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Chapter 7 Risk and return 代寫

Chapter 7

 
Risk and return
 

Solutions to questions

 
1.    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 diversification, and rather than face the prospect of many simultaneous payouts, insurance companies may decide not to offer insurance against the risk of flood.
2.    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 risk-seeking 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.
3.    Purchasing securities with rates of return that are less than perfectly positively correlated, provides an investor with the benefit of risk reduction. The amount of risk reduction 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.
7.    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 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.
8.    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.
9.    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.
10.  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.

Solutions to problems

 
1.    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 RussellComputing is 0.675. Using Equation 7.4, the variance of the returns on the portfolio will be:
         =   (0.375)2(0.4)2 + (0.675)2(0.25)2 + 2(0.375)(0.675)(0.7)(0.4)(0.25)
              =   0.086414
       The standard deviation of portfolio returns, sp, is therefore 0.2939625, or 29.39625per cent 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 x 2.327 = $5.4724 million. This calculation assumes that returns follow a normal probability distribution.
2.    (a)   A risk-averse investor requires a higher return to compensate for additional risk. In this situation, Mr Bob Neil would prefer either X or Z to Y.
       (b)   A risk-neutral investor ignores risk when making investment decisions. In this case, Mr Bob Neil would rank Z first, and then rank X and Y equally.
       (c)   A risk-seeking investor obtains utility from both expected return and risk. Therefore, Mr Bob Neil would prefer Z to Y and, in turn, prefer Y to X.
3.   
       (a)     =  1
                   =  (0.4)2 (0.08)2 + (0.6)2 (0.12)2 + 2(0.4)(0.6)(1.0)(0.08)(0.12)
                        =  0.010816
              sp      =  0.104, or 10.4 per cent
       (b)         =0.4
                   =  (0.4)2 (0.08)2 + (0.6)2 (0.12)2 + 2(0.4)(0.6)(0.4)(0.08)(0.12)
                        =  0.008051
              sp      =  0.089728, or 8.97 per cent
       (c)     =  0
                   =  (0.4)2 (0.08)2 + (0.6)2 (0.12)2 + 2(0.4)(0.6)(0)(0.08)(0.12)
                        =  0.006208
              sp      =  0.078791, or 7.88 per cent
       (d)     =  –1
                   =  (0.4)2 (0.08)2 + (0.6)2 (0.12)2 + 2(0.4)(0.6)(–1)(0.08)(0.12)
                        =  0.001600
              sp      =  0.04, or 4 per cent
Comment:
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(RL )  = 0.5(–0.12) + 0.5(0.24)
                          = 0.06, or 6 per cent
              E(RM)   =   0.4(–0.12) + 0.6(0.24)
                          =   0.096, or 9.6 per cent
                     =   0.5(–0.12 – 0.06)2 + 0.5(0.24 – 0.06)2
                          =   0.0324
                     =
                          =   0.18, or 18 per cent
                    =   0.4(–0.12 – 0.096)2+ 0.6(0.24 – 0.096)2
                          =   0.0311
                    =  
                          =   0.1764, or 17.64 per cent
       (b)       = rL,MsLsM
                          =   0.75(0.18) (0.1764)
                          =   0.0238
5.    (a)        = 
                 =  +1.0:
                   =  (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 (sp = 0.14, or 14 per cent)
       (b)      =  0.5:
                   =  (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 (sp = 0.1217, or 12.17 per cent)
       (c)      =  0:
                   =  (0.4)2 (0.2)2 + (0.6)2(0.1)2
                        =  0.0100 (sp = 0.1, or 10 per cent)
       (d)      =  –0.5:
                   =  (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 (sp = 0.0721, or 7.21 per cent)
7.    (a)   E(R1)   =   wAE(RA) + wBE(RB)
                          =   0.4 E(RA) + 0.6E(RB)
                          =   0.4(12.5) + 0.6(16)
                          =   14.6 per cent
                    =  
                          =   (0.4)2(40)2 + (0.6)2(45)2 + 2(0.4)(0.6)(0.2)(40)(45)
                          =   1157.8
              s1        =   34.026 per cent
       (b)   E(R2)   =  
                          =   0.6(12.5) + 0.225(16.0) + 0.175(20)
                          =   14.6 per cent
                    =  
                              
                          =   (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)
                          =   1083.628
              s2        =   32.9185 per cent
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)   =   wAE(RA) + wBE(RB) + wFE(RF)
                          =   (0.048)(12.5) + (0.75)(16) + (0.202)(9.9)
                          =   14.6 per cent
                    =  
                              
                          =   (0.048)2 (40)2 + (0.75)2 (45)2 +0 + 2(0.048)(0.75)(0.2)(40)(45) + 0 + 0
                          =   1168.67
              s3        =   34.186 per cent
      
       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)   =   wAE(RA) + wBE(RB) + wCE(RC)
                          =   (1/3)(12.5) + (1/3)(16.0) + (1/3)(20.0)
                          =   16.16 per cent
                    =   (1/3)2 (40)2 + (1/3)2 (45)2 + (1/3)2(60)2 + 2(1/3)(1/3)(0.2)(40)(45)
                               + 2(1/3)(1/3)(0.35)(40)(60) + (1/3)(1/3)(0.1)(45)(60)
                          =   1129.444
              s4        =   33.607 per cent
 
Comment:     
Portfolio 4 has a higher expected return, and lower risk, than Portfolios 1 and 3. All risk-averse 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.
7.    (a)   E(R1)   =   wAE(RA) + wBE(RB)
                          =   0.4 E(RA) + 0.6E(RB)
                          =   0.4(12.5) + 0.6(16)
                          =   14.6 per cent
                    =  
                          =   (0.4)2(40)2 + (0.6)2(45)2 + 2(0.4)(0.6)(0.2)(40)(45)
                          =   1157.8
              s1        =   34.026 per cent
       (b)   E(R2)   =  
                          =   0.6(12.5) + 0.225(16.0) + 0.175(20)
                          =   14.6 per cent
                    =  
                              
                          =   (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)
                          =   1083.628
              s2        =   32.9185 per cent
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)   =   wAE(RA) + wBE(RB) + wFE(RF)
                          =   (0.048)(12.5) + (0.75)(16) + (0.202)(9.9)
                          =   14.6 per cent
                    =  
                              
                          =   (0.048)2 (40)2 + (0.75)2 (45)2 +0 + 2(0.048)(0.75)(0.2)(40)(45) + 0 + 0
                          =   1168.67
              s3        =   34.186 per cent
      
       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)   =   wAE(RA) + wBE(RB) + wCE(RC)
                          =   (1/3)(12.5) + (1/3)(16.0) + (1/3)(20.0)
                          =   16.16 per cent
                    =   (1/3)2 (40)2 + (1/3)2 (45)2 + (1/3)2(60)2 + 2(1/3)(1/3)(0.2)(40)(45)
                               + 2(1/3)(1/3)(0.35)(40)(60) + (1/3)(1/3)(0.1)(45)(60)
                          =   1129.444
              s4        =   33.607 per cent
 
Comment:     
Portfolio 4 has a higher expected return, and lower risk, than Portfolios 1 and 3. All risk-averse 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)           =   wA E(RA) + wB E(RB) + wC E(RC) + wF E(RF)
                          =   (0.25)(12.5) + (0.25)(16) + (0.25)(20) + (0.25)(9.9)
                          =   14.6 per cent
              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:
             
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.
 

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