Solutions for Questions and Problems #1(1).pdf

7/25/2019 Solutions for Questions and Problems #1(1).pdf

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SOLUTIONS FOR QUESTIONS AND PROBLEMS #1:

RISK AND RETURN I

Finance 321 Frank LaMay

Page 1

QUESTIONS

1.The arithmetic average is the average return that the investment itself produces each year. It treats every

years result independently, and doesnt reflect any carryover or accumulation of return from year to year.

The only way an investor can earn the arithmetic average over several years would be to start fresh each

year, i.e., begin each year with the same amount of money invested.The geometric average is the average compound return an investor earns from an investment. It assumes

that one years result carries over to the next year, i.e., that the investors balance at the end of the first

year is the beginning balance for the second, etc. It can be thought of as the result you would get from a

buy and hold strategy in which you put an amount of money into an investment at the beginning of a

period and dont make deposits or withdrawals during the investment period.

Note that whenever there is any variability in returns from year to year (which is true for most investments)

the geometric average will be lower than the arithmetic average.

The dollar weighted average is the actual return an individual investor earns on an investment. Like the

geometric average, it reflects compounding from year to year, but it also reflects the impact of the

investors decisions to make deposits and withdrawals: whether the investor adds cash or removes cash,how much she adds or removes, and when. For example, if an investor makes a large deposit right before

the investment earns very high returns (favorable timing), the dollar weighted average will be higher than

the geometric average; the opposite happens if the investor makes a large deposit right before the

investment earns low returns. Similarly, if an investor withdraws a large amount right before a period of

high returns, that will make the dollar weighted average lower than the geometric average; on the otherhand, withdrawals before periods of low return will increase the dollar weighted average.

2.The standard deviation is a measure of how much the yearly returns on the stock market varied from the

average return.

3.

a. One way of thinking about the expected return is that it is a forward-looking average. The textsays: You can think of [the expected return] as the average holding period return you would

earn if you were to repeat an investment in the asset many times. If you invested in this stock

over a number of years and your assumptions about returns and probabilities remained valid,

youd expect your average return to be the expected return (or very close).

b. The standard deviation is a measure of how much the yearly returns on the stock would vary from

the expected return. (Remember that the expected return is a kind of average; you would almost

never earn the expected return in any given year.)

c. The standard deviation allows us to talk about confidence intervals e.g., we could say that about

two-thirds of the time (68.26% to be exact) youd expect this stocks return to be between (15%)

(expected return minus one standard deviation) and 45% (expected return plus one standard

deviation).

4.

Treasury bills are considered risk-free and are the foundation on which other rates of return are built.

Investors know what rate of return they can get without taking on risk (the risk-free rate) and for risky

investments they require a risk premium over and above the risk-free rate.

5.Excess return is the difference between an investments return and the risk-free rate. Its the risk premiumor reward that investors receive for bearing the investments risk. The idea applies to historical returns

(that is, how much excess return did investors get on a risky investment) and to expected returns (how

excess return do they expect to get on a risky investment in the future).


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RISK AND RETURN I


Page 2

6.One possible explanation for high returns on risky investments is high risk premiums; the other is high

risk-free rates. Its possible that during the several years the risk-free rate and other interest rates were

quite high, and that the risky investments did not offer much of a risk premium.

7.The Sharpe measure captures the risk premium an investment offers relative to its risk. Because it captures

reward per unit of risk it allows us to compare the risk premiums across investments or portfolios.

PROBLEMS

1.

Beginning Year-end Change Dividend Total $ Return % Return

(a) (b) (c) (d) = (b) + (c) (d) / (a)

75.00 60.00 (15.00) 2.00 (13.00) (0.17)

60.00 90.00 30.00 2.10 32.10 0.54

90.00 80.00 (10.00) 2.10 (7.90) (0.09)

0.27

Arithmetic average = 0.09

Price

Geometric average return:

r05.

0516

1]163162.1[1)]91)(.54.1)(83[(.

1))]09.(1()54.1))(17(.1[(

1)r1(...)r1)(r1)(r1(

33333.

3

1

3

1

N

1

N321

See question #1 above for discussion of difference between arithmetic and geometric averages.

2.a. The three month holding return (HPR) on the Treasury bill is:

031.700,9

700,9000,10

priceStart

pricestart-flowscashteintermediaofvalueendPriceEnd

The annualized holding period return is:

1299.11299.11)031.1(1)031.1(

1HPR)(1AHPR

425.1

T1

(b) For the mutual fund, the four-year holding period return (HPR) is:


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RISK AND RETURN I


Page 3

80.000,1

000,1800,1

priceStart


and the annualized holding period return (AHPR) is:

1583.11583.11)8.1(1)8.1(

1HPR)(1AHPR

25.41

T1

(c) For the stock, the three-year holding period return (HPR) is:

222.000,9

000,9000,7

priceStart


and the annualized holding period return (AHPR) is:

08.0803..19197.1)778(.1)222.1(

1HPR)(1AHPR

333.31

T1

(Note: Since the number of shares didnt change over the three years, youd get the same result by

calculating the return on one share i.e., using the price per share of 90 at the beginning and 70 at the

end to calculate the holding period return.)

3.

(a) 025.4

10.20.)25(.05.

(b)

r01.

r0097.

1]0395.1[

1)]1.1)(2.1)(75)(.05.1[(

1)]101)(20.1())25(.1)(05.1[(

1)r1(...)r1)(r1)(r1(

25.

25.

4

1

N

1

N321


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RISK AND RETURN I


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(c)

2006 2007 2008 2009 2010

5% -25% 20% 10%

Starting value 2,000 2,000 2,500 3,200

$ return 100 (500) 500 320

Ending value 2,100 1,500 3,000 3,520

Net inflow 2,000 (100) 1,000 200

(Net deposits/withdrawals)

(at year-end)

Arithmetic average 2.5%

Geometric average 1.0%

0 1 2 3 4

Cash flows (2,000) 100 (1,000) (200) 3,520

Dollar-weighted return 4.0%

(EXCEL IRR FUNCTION applied to cash flows above)

4.

The dollar weighted return is different because it reflects the amount and timing of the deposits you

made: you withdrew money before the year with the worst return (2008), made a large deposit just

before the year with the strongest return (2009) and made another deposit before the year with the

second strongest return (2010). This is an example of favorable market timing (something which is

hard to achieve other than by luck).

5.

The average return is 178.1775.4

20.13.)20(.58.

The standard deviation is .32:

Standard

Variance Deviation

Year Return - Average =

1 ( .58 - .178 )2 = .1616

2 ( (.20) - .178 )2 = .1429

3 ( .13 - .178 )2 = .0023

4 ( .20 - .178 )2 = .0005

.3073 3 = .10243 .320

Deviation2

Variance


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RISK AND RETURN I


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6. (a) The expected return is .105 and the standard deviation is ~.25:

Probability

Probability Return x Return

Very good year .20 .45 .090

Good yeaer .30 .20 .060Mediocre year .30 .05 .015

Poor year .20 (.30) (.060)

.105

Probability Return - Expected =

Return

Very good year .20 x ( .45 - .105 )2 = .02378

Good year .30 x ( .20 - .105 )2 = .00271

Mediocre year .30 x ( .05 - .105 )2 = .00091

Poor year .20 x ( (.30) - .105 )2 = .03281

VARIANCE .06020

STANDARD DEVIATION .24535

(square root of variance)

~.245 rounded

Deviation2

(b) The standard deviation is a measure of the risk of an investment in the fund. It reflects the

volatility that you expect in yearly returns for the fund how much the yearly result on the fundis likely to vary from its expected return. Using the idea of confidence intervals, you could

estimate that about two-thirds of the time (actually 68% of the time), the funds result in any year

will be in a range of .35 [expected return + one standard deviation] to (.14) [expected return one

standard deviation]. You could estimate a 95% confidence interval by adding or subtracting two

standard deviations.

(c)

298.

403.105.

)245(.64485.1105.

1.64485rVaR

(d) The VaR indicates that 5% of the time wed expect the return on this fund to be below .298 (i.e.,

the probability that its return will be below .298 is 5%). Put differently, it indicates that 95% of

the time wed expect the return on this fund to be above .298 (i.e., the probability that its returnwill be better than .298 is 95%).


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RISK AND RETURN I


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7.For stocks:

32.19.

06.

19.

02.08.rrS

P

FP

For bonds:

00.103.

03.

03.

02.05.rrS

P

FP

The Sharpe ratio for stocks is much lower for these fifteen years than the longer-term ratio (which was

.40 on a rounded basis), while the ratio for bonds is much higher than the longer-term ratio. Over short

periods of time, investment results can vary significantly from longer-term patterns.

8.The approximation is rnominal i = .07 - .03 = .04.

The exact calculation is 0388.1

03.1

07.1

i1

r1r1 alminnoreal

rreal= .0388

The exact calculation takes into account the compounding of inflation, while the approximation

ignores compounding.

The approximation is less useful (i.e., the difference between the approximation and the exact

calculation is bigger) when inflation and returns are high.

The approximation is also less useful when youre thinking about the impact of returns over a long

period of time: a small difference in your estimate of the real return can have a big impact if it

compounds over many years. For example, if you are estimating how much your retirement account

will be worth 40 years from now, a .5% difference in the estimate of your expected real return willhave a big impact on your answer.

EXTRA PRACTICE PROBLEMS

9.

(a) 055.4

05.)20(.30.07.

(b)

r04.

r0397.

1]16844.1[

1)]05.1)(8)(.30.1)(07.1[(

1)]05.1))(20(.1()30.1)(07.1[(

1)r1(...)r1)(r1)(r1(

25.

25.

4

1

N

1

N321


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RISK AND RETURN I


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(c)

2011 2012 2013 2014 2015

7% 30% -20% 5%

Starting value 5,000 4,850 7,305 6,044

$ return 350 1,455 (1,461) 302

Ending value 5,350 6,305 5,844 6,346

Net inflow 5,000 (500) 1,000 200

(Net deposits/withdrawals)

(at year-end)

Arithmetic average 5.5%

Geometric average 4.0%

0 1 2 3 4Cash flows (5,000) 500 (1,000) (200) 6,346

Dollar-weighted return 3.0%

(EXCEL IRR FUNCTION applied to cash flows above)

10.

The average return is 0825.4

10.40.08.)25(.

The standard deviation is .266:

Standard

Variance Deviation

Year Return - Average =

1 ( (.25) - .0825 )2

= .1106

2 ( .08 - .0825 )2

= .0000

3 ( .40 - .0825 )2

= .1008

4 ( .10 - .0825 )2

= .0003

.2117 3 = .07056 .266

Deviation2

Variance

11. (a) The expected return is .114 and the standard deviation is ~.284:

Probability

Probability Return x Return

Very good year .20 .50 .100

Good yeaer .30 .25 .075

Mediocre year .30 .03 .009

Poor year .20 (.35) (.070)

.114


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RISK AND RETURN I


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Probability Return - Expected =

Return

Very good year .20 x ( .50 - .114 )2

= .02980

Good year .30 x ( .25 - .114 )2

= .00555

Mediocre year .30 x ( .03 - .114 )2

= .00212

Poor year .20 x ( (.35) - .114 )2

= .04306

VARIANCE .08052

STANDARD DEVIATION .28377

(square root of variance)

~.284 rounded

Deviation2

(b) The standard deviation measures the risk of an investment: it reflects the volatility that you expect

in yearly returns for the fund how much the yearly result on the fund is likely to vary from itsexpected return. Using the idea of confidence intervals, you could estimate that about two-thirds

of the time (actually 68% of the time), the funds result in any year will be in a range of .398

[expected return + one standard deviation] to (.17) [expected return one standard deviation].

(c)

353.

3531374.

4671374.114.

)284(.64485.1114.

64485.1rVaR

(d) The VaR indicates that 5% of the time wed expect the return on this fund to be below .353 (i.e.,

the probability that its return will be below .353 is 5%). Put differently, it indicates that 95% of

the time wed expect the return on this fund to be above .353 (i.e., the probability that its return

will be better than .353 is 95%).

Solutions for Questions and Problems #1(1).pdf

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