.5% + (5.0%)(0.00) = 5.50%

  • rColl = 5.5% + (5.0%)(-0.87) = 1.15%


Illustrating the security market line
Illustrating the returnSecurity Market Line

SML: ri = 5.5% + (5.0%) bi

ri (%)

SML

.

HT

.

.

rM = 10.5

rRF = 5.5

.

USR

T-bills

.

Risk, bi

-1 0 1 2

Coll.



An example equally weighted two stock portfolio
An example: returnEqually-weighted two-stock portfolio

  • Create a portfolio with 50% invested in HT and 50% invested in Collections.

  • The beta of a portfolio is the weighted average of each of the stock’s betas.

    bP = wHT bHT + wColl bColl

    bP = 0.5 (1.32) + 0.5 (-0.87)

    bP = 0.225


Calculating portfolio required returns
Calculating portfolio required returns return

  • The required return of a portfolio is the weighted average of each of the stock’s required returns.

    rP = wHT rHT + wColl rColl

    rP = 0.5 (12.10%) + 0.5 (1.2%)

    rP = 6.6%

  • Or, using the portfolio’s beta, CAPM can be used to solve for expected return.

    rP = rRF + (RPM) bP

    rP = 5.5%+ (5.0%) (0.225)

    rP = 6.6%


Factors that change the sml
Factors that change the SML return

  • What if investors raise inflation expectations by 3%, what would happen to the SML?

ri (%)

SML2

D I = 3%

SML1

13.5

10.5

8.5

5.5

Risk, bi

0 0.5 1.0 1.5


Y = C + mx return

ri = rf + (Rm - Rf)b

RPm

SML line

  • - Inflation is captured by rf

    - Inflation - rf

  • - Risk factor is captured by (Rm – Rf)

    - Risk factor - (Rm – Rf)


Factors that change the sml1
Factors that change the SML return

  • What if investors’ risk aversion increased, causing the market risk premium to increase by 3%, what would happen to the SML?

ri (%)

SML2

D RPM = 3%

SML1

13.5

10.5

5.5

Risk, bi

0 0.5 1.0 1.5


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