Chapter 7 Risk and return Solutions to questions 1.
It is suggested that investors behave as though they are risk-averse when an investment involves a significant proportion of their wealth. However, investors may exhibit riskseeking behaviour where an investment involves only a small outlay, but offers a small probability of a very large return. For a risk-averse investor, the standard deviation (or variance) of the return distribution is a relevant measure of risk if returns are normally distributed.
2.
Purchasing securities with rates of return that are less than perfectly positively correlated, provides an investor investo r with the benefit of risk reduction. The amount of risk reduction reduct ion that can be achieved by adding a new security to an existing portfolio increases as the correlation between the expected returns on the new security and the expected returns on the existing portfolio decreases.
3.
The combination of two assets whose returns are perfectly negatively correlated—that is, ρ1,2 = –1.0—can produce a portfolio with zero variance. However, in practice, it is unlikely that two (or more) such securities can be found.
4.
This statement is not necessarily true. For example, assume that we have all of our wealth invested in a very low-risk asset. We then sell off half of the initial portfolio and invest these proceeds into a relatively high-risk asset. The total risk of the resultant portfolio may well have increased, even where the returns of the constituent assets are less than perfectly correlated. This highlights the point that we do not demonstrate the benefits of diversification diver sification by comparing comparin g the risk of the diversified div ersified portfolio p ortfolio with the risk of the portfolio prior to the addition of the new assets. Instead, a diversification benefit is demonstrated by comparing the risk of the portfolio with the weighted average risk of the individual assets. Provided that the returns of the new assets are less than perfectly correlated with the returns of the initial portfolio, the risk of the portfolio will be less than the weighted average risk of all assets in the portfolio.
5.
These terms are explained in the chapter.
6.
The total risk of a portfolio (or a security) is measured by the standard deviation (or variance) of its returns. The total risk of a portfolio can be reduced by increasing the number of securities in the portfolio. The systematic risk of a security is measured by its beta value. This is the relevant measure of a security’s risk for an investor who holds the security as part of an efficient
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portfolio. Systematic risk is that component of a security’s total risk that cannot be diversified away, and it is totally dependent on market factors. Unsystematic risk is the difference between a security’s total risk and its systematic risk, and it is that part of total risk that is specific to the security. Unsystematic risk is sometimes referred to as the security’s diversifiable risk, as it can be completely removed by holding the security as part of an efficient portfolio. 7.
Discussion in the chapter indicates that a security’s unsystematic risk can be removed by diversification. It is suggested, therefore, that the market will not compensate an investor for unsystematic risk. As a result, the market price of securities will reflect only their systematic risk. As it is assumed that management’s objective is to maximise the market value of the company’s shares, then the impact of any financial decision on the company’s systematic risk is an important consideration for the financial decision-maker. Managers should also ensure that proposed investments by a company offer expected returns that are adequate, given the investment’s systematic risk.
8.
There are few, if any, shares with negative betas, because the returns on most businesses are positively related to the state of the economy that is reflected in the returns on the market portfolio—that is, when the economy is growing strongly, the profitability of most companies will increase, and the prices of their shares will increase. Conversely, during a recession or depression, the profitability of most companies will fall, as will the prices of their shares and the value of the market portfolio. Therefore, the returns on most shares will be positively correlated with the returns on the market portfolio, which means that most shares will have positive betas.
9.
The statement is false. It is true that diversification is good for investors, but investors can easily diversify their portfolios by purchasing the shares of several companies. Therefore, diversification at the company level does not create any new investment opportunity, and there is no reason for investors to pay a premium for the shares of companies that diversify.
10. Minco’s employees should not endorse the fund’s investment policy because it involves the failure to diversify. The wealth of the fund’s members is already dependent on the prosperity of the mining industry, in that they are employees of a mining company. If their superannuation fund also invests heavily in mining company shares, many members will have most of their wealth ‘invested’ in one industry. Therefore, their risk can be reduced by investing in a wider range of industries. 11. Hailstorms tend to be localised, and are likely to affect only a small proportion of farms in any given time period. Therefore, the risk of hail damage to each crop is largely independent of the risk of such damage to other crops. In effect, an insurance company is able to diversify away much of the risk associated with claims for hail damage. However, when a flood occurs, it may affect a large area, so that most or all of the flood-prone land is flooded at the same time. Therefore, an insurance company that offered flood insurance could expect that whenever a flood occurred, it would have a large number of costly claims at the same time. It would be much more difficult to reduce this risk by Solutions manual to accompany Business Finance 11e by Peirson, Brown, Easton, Howard and Pinder ©McGraw-Hill Australia 2012
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diversification, and rather than face the prospect of many simultaneous payouts, insurance companies may decide not to offer insurance against the risk of flood. 12. Both types of models are based on the principles that investors require compensation for taking on risk, and that the market will only reward investors for bearing risks that cannot be eliminated by diversification. The major difference is that in the CAPM, the market portfolio is identified as ultimately the single source of risk, whereas, with other models, there are additional risk factors that are accounted for. For example, the FamaFrench three factor model of expected returns is provided in equation 7.16 on page 196, and is specified as: E ( Rit ) – R ft = βiM [ E ( R Mt ) – R ft ] + βiS E (SMBt) + βiH E ( HMLt )
As is apparent from the equation above, the first factor, reflecting the sensitivity of asset i’s returns to the returns from the market portfolio, is identical to that specified in the CAPM. In addition to this, an asset’s expected return is also linked to the sensitivity of its returns to the size factor which is measured by the returns on Small Minus Big firms (SMB), and to the book-to-market factor which is measured by the returns on High Minus Low (HML) book-to-market ratio firms. 13. The use of a simple benchmark equity index, such as the S&P/ASX 200 Index, to assess the performance of a portfolio is only really ever appropriate when the portfolio being assessed is a diversified portfolio of shares that closely matches the risk profile of the benchmark index. 14. As described in Section 7.8 (and illustrated in Example 7.3) the Sharpe ratio and the Treynor ratio each measure the risk-return performance of a portfolio using alternative measures of risk. The Sharpe ratio assumes that the relevant measure of risk is the total risk of the portfolio as measured by the ex-post standard deviation of returns of the portfolio over the investment period. Consequently, the Sharpe ratio is only an appropriate tool for performance measurement when dealing with well-diversified portfolios. Conversely, the Treynor ratio utilises the ex-post systematic risk of the portfolio is benchmarking the risk-return performance of the portfolio against a market proxy and is a more appropriate tool to use when assessing the performance of an individual asset or undiversified portfolio (although of course it is still a suitable measure of performance for well-diversified portfolios).
Solutions to problems
1.
(a) A risk-averse investor requires a higher return to compensate for additional risk. In this situation, Mr Barlin would prefer either X or Z to Y . (b) A risk-neutral investor ignores risk when making investment decisions. In this case, Mr Barlin would rank Z first, and then rank X and Y equally.
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(c) A risk-seeking investor obtains utility from both expected return and risk. Therefore, Mr Barlin would prefer Z to Y and, in turn, prefer Y to X. 2.
(a)
2
σ p
= w12 σ 12 + w22 σ 22 + 2w1 w2 ρ1 ,2 σ 1σ 2
ρ 1, 2 = +1.0: 2
σ p
= (0.4)2 (0.2)2 + (0.6)2 (0.10)2 + 2(0.4)(0.6)(1.0)(0.2)(0.1) = 0.0196 (σ p = 0.14 or 14%)
(b) ρ 1, 2 = 0.5: 2
σ p
= (0.4)2 (0.2)2 + (0.6)2(0.1)2 + 2(0.4)(0.6)(0.5)(0.2)(0.1) = 0.0148 (σ p = 0.1217 or 12.17%)
(c) ρ 1, 2 = 0: 2
σ p
= (0.4)2 (0.2)2 + (0.6)2(0.1)2 = 0.0100 (σ p = 0.1 or 10%)
(d) ρ 1, 2 = –0.5: 2
σ p
= (0.4)2 (0.2)2 + (0.6)2(0.1)2 + 2(0.4)(0.6)(–0.5)(0.2)(0.1) = 0.0052 (σ p = 0.0721 or 7.21%)
3.
2
σ p
= w A2 σ A2 + w B2 σ B2 + 2 w A w B ρ A, B σ A σ B
(a) ρ A,B = +1.0: σ p
= (0.4)2 (0.08)2 + (0.6)2 (0.12)2 + 2(0.4)(0.6)(1.0)(0.08)(0.12)
σ p
= 0.010816 = 0.104 or 10.4%
2
(b) ρ A,B =
0.4:
σ p
= (0.4)2 (0.08)2 + (0.6)2 (0.12)2 + 2(0.4)(0.6)(0.4)(0.08)(0.12)
σ p
= 0.008051 = 0.089728 or 8.97%
2
(c) ρ A,B = 0: σ p
= (0.4)2 (0.08)2 + (0.6)2 (0.12)2 + 2(0.4)(0.6)(0)(0.08)(0.12)
σ p
= 0.006208 = 0.078791 or 7.88%
2
(d) ρ A,B = –1.0: σ p
= (0.4)2 (0.08)2 + (0.6)2 (0.12)2 + 2(0.4)(0.6)(–1)(0.08)(0.12)
σ p
= 0.001600 = 0.04 or 4%
2
Comment: Solutions manual to accompany Business Finance 11e by Peirson, Brown, Easton, Howard and Pinder ©McGraw-Hill Australia 2012
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The risk of a portfolio depends significantly on the correlation between the returns on the assets in the portfolio. When the correlation is +1, the standard deviation of the portfolio is the weighted average of the standard deviations of the individual assets. When the correlation is less than +1, the standard deviation of the portfolio is less than the weighted average of the standard deviations of the individual assets. The question illustrates the fact that the benefits of diversification arise from combining assets whose returns are less than perfectly positively correlated. 4.
(a) E ( R L ) = = E ( R M ) = = 2 σ L = = σ L = 2 σ M
σ M
(b)
5.
σ L, M
= = = = = = = =
(a) E ( R p) = = 2 σ p = = σ p = =
0.5(–0.12) + 0.5(0.24) 0.06 (or 6%) 0.4(–0.12) + 0.6(0.24) 0.096 (or 9.6%) 0.5(–0.12 – 0.06) 2 + 0.5(0.24 – 0.06)2 0.0324 0.0324 0.18 or 18% 0.4(–0.12 – 0.096)2 + 0.6(0.24 – 0.096) 2 0.0311 0.0311 0.1764 or 17.64% ρ L,M σ Lσ M 0.75(0.18) (0.1764) 0.0238 (1/3) 0.15 + (2/3) 0.21 0.19 or 19% (1/3)2 (0.18)2 + (2/3)2(0.25)2 + 2(1/3)(2/3)(0.5)(0.18)(0.25) 0.0414 0.0414 0.2034 (or 20.34%)
(b) The expected rate of return in each case is 19%. = 0: ρ 1,2 2 σ p = (1/3)2 (0.18)2 + (2/3)2 (0.25)2 + 2(1/3)(2/3)(0)(0.18)(0.25) σ p
= 0.0314 = 0.1772 (or 17.72%)
ρ 1, 2
= 1.0:
σ p
= (1/3)2 (0.18)2 + (2/3)2 (0.25)2 + 2(1/3)(2/3)(1.0)(0.18)(0.25)
σ p
= 0.0514 = 0.2267 (or 22.67%)
2
(c) By investing in two shares that are less than perfectly correlated, Harry has achieved a diversification benefit. This is demonstrated by the fact that the risk of his portfolio (20.34%) is less than the weighted average risk of the individual assets in the portfolio (22.67%). Solutions manual to accompany Business Finance 11e by Peirson, Brown, Easton, Howard and Pinder ©McGraw-Hill Australia 2012
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6.
(a) E ( R1) = = = = σ 12 = = = σ 1 =
w A E ( R A) + w B E ( R B) 0.4 E ( R A) + 0.6 E ( R B) 0.4(12.5) + 0.6(16) 14.6% 2 2 2 2 w Aσ A + w Bσ B + 2 w A w B ρ ABσ Aσ B (0.4)2(40)2 + (0.6)2(45)2 + 2(0.4)(0.6)(0.2)(40)(45) 1 157.8 34.026%
(b) E ( R2) = = = σ 22 =
w A E ( R A ) + w B E ( R B ) + wC E ( RC ) 0.6(12.5) + 0.225(16.0) + 0.175(20) 14.6% 2 σ C 2 + 2 w A w B ρ A, Bσ Aσ B + 2 w A wC ρ A,C σ Aσ C w A2 σ A2 + w B2 σ B2 + wC
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+ 2w B wC ρ B ,C σ Bσ C
σ 2
= (0.6)2 (40)2 + (0.225)2 (45)2 + (0.175)2(60)2 + 2(0.6)(0.225)(0.2)(40)(45) + 2(0.6)(0.175)(0.35)(40)(60) + 2(0.225)(0.175)(0.1)(45)(60) = 1 083.628 = 32.9185%
Comment: (a) and (b) Both portfolios have the same expected return, but Portfolio 2 has the lower risk, despite the fact that asset C has the highest standard deviation of the three assets. Portfolio 2 is more diversified than Portfolio 1, because it contains all three assets. (c) E ( R3) = = = = σ 32
w A E ( R A) + w B E ( R B) + w F E ( R F ) (0.048)(12.5) + (0.75)(16) + (0.202)(9.9) 14.6% 2 w A2 σ A2 + w B2 σ B2 + w F σ F 2 + 2 w A w B ρ A, Bσ Aσ B + 2 w A w F ρ A, F σ Aσ F + 2w B w F ρ B , F σ Bσ F
σ 3
= (0.048)2 (40)2 + (0.75)2 (45)2 + 0 + 2(0.048)(0.75)(0.2)(40)(45) + 0 + 0 = 1 168.67 = 34.186%
Comment: (a) (b) and (c): All three portfolios have the same expected return, but they differ in risk. Portfolio 3 has the highest risk despite the fact that it includes an investment in the risk-free asset F. Portfolio 3 is not well diversified, because 75 per cent of the portfolio is invested in one asset and this largely accounts for its greater risk. This highlights the point that the benefits of diversification are dependent not only on the correlation between assets, but also on the relative weights invested in the assets. (d) E ( R4) = = = 2 σ 4 =
w A E ( R A) + w B E ( R B) + wC E ( RC ) (1/3)(12.5) + (1/3)(16.0) + (1/3)(20.0) 16.16% (1/3)2 (40)2 + (1/3)2 (45)2 + (1/3)2(60)2 + 2(1/3)(1/3)(0.2)(40)(45)
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+ 2(1/3)(1/3)(0.35)(40)(60) + (1/3)(1/3)(0.1)(45)(60) = 1 129.444 = 33.607%
Comment: Portfolio 4 has a higher expected return, and lower risk, than Portfolios 1 and 3. All riskaverse investors will prefer Portfolio 4 to Portfolios 1 and 3. Portfolio 4 has a higher risk, and a higher expected return, than Portfolio 2. Depending on the investor’s preferences, a risk-averse investor may prefer Portfolio 2 or Portfolio 4. Note also that Portfolio 4 is well diversified, because it contains the three risky assets in equal proportions. Because of its better diversification, Portfolio 4 is probably close to the efficient frontier. (e) E ( R5) = w A E ( R A) + w B E ( R B) + wC E ( RC ) + w F E ( R F ) = (0.25)(12.5) + (0.25)(16) + (0.25)(20) + (0.25)(9.9) = 14.6% Portfolio 5 is equivalent to a combination of Portfolio 4 (75%) plus the risk-free asset F (25%). Therefore, its standard derivation can be calculated as: 2 2 2 σ 5 = w4 σ 4 2
2
= (0.75 ) (33.60 ) = 635.312 σ 5 = 25.205%
Comment: Portfolio 5 effectively consists of Portfolio 4 plus the risk-free asset. It has the same expected return as Portfolios 1, 2 and 3, but a much lower risk. The results show that a favourable risk-return combination can be obtained by combining a well-diversified portfolio of risky assets with an investment in the risk-free asset. 7.
(a) E ( Ri) = R f + βi[ E ( R M ) – R f ] = 0.08 + 1.25 (0.06) = 0.155 or 15.5% (b) 0.11 = 0.08 + 0.75 ( E ( R M ) – 0.08) E ( R M ) = 0.12 or 12% (c) 0.14 βi
8.
= 0.10 + βi(0.05) = 0.8
(a) Portfolio of A and B 12 = w A E ( R A) + w B E ( R B) but w B = (1 – w A) ∴ 12 = w A(10.8) + (1 – w A)(15.6) ∴w A = 0.75, w B = 0.25 σ a2 = w A2 σ A2 + w B2 σ B2 + 2 w A w B Cov( R A , R B ) = (0.75)2 (324) + (0.25)2 (289) + 2(0.75) (0.25) (60)
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σ a
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= 222.8125 = 14.9269%
(b) Portfolio of B and F 12 = w B (15.6) + (1 – w B)(6) w B = 0.625 2 σ B = w B2 σ 2B
σ b
= (0.625)2 (289) = 112.89 = 10.625%
(c) Portfolio of M and F = w B(14) + (1 – w M ) (6) 12 ∴ w M = 0.75 2 σ 2 = W M 2 σ M c
σ c
= (0.75)2 (80) = 6.708%
Pricing of assets A and B. The CAPM is: E ( R) + R f + β( E ( R M ) – R f ) = 6 + β(14 – 6) = 6 + 8β For asset A: E ( R A) = 6 + 8 β A where Cov( R A , R M ) β A = Var ( R M ) 48 = 80 = 0.6 ∴ E ( R A) = 6 + 8(0.6) = 10.8% 10.8% agrees with the data given. Therefore, asset A is priced according to the CAPM. For asset B: β B
96 80 = 1.2
=
Therefore, the CAPM gives: E ( R B) = 6 + 8(1.2) = 15.6%, which agrees with the data given, and shows that B is priced according to the CAPM. Solutions manual to accompany Business Finance 11e by Peirson, Brown, Easton, Howard and Pinder ©McGraw-Hill Australia 2012
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Comments: All three portfolios have an expected return of 12%. As the CAPM holds in this market, all three portfolios should have the same beta. This is shown below: First portfolio [part (a)]:
β = w Aβ A + w Bβ B = (0.75)(0.6) + (0.25)(1.2) = 0.75
portfolio beta
Second portfolio [part (a)]: β = w Bβ B + w F β F = (0.625)(1.2) + (0.375)(0) portfolio beta = 0.75 Third portfolio [part (a)]:
β = w M β M + w F β F = (0.75)(1) + (0.25)(0) = 0.75
portfolio beta
However, the standard deviations of the portfolios differ because of the different levels of diversification. The first two portfolios are poorly diversified. Because the CAPM holds, the diversifiable risk in the first two portfolios is not rewarded with increased expected return. 9.
(a) E ( R P ) = w BHZ E ( R BHZ ) + w ANB E ( R ANB) = 0.3(9) + 0.7(13) = 11.8% (b) σ P 2
σ P
2 2 2 2 = w BHZ σ BHZ + w ANB σ ANB + 2 w BHZ w ANB ρ BHZ , ANBσ BHZ σ ANB
= (0.3)2 (8)2 + (0.7)2 (48)2+ 2(0.3)(0.7)(0.8)(8)(48) = 1 263.73 = 35.55%
(c) You should point out to your client that the benefits of diversification are not measured by comparing the risk of your portfolio prior to the addition of new assets, with the risk of the portfolio after the their addition. Instead, we can demonstrate a diversification benefit by comparing the risk of a portfolio with the weighted average risk of the individual assets. In this case, the standard deviation of the portfolio’s returns ( σ P =35.55%) is slightly less than the weighted average of the asset’s individual standard deviations ( σ AVERAGE = (0.3)(8)+(0.7)(48) = 36%). 10. The weight of the investment in Outlook Publishing is $3 million of $8 million, or 0.375 of the portfolio. The weight of the investment in Russell Computing is 0.675. Using Equation 7.4, the variance of the returns on the portfolio will be: 2
2
2
2
σ p2 = (0.375) (0.4) + (0.675) (0.25) + 2(0.375)(0.675)(0.7)(0.4)(0.25)
= 0.086414
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The standard deviation of portfolio returns, σ p, is, therefore, 0.2939625 or 29.39625%, and the standard deviation on the investment is $8 million × 0.2939625 = $2.3517 million. The value at risk of the portfolio is $2.3517 million multiplied by 2.327 (i.e. the critical value from a standard normal distribution that equates to a 1% probability of the abnormally bad market conditions) = $5.4724 million. This calculation assumes that returns follow a normal probability distribution. 11. (a) The simple benchmark index appears to have outperformed the Fort Knox Fund by 2% over the year. This measure of course fails to account for any differences in the risk profile of the fund and the benchmark index. (b) The Sharpe ratios for the Fort Knox Fund and the S&P/ASX 200 index are calculated as follows: r P − r f
S F or tK no x =
σ P
SS&P/ ASX 200 =
r P − r f σ P
=
=
0 . 10 − 0 . 03 0 .1 5
= 0 .4 6 6 7
0.12 − 0.03 = 0.3000 0.30
As the Sharpe ratio for the fund exceeds that of the index—the fund appears to have outperformed the index.
(c) The Treynor ratios for the Fort Knox Fund and the S&P/ASX 200 index are calculated as follows:
T F or tK no x =
r P − r f
TS&P/ ASX 200 =
β P
r P − r f β P
=
=
0 . 10 − 0 . 03 = 0 .9 3 3 3 0 .7 5
0.12 − 0.03 = 0.9000 1
As the Treynor ratio for the fund exceeds that of the benchmark index, this final result confirms the suggestion that the fund has outperformed the index after allowing for the differences in systematic risk between the two assets.
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