(a) The retirement fund of Ken consists of the following two investment opportunities.
Investment INV1:
P: Principal Sum
R: Rate of Interest
T: Number of years
N: Number of times of compounding per year
I: Interest accumulated
A: Amount
Phase 1 –
P1 = $115,000
R1 = 13% p.a. = 0.13 p.a. (simple interest)
T1 = 2 years
? I1 = (P1 * R1 * T1) / 100 = $ (115,000 * 0.13 * 2) / 100 = $299
? A1 = $ (115,000 + 299) = $115,299
Phase 2 –
P2 = $115,299
R2 = 10.5% p.a. = 0.105 p.a. (monthly compound interest)
T2 = 3 years
N2 = 12
? A2 = P2 * [(1 + R2 / N2) ^ (N2 * T2)]
= $ 115,299 * [(1 + 0.105 / 12) ^ (12 * 3)]
= $115,299 * (1.00875 ^ 36)
= $115,299 * 1.37 = $157,959.63 ≅ $157,960
Phase 3 –
P3 = $157,960
R3 = 7.25% p.a. = 0.0725 p.a. (annually compound interest)
T3 = 3 years
N3 = 1
? A3 = P3 * [(1 + R3 / N3) ^ (N3 * T3)]
= $157,960 * [(1 + 0.0725 / 1) ^ (1 * 3)]
= $157,960 * (1.0725 ^ 3) = $157,960 * 1.23 = $194,290.80 ≅ 194,291
Phase 4 –
P4 = $194,291
R4 = 9% p.a. = 0.09 p.a. (daily compound interest)
T4 = 2 years
N4 = 365
? A4 = P4 * [(1 + R4 / N4) ^ (N4 * T4)]
= $194,291 * [(1 + 0.09 / 365) ^ (365 * 2)]
= $ 194,291 * (1.00025 ^ 730)
= $ 194,291 * 1.2 = $233,149.20 ≅ $233,149
? the total amount accumulated by Ken from Investment INV1 at the end of 10 years is $233,149.
Investment INV2:
P5 = $10,000
R5 = 11.5% p.a. = 0.115 p.a. (quarterly compound interest)
T5 = 5 years
N5 = 4
The investment is made at the end of each quarter and is compounded quarterly. Thus, it is a perpetuity scheme for Ken.
The amount accumulated at the end of 5 years would be
A5 = P5 * [{(1 + R5 / N5) ^ (N5 * T5)} – 1] / (R5 / N5)
Or, A5 = $10,000 * [{(1 + 0.115 / 4) ^ (4 * 5)} – 1] / (0.115 / 4)
Or, A5 = $10,000 * [(1.029) ^ 20] / (0.115 / 4)
Or, A5 = $10,000 * (1.77 / 0.02875)
Or, A5 = $10,000 * 61.56 = $615,600
The accumulated value of Ken’s investment at the end of 10 years is,
$(233,149 + 615,600) = $848,749
(b) Let the annual interest rate required to make $1,000,000 from $115,000 in 10 ears be r% p.a. compounded annually.
P = $115,000
R = r% p.a. (compounded annually)
N = 10 years
? A = P * [(1 + R / 100) ^ 10]
Or, A = $115,000 * [(1 + r / 100) ^ 10]
? 115,000 * [(1 + r / 100) ^ 10] = 1,000,000
Or, [(1 + r / 100) ^ 10] = 1,000,000 / 115,000
Or, [(1 + r / 100) ^ 10] = 8.69
Or, 1 + r / 100 = 1.24
Or, r / 100 = 0.24
Or, r = 0.24 * 100 = 24
Thus the annual rate of interest required to achieve the target is 24%.
The estimated annual rate of return is not realistic. This is because no financial institution will be able to offer such a high return on any investment for such a long period of time (Brealey, Myers and Marcus, 2014).
(c) Let the time required for the initial investment of $115,000 to turn into $1,000,000 be T years.
P = $115,000
R = 8.35% = 0.0835 p.a. (weekly compound interest)
N = 52
? A = P * [(1 + R / N) ^ (N * T)]
Or, A = $115,000 * [(1 + 0.0835 / 52) ^ (52 * T)]
Or, 115,000 * [(1 + 0.0835 / 52) ^ (52 * T)] = 1,000,000
Or, [(1 + 0.0835 / 52) ^ (52 * T)] = 1,000,000 / 115,000
Or, [(1 + 0.0835 / 52) ^ (52 * T)] = 8.70
Or, (1.002) ^ (52 * T) = 8.70
Or, (52 * T) * ln(1.002) = ln(8.70)
Or, (52 * T) * 0.001998 = 2.1633
Or, 52 * T = 2.1633 / 0.001998 = 1082.7
Or, T = 20.82 ≅ 21
? it will take 21 years for his initial investment to grow into his target amount.
The table for All Ordinaries Index, Wesfarmers Limited and Woolworths Limited is given as follows:
|
Year |
All Ordinaries Index |
Wesfarmers Limited |
Woolworths Limited |
Market Returns |
Wesfarmers Returns |
Woolworths Returns |
|
(MARKET RETURNS) |
||||||
|
2005 |
4708.8 |
18.9922 |
9.61035 |
|||
|
2006 |
5461.6 |
21.07258 |
14.22607 |
0.14831 |
0.103944301 |
0.392235554 |
|
2007 |
6421 |
23.4822 |
21.01378 |
0.16183 |
0.108270018 |
0.390102216 |
|
2008 |
3659.3 |
11.60563 |
17.27901 |
-0.5623 |
-0.70475236 |
-0.19568594 |
|
2009 |
4882.7 |
21.65905 |
19.20301 |
0.28843 |
0.623933056 |
0.105574568 |
|
2010 |
4846.9 |
23.41564 |
19.6086 |
-0.00736 |
0.077980794 |
0.020901208 |
|
2011 |
4111 |
23.08883 |
19.55875 |
-0.16467 |
-0.01405522 |
-0.00254549 |
|
2012 |
4664.6 |
30.98662 |
24.66703 |
0.12634 |
0.294206548 |
0.232044778 |
|
2013 |
5353.1 |
38.93328 |
30.05066 |
0.13767 |
0.228293913 |
0.19741709 |
|
2014 |
5388.6 |
38.97753 |
28.76203 |
0.00661 |
0.001135914 |
-0.04382851 |
|
2015 |
5174.3 |
37.06 |
23.1 |
-0.04058 |
-0.05044711 |
-0.2192235 |
|
2016 |
5719.1 |
42.14 |
24.49 |
0.10011 |
0.128459188 |
0.058432253 |
|
E(R) |
0.01767 |
0.072451731 |
0.085038566 |
|||
|
Risk |
0.21699 |
0.303727674 |
0.196000034 |
(Greene, 2011)
(b) (i) Wesfarmers Limited is a conglomerate industry catering to the markets of Australia and New Zealand and their business is based on retail, fertilizers, industrial and safety products, coal mining and chemicals. In terms of revenue it is the largest company in Australia. The ASX code of Wesfarmers Limited is 43.23 AUD. Thus Wesfarmers caters to a variety of demands in the market and will thus be affected by any changes in any of these sectors (Brealey, Myers and Marcus, 2014).
Woolworths Limited primarily deals in retail in the markets of Australia and New Zealand. It is the second largest company in Australia in terms of revenue. The ASX code of Woolworths Limited is 26.25 AUD. Thus Woolworths is only focused in the retailing industry and any changes in the performance of the company will mainly originate from changes in the retail sector (Shim and Siegel, 2008).
(ii) Wesfarmers’ performance in terms of the stock is relatively better than that of the market. The expected return from the market over the period of years is 1.77%. On the other hand, the annual expected return from Wesfarmers Limited is 7.25%. Thus Wesfarmers Limited yields a much higher return than the market portfolio. The market portfolio consists of a number of assets and hence the returns vary according to the performance of the different assets. However, the risk associated with the market is 21.7% and the risk from the Wesfarmers Limited is 30.4%. Thus though Wesfarmers Limited yields a higher return than the market, the risk associated with investing in Wesfarmers Limited is higher than that associated with the market (Brealey, Myers and Marcus, 2014). The higher risk is compensated for by the higher return. The market index is comprises many different assets and hence the risk is relatively more diversified.
Woolworths provides an even higher return with respect to the market over the period of time concerned. The expected market return over the 11 years is 1.77% and the expected return from Woolworths Limited over the 11 years concerned is 8.50%. Thus, in terms of returns Woolworths Limited has been performing better than the market. The market index consists of a number of different assets and hence the relative returns on all these assets are combined to arrive at the market return (Ross, Westerfield and Jordan, 2015). On the other hand, the risk associated with the market is 21.7% and the risk associated with investing in the stocks of Woolworths Limited is 19.6%. Thus the market is riskier than Woolworths Limited. This is because the market index consists of a variety of assets and hence the risk originates in all these assets separately. However, Woolworths deals only in retail and hence the risk will originate only from the retail sector. Thus, Woolworths Limited provides a higher return than that of the market index and the risk associated with Woolworths Limited is lower than that associated with the market. This evidently indicates that Woolworths Limited provides a much better investment opportunity than the market index.
The expected return yield of Wesfarmers over the 11 years considered is 7.25% and the expected return from Woolworths Limited over the 11 years is 8.50%. Thus, in terms of returns Woolworths performs and yields much better than Wesfarmers. The primary reason could be that Wesfarmers is a conglomerate industry and has a diversified sector which deals in a wide range of products and services. Thus the returns would depend on the performance of each of these sectors and will be accordingly determined (Ross, Westerfield and Jordan, 2015). On the other hand, Woolworths only deals in retail and if the retail sector performs well, the returns from Woolworths would be high. Retail being an essential sector of the Australian economy has consistently high demand and hence performs well consistently. Thus the returns are high. Again, the risk associated with investing in Wesfarmers is 30.4% whereas the risk associated with Woolworths Limited is 19.6%. Thus Woolworths performs much better than Wesfarmers even in terms of the risk because the risk from the former is much lower than that of the latter. Since Wesfarmers deals in a variety of sectors, any change in any of these sectors would affect the Wesfarmers stock. However, the performance of Woolworths is based only on the retail sector. Thus both in terms of risk and return the performance of Woolworths Limited over the 11 years given is much better than that of Wesfarmers.
(a)
|
Year |
Asset A |
Asset B |
Asset C |
Asset D |
Asset E |
Portfolio 1 (A, B) |
Portfolio 2 (C, D) |
Portfolio 3 (A, E) |
|
1997 |
9.62% |
10.42% |
134.78% |
27.85% |
21.87% |
10.02% |
81.31% |
15.75% |
|
1998 |
0.04% |
20.00% |
1.79% |
-0.50% |
-24.51% |
10.02% |
0.65% |
-12.23% |
|
1999 |
20.00% |
0.03% |
2.11% |
-0.40% |
52.18% |
10.02% |
0.85% |
36.09% |
|
2000 |
12.88% |
7.15% |
136.54% |
28.22% |
3.98% |
10.02% |
82.38% |
8.43% |
|
2001 |
20.00% |
0.01% |
3.92% |
-0.05% |
2.56% |
10.00% |
1.93% |
11.28% |
|
2002 |
20.00% |
0.03% |
15.05% |
2.34% |
2.73% |
10.01% |
8.70% |
11.37% |
|
2003 |
1.99% |
18.04% |
4.14% |
0.00% |
87.39% |
10.01% |
2.07% |
44.69% |
|
2004 |
8.75% |
11.28% |
5.75% |
0.36% |
36.04% |
10.01% |
3.06% |
22.39% |
|
2005 |
0.49% |
19.54% |
3.38% |
-0.18% |
43.30% |
10.01% |
1.60% |
21.89% |
|
2006 |
7.09% |
12.94% |
3.83% |
-0.07% |
11.36% |
10.02% |
1.88% |
9.22% |
|
2007 |
6.95% |
13.08% |
5.35% |
0.27% |
21.28% |
10.02% |
2.81% |
14.11% |
|
2008 |
15.01% |
5.02% |
6.02% |
0.42% |
-0.22% |
10.02% |
3.22% |
7.39% |
|
2009 |
19.88% |
0.16% |
2.45% |
-0.34% |
24.07% |
10.02% |
1.06% |
21.97% |
|
2010 |
19.98% |
0.05% |
-15.83% |
-4.25% |
41.53% |
10.02% |
-10.04% |
30.76% |
|
2011 |
0.05% |
19.99% |
8.88% |
1.01% |
30.51% |
10.02% |
4.94% |
15.28% |
|
2012 |
20.00% |
0.04% |
1.89% |
-0.47% |
16.61% |
10.02% |
0.71% |
18.30% |
|
2013 |
0.02% |
20.00% |
-4.77% |
-1.90% |
29.47% |
10.01% |
-3.34% |
14.74% |
|
2014 |
5.33% |
14.71% |
17.00% |
2.75% |
84.34% |
10.02% |
9.87% |
44.83% |
|
2015 |
0.02% |
20.00% |
10.44% |
1.35% |
-73.72% |
10.01% |
5.89% |
-36.85% |
|
2016 |
19.93% |
0.09% |
1.42% |
-0.56% |
3.79% |
10.01% |
0.43% |
11.86% |
|
E(R) |
10.40% |
9.63% |
17.21% |
2.79% |
20.73% |
10.02% |
10.00% |
15.56% |
|
Risk |
0.08122 |
0.08123 |
0.400248 |
0.085295 |
0.34331 |
5.8949E-05 |
0.24276563 |
0.17638667 |
|
Correlation coefficient |
-1 |
1 |
0 |
(Wooldridge, 2015)
(b) The correlation coefficient of a portfolio shows the degree of association between the assets that construct the portfolio. The association between the assets will determine the overall return and the risk associated with the portfolio. A high degree of positive correlation among the assets makes the portfolio very risky. This is because when there is any change in any one asset, the other assets will change in the exact direction (Berman, Knight and Case, 2013). Thus if due to some market event, the return on a certain stock falls the return on all the other assets will fall. On the other hand, a high degree of negative correlation will imply that if the returns from one asset falls, the returns from all the other assets will increase (Ross, Westerfield and Jordan, 2015). Thus the return from the entire portfolio will balance out. Hence a negative correlation is better than a positive correlation in terms of risk mitigation. If there is no association between the assets of a portfolio, then any market movement that affects one asset will leave the other asset unaffected. Hence there is no risk associated with the portfolio.
In Portfolio 1, which consists of assets A and B, the correlation coefficient is -1. This implies that when the return from A falls by 1%, the return from B will increase by exactly 1%. Thus the two assets are perfectly negatively correlated to each other. For portfolio 2, consisting of assets C and D, the correlation coefficient is 1. This implies that when the return from C falls by 1%, the return from D falls by exactly 1%. Thus the portfolio 2 is a much riskier investment option. Portfolio 3 consisting of A and E has 0 correlation coefficient. Thus if the return on asset A increases by 1% there will be no change in the return of asset E and vice versa. Thus, the investment is even less risky than portfolio 1 and 2 (Ross, Westerfield and Jordan, 2015)
(c) Diversification of assets is a financial technique used for the purpose of risk management when constructing a portfolio (Brealey, Myers and Marcus, 2014). This is because a variety of assets will, on average, generate higher returns and the risk associated would be relatively low. The main purpose of diversification is to even out the unsystematic risk associated with the portfolio. For diversification to be effective, the different assets should be uncorrelated to each other, otherwise, the purpose of higher return and lower risk would not be served. In Portfolio 1, the correlation coefficient is 1. Thus, there is no diversification of assets in portfolio 1 because the assets are perfectly positively correlated (Ross, Westerfield and Jordan, 2015). In portfolio 2, the assets are perfectly negatively correlated because the correlation coefficient is -1. There is no diversification in portfolio 2 also because the returns and risk would vary according to each other. Portfolio 3 has a correlation coefficient of 0. This means that the assets are not correlated to each other at all (Berk and DeMarzo, 2013). Hence, portfolio 3 fulfills the requirements of diversification. Since the assets are not associated, the average return would be high and the risk would be low.
(a) Beta measures the degree of volatility or the systematic risk associated with a certain industry, a company, a portfolio or a security with respect to the market. When the value of Beta for a certain asset is equal to 1, it indicates that the price of the asset moves with the market. Thus the volatility of the asset is exactly equivalent to that of the market. If the value of beta is less than 1, it indicates that the asset is less volatile than the market. Thus the price of the asset will be less than proportionately responsive to the market. When the market price changes by 1%, the price of the asset will change by less than 1%. Again, if the beta is more than 1, it indicates that the asset is more volatile or riskier than the market, that is, the price of the asset is more than proportionately responsive to the market. When the market price changes by 1%, the price of the asset changes by more than 1% (Brigham and Ehrhardt, 2016).
In the examples given, the bank and food and staples retailing industries are less volatile than the market. On the other hand, the materials and retailing are more volatile than the market. Of the given companies, Westpac is the least responsive to the market followed by Woolworths which is still less volatile than the market. However, Woodside is more volatile than the market (Ross, Westerfield and Jordan, 2015).
(b) The standard deviation of a certain asset is specific to that particular asset. It measures the risk associated with that asset. The Beta on the other hand measures the responsiveness of the asset to the market, that is, it measures the market risk or volatility associated with that asset. Beta represents the market risk premium whereas the standard deviation represents the risk. Beta accounts for the systematic risk only whereas the standard deviation accounts for both systematic and unsystematic risks (Stephen, 2017).
If a certain company has a high standard deviation, it implies that the returns from the company fluctuate rapidly, that is, the company is subject to some risk specific to itself. However, this does not necessarily imply that the stock price would be responsive to the market and will hence be subject to market risk (Ross, Westerfield and Jordan, 2015). Thus higher individual risk of a company does not imply higher market risk. Accordingly, a company having a high standard deviation does not necessarily have a high beta (Brealey, Myers and Marcus, 2014).
References
Wooldridge, J. (2015). Introductory Econometrics: A Modern Approach. 6th edn. USA: South-Western College.
Greene, W. (2011). Econometric Analysis. 7th edn. New York: Pearson Education.
Ross, S., Westerfield, S. and Jordan, B. (2015). Fundamentals of Corporate Finance. 11th edn. New York: McGraw-Hill Education.
Brealey, R. A., Myers, S. C., & Marcus, A. J. (2014). Fundamentals of corporate finance. 8th edn. New York: McGraw-Hill.
Shim, J. K., & Siegel, J. G. (2008). Financial management. 3rd edn. New York: Barron’s Educational Series.
Berman, K., Knight, J. & Case, J. (2013). Financial Intelligence: A manager’s guide to knowing what the numbers really mean. 2nd edn. New York: Harvard Business Review Press.
Berk, J. & DeMarzo, P. (2013). Corporate Finance. 3rd edn. New York: Pearson.
Brigham, E. & Ehrhardt, M. (2016). Financial Management: Theory and Practice. 15th edn. Boston: Cengage Learning.
Stephen, R. A. (2017). Fundamentals of corporate finance. New South Wales: Mcgraw-Hill Education Australia.
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