Estructura Básica de las matrices

(a) Given:

βSHARES

Systematic risk of shares Risk of market po =

rtfolio then:

Systematic risk of shares =βSHARES× Risk of market portfolio

Also given: βx y βx βy x x y y x y + = ⋅ ₊ + ⋅ ₊ Therefore: βV A+ =  0.67 +0.67 =0.67 62 66 4 66 βV B+ =_0.67 _+_1.14 _=0.698 62 66 4 66 βV C+ =  0.67 +0.88 =0.683 62 66 4 66 C H A P T E R 1 2 T H E C A P I T A L A S S E T P R I C I N G M O D E L 7 2 9

Chapter 12

The capital asset pricing model

problem

1

And so: Systematic β × σ_M risk V + A = 0.67 × 16% = 10.72% V + B = 0.698 × 16% = 11.17% V + C = 0.683 × 16% = 10.93%

In all three cases, the new market value of the company would be expected to be: £62 million + £4 million = £66 million.

(b) Portfolio theory shows that when assets are combined, the total risk of the combina- tion (measured as standard deviation of returns) is less than a weighted average of the risks of the individual assets, as long as the assets are less than perfectly positively corre- lated with each other. The further away the correlation coefficient is from being per- fectly positive (i.e. +1), the greater will be the amount of risk reduction.

The simplest type of portfolio is a two-asset portfolio. Vanhal, plus an acquired company, could be viewed as such a portfolio. For example, if Vanhal were to acquire company A, then the total risk of the enlarged company would be given by:

σV+A=√x 2

σ2

V+ 2x(1 – x)σVσAρV,A

where x and (1 – x) represent the proportions of Vanhal and A represented in the enlarged company. As long as the correlation coefficient (ρV,A) is less than + 1, thenσV+A

will be less than (σV⋅ x) + σA⋅ (1 – x). In other words, Vanhal will have been able to

reduce its total risk (in the sense that the resulting total risk will be less than a weighted average of the total risk of the components) through the diversification process. (c) There could be a large number of possible reasons why the directors of Vanhal might wish to diversify, but the question indicates that the primary reason is to broaden the company’s activities. In this respect, the desire of the directors to diversify arises from their wish to reduce the total risk of the company. How this may be done – in rela- tion to the company’s stock exchange return – has been shown in the answer to part (b). However, from the directors’ viewpoint, risk reduction through diversification would manifest itself in a reduction in the variability of the company’s operating cash flow and therefore resulting (in stock market parlance) in an increase in the perceived ‘quality’ of the company’s earnings.

The directors would be interested in trying to bring about such an effect for two reasons. One would be to hope for an enhancement of the stock market price of the company’s shares through the increased earnings quality leading to a higher price–earnings ratio multiple being applied to the company’s earnings per share. However, such an effect would only come about if the market valued total risk, rather than just systematic risk.

The second reason for the directors’ interest in such a policy would be the benefits that a more stable corporate cash flow would bring to the job of management. For example, there would be a reduced probability of insolvency (and the consequential costs for directors); there may be opportunities for increasing the company’s gearing; a stable dividend policy might be able to be maintained with greater ease; and generally the task of managing the overall company would become less demanding.

To suggest which of the three companies under consideration best meets the direc- tors’ requirements is difficult, given the information available. As all three are in the same area of industry, any one of the three would presumably provide the required broadening of the company’s activities. However, assuming that the directors are inter- ested in such a move in order to reduce total corporate risk, then they may not be indif- ferent between the three companies. Given that all three companies have the same value,

and assuming that they all have the same correlation coefficient with Vanhal (which is not unrealistic, given the circumstances), then company A is likely to be preferable as it has the smallest amount of total risk and specific risk. (Non-specific or systematic risk cannot be diversified away.) If the assumption about the correlation coefficient is unsafe, then the company best suiting the directors’ requirements would be that whose combi- nation of total risk and correlation coefficient – used in the expression given in answer to part (b) above – would result in the lowest level of total risk for the enlarged company. (d) Portfolio theory and the capital asset pricing model suggest that investors should only be interested in systematic risk. As systematic risk cannot be diversified away, there would be no risk reduction benefits accruing to shareholders as a result of the merger, assuming that they already hold well diversified investment portfolios.

In fact such a move as that contemplated by Vanhal may be unwelcome to share- holders if it were to significantly change the total market value and beta of the company. In such circumstances, a shareholder holding a diversified portfolio with a desired beta value would have to adjust his/her portfolio (and so incur transaction costs) in the light of the change to Vanhal.

Despite the foregoing, shareholders might still welcome the takeover, even given the assumption in part (a) of the question that there are no synergy benefits. For example, shareholders holding non-fully diversified portfolios would gain some risk reduction effect. Alternatively, if the company taken over was worth more than the £4 million pur- chase price, then Vanhal’s shareholders’ wealth would increase. Again if, as a result of the takeover, Vanhal were able to increase their debt capacity, then the tax shield benefits (if they exist) would also accrue to the shareholders (see Chapter 19).

However, given a reasonably efficient capital market with shareholders holding well diversified investment portfolios, the value to shareholders of such a takeover as that proposed is likely to be minimal. In fact, the costs (both internally and externally to Vanhal) associated with the takeover may result in shareholders suffering an actual reduction in their wealth.

problem

2

(a) Given that Mr Swift has a well diversified portfolio, it will be safe to assume that most of the unsystematic risk that is attached to the individual securities in the portfolio will have been diversified away. Thus his portfolio risk will largely consist of systematic risk and so the variance of returns on his portfolio will essentially measure systematic risk.

If he wishes to include shares that will reduce his portfolio variance, then he will be concerned with the covariance of returns between any new share and his existing port- folio. In other words, he is interested in selecting the shares of that company which would help to bring the greatest reduction in portfolio risk. This would be indicated by looking at the product of the standard deviation of returns and the correlation with Mr Swift’s existing portfolio:

Dove: σD×ρD,S= 35% × 0.16 = 5.6%

Jay: σJ ×ρJ,S = 30% × 0.21 = 6.3%

Under these circumstances, the optimal course of action for Mr Swift – given his objective – would be to invest in the shares of Dove plc.

(b) The CAPM shows that there is a positive relationship between the expected return on a security and its degree of systematic risk – which is normally measured by its beta value. Thus, the greater the amount of systematic risk the greater will be the expected return demanded by investors in an equilibrium stock market. The systematic risk of C H A P T E R 1 2 T H E C A P I T A L A S S E T P R I C I N G M O D E L 7 3 1

individual securities can be measured as the product of their standard deviation of return and their correlation coefficient with the market portfolio. In the case of Dove and Jay, this gives values for systematic risk of:

Dove: σD×ρD,M= 35% × 0.3 = 10.5%

Jay: σJ ×ρJ,M = 30% × 0.25 = 7.5%

As Dove plc has the higher level of systematic risk, it would follow that Dove should have the higher expected return, as indeed it does: 9% as against only 7% for Jay plc. (c) If Mr Swift’s portfolio contained shares in a few companies only, then he would be holding a largely undiversified portfolio. Hence, the variance of returns of the portfolio would reflect both the systematic and unsystematic risk components as there would be insufficient diversification to wash out the unsystematic risk.

Without passing comment on whether such a portfolio is wise (although it would appear sensible for Mr Swift to diversify further), in order to meet his objective on port- folio variance Mr Swift would be most interested in selecting that company which would help to bring the greatest reduction to the total risk of his existing portfolio.

However, in these particular circumstances a problem arises from the fact that with small portfolios a security’s contribution to portfolio risk can arise out of its own vari- ance, as well as from its covariance with the existing portfolio. Thus, although Dove plc, as was seen in the answer to (a), has the smaller covariance with the existing portfolio, it has a higher variance (or standard deviation) than Jay:

σ2

D= 0.1225 σ 2

J= 0.09

Therefore the final choice between the two companies will depend upon the existing components of Mr Swift’s portfolio and their weights and the resulting changes brought about through the introduction of the new security.

(d) Shareholders, assuming that they hold well diversified efficient portfolios, will be interested in the effect on the risk of their portfolio of the addition to it of any particular security. However, because the portfolio is fully diversified, when a new share is added (again, assuming that only a marginal investment is made), then that new share’s unsys- tematic risk is eliminated, or is washed out, and it is only the systematic risk that is added to the portfolio. It is therefore for this reason that the relevant measure of risk for a com- pany’s shareholders is the amount of systematic risk. This is most conveniently mea- sured in relative terms via the beta value.

Debt holders are also interested in the systematic risk of their investment which, again, could be measured by beta. However, as most debt is unquoted, beta does not provide a convenient measure of risk. Hence debt holders attempt to measure the risk of a company’s debt through a series of alternative measures including the degree of capital gearing, the interest cover ratio, the amount of tangible assets held by the company and the stability, or otherwise, of the company’s annual net cash flow. Just which of these factors contribute to systematic risk and which to unsystematic risk is somewhat unclear. What evidence there is available suggests that all four factors – with the possible excep- tion of the amount of tangible assets held – are likely to contribute to systematic risk.

Finally, managers, as far as their labour is concerned, hold undiversified portfolios. Therefore, unlike outside investors, they are interested in the total risk of a company and – in particular – the likelihood that it will fall. Thus, managers are likely to measure risk by the variability of net annual cash flows (i.e. by the variance of corporate net cash flows), the skew of those cash flows and by the degree of capital gearing. All three factors will have a bearing on the riskiness of the company as seen from management’s viewpoint.

problem

3

(a) (i) Expected return on Cemenco equity Average % annual capital gain:

[16.42 ÷ 9.50]1

3_{– 1= 20%}

Average % dividend yield:

[10% + 12% + 8% + 10%] ÷ 4 = 10% Therefore, expected return on Cemenco shares = 20% + 10% = 30% (ii) Expected return on TSE Index

Average % annual capital gain:

[1983 ÷ 1490]1

3_{– 1 = 0%}

Average % dividend yield:

[16% + 15% + 10% + 18%] ÷ 4 = 15% Therefore, expected return on the TSE Index = 10% + 15% = 25% (iii) Return on government stocks

15% + 16% + 14% + 15% ÷ 4 = 15% Therefore, risk-free return = rf= 15%

(iv) Beta value of Cemenco equity

E[r_c] = r_f+ (E[r_m] – r_f) .βc Therefore: E r r E r r c f m f c [ ] – [ ] – % – % % – % – =β 30 15 = 25 15 15 10 1.50

(b) It is difficult to predict with any accuracy whether the government’s action will make Cemenco Ltd more or less systematically risky. Although in total risk terms the risk of the company will be reduced, it is difficult to be certain what will be the effect on sys- tematic risk. The company’s revenues will, as always, be fairly sensitive to the level of Trinka’s economic activity and this is unlikely to change by being given a monopoly. However, there will be price control and this may therefore result in increasing the com- pany’s systematic risk exposure.

Chapter 13

Option valuation

problem

1

(i) In order to answer this question, we need to use the Black and Scholes model to value the Pear call options, where:

S = 415p X = 400p; T = 0.25 years R_f= 0.05 ␴ = 0.22. C H A P T E R 1 3 O P T I O N V A L U A T I O N 7 3 3

problem

3

Chapter 13

Option valuation

and the Black and Scholes model is: C = S × N (d₁) – [X × e–Rf × T] × N (d₂) where:

(

)

( ) log ( ) d R T T T 1 =    +   +   e S X f 0.5 σ σ and (d2) = (d1) –␴ × √T

We will do the calculations, item by item:

logeS/X = loge415/400 = loge1.0375 = 0.0368 Rf × T = 0.05 × 0.25 = 0.0125 ␴ × √T = 0.22 × √0.25 = 0.22 × 0.50 = 0.11 0.5 ×␴ × √T = 0.5 × 0.11 = 0.055 (d1) = 0.0368 + 0.0125 0.11 + 0.055 = 0.5032 (d₂) = 0.5032 – 0.11 = 0.3932

Using the area under the normal curve tables:

N (d₁) = N (0.50) = 0.5 + 0.1915 = 0.6915 N (d₂) = N (0.39) = 0.5 + 0.1517 = 0.6517 and so finally: C = 415p × 0.6915 – (400p × e–0.25× 0.05) × 0.6517 C = 286.97p – (400p × 0.9876) × 0.6517 C = 286.97p – (395.03p × 0.6517) C = 286.97p – 257.44p = 29.5p

The Black and Scholes model suggests that the Pear calls should have a value of 29.5p. (ii) We can use the put–call parity equation to value the pear put options:

P = C + [X × e–Rf × T] – S where: C = 29.5p X = 400p S = 415p e–Rf × T= e–0.05 × 0.25= 0.9876 Therefore: P = 29.5p + [400p × 0.9876] – 415p = 9.5p

(iii) The delta or hedge ratio for the Pear calls is given by the value of N (d₁) = 0.6915. Therefore, in order to construct a delta neutral hedge on the holding of 50 000 Pear shares, the investor would need to sell/write:

50 000 shares ÷ 0.6915 = 72 307 call options.

As a result, any gains or losses on the shareholding should be exactly matched by off- setting losses or gains on the written call options.

problem

2

One plc Two plc Three plc

e–Rf.T 0.988 0.951 0.963 √T. ␴ 0.15 0.177 0.173 Rf.T 0.0125 0.05 0.0375 In S/X 0.0513 0 –0.0954 d₁ 0.5003 0.3710 –0.2482 d₂ 0.3503 0.1940 –0.4212 N(d₁) 0.6915 0.6443 0.4013 N(d₂) 0.6368 0.5753 0.3372 C 9.4p 9.7p 4.4p P 3.3p 4.8p 10.3p

Chapter 14

Interest rate risk

In document AVANCES DE ACUICULTURA Y PESCA EN COLOMBIA VOLÚMEN I (página 32-36)