CHAPTER 11
RISK AND EXPECTED RETURN
SUGGESTED SOLUTIONS END OF CHAPTER QUESTIONS
11.1 (a)
Linda Msepe: ROI
Bank investment = R450 / R3 000 X 100 = 15% In terms of the Income Tax Act, a specified amount of interest earned is tax free, where after, once that amount is exceed, tax is payable. If the interest is subject to tax in the hands of Horace, his ROI will be ROI = R450 (,7)/R3 000 X 100 = 10,5%
(b)
Jo Naidoo: ROI
(c)
Maria Nkosi: ROI
(d)
Peter Simpson: 200 shares ROI = [240 + 200 (20-18)]/ (200 x R18) = (240 + 400)/3 600 = 17,8% Both the potential capital gain of R2 per share and the dividends received of R240 comprise a return on Peter's investment. If he decided to sell his shares at the end of the year he would realise income of R400 (capital gain) and R240 (dividends). The capital gain’s tax depends to great extent on individual circumstances, and is unlikely to exceed 10% to 15% of the return. His return on investment after capital gains tax is likely to be in the region of 14% to 15%.
Sole proprietorship = R1 200 (,7)/R9 000 = 9,3% The profits which are earned by a sole proprietorship are taxable in the hands of the owner. Flo will also be taxed on her monthly salary.
Close corporation = R20 000 (,10) / R10 000 = 20% Although no distribution of profits has been made, the profits, which have been reinvested in the business represent a return on the original investment. As the CC has already been subjected to tax, no tax is payable in the hands of the member Nicola, regardless of whether the profits have been distributed or not.
11.2 There is no correct answer to this question and it forms a basis for discussion in order to identify the principles, which underlie investment decisions. The first principle is that all investments are dependant upon the required rate of return, which in turn depends on the risk free rate (time value of money plus expected inflation rate at the time) plus the reward for the risk of the particular investment. The starting point is to assume a reasonable risk free rate – say 10%. [The variable part of this is totally dependant upon the current inflationary expectations]. Thereafter, to add to that a premium that is considered to be an estimate of the risk of each type of investment, relative to each other.
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(a)
This is a low risk investment, so a premium of around 2% to 3% is appropriate – resulting in a required rate of return of 12% to 13%.
(b)
Purchasing and owning a flat for lease offers two potential returns, one in the form of the monthly rent received and the other in the form of the capital gain when the flat is sold at a later date for a higher price. It is a relatively low risk investment and the expected return is likely to be in the region of 15% to 20% p.a. [The calculation of actual return on an annualised basis can only be done once the flat is sold].
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(c)
A general equity unit trust would be expected to deliver the average annual return on equities. This, over the long term, is in the region of 12%, thus offering an expected return of around 22%.
(d)
Placing all your funds into a single share is more risky because of the absence of diversification, and the expected return is thus fully dependant upon the fortunes of that particular company. A retail chain store is relatively less risky than the average return on all types of companies, thus the expectation is likely to be around 20%.
(e)
The return on shares of an information technology company tend to be more variable over time and therefore more risky than the average return on all types of companies, lifting the required rate of return (the expected return which will induce an investor to invest), to 25% and above.
11.3 Purchase price [share in DD] Price one year later Capital gain/loss % Return(loss) to shareholder
= = = =
R 960.00 R 840.00 -R 120.00 -12.50%
(-110 / 950)
R 3.20 R 16.00
(320/.20)
11.4 Expected returne Principal [Price]
11.5 Purchase price [share in Foschini] Price one year later Capital gain/loss Dividend Increase in wealth per share % Return(loss) to shareholder
= = = = = =
R 7.70 R 8.60 R 0.90 R 0.90 R 1.80 23.38%
(180 / 770)
Return in perpetuity Required rate of return Principal [Maximum investment]
= = =
R 120 20.00% R 600.00
(R120 /.2)
11.6
11.7 Principal R 8,000 R 8,000 R 8,000 R 20,000 R 60,000 R 4,000 R 10,000
Rate 12.00% 12.00% 12.00% 10.00% 12.00% 15.00% 18.00%
Time 8 years 8 months 8 days 3 years 10 years 152 days 40 months
Interest R 76.80 R 6.40 R 0.21 R 6,000 R 72,000 R 250 R 6,000
11.8 Principal FV @ 12% for 9 years Future value
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= R 8,000 = 2.773 = R 22,184.63
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11.9 Principal FV @ 14% for 15 years Future value
= R 250,000 = 7.138 = R 1,784,484.49
Future value required PV @ 16% for 6 years Amount to invest now
= R 40,000 = 0.410 = R 16,417.69
11.10.
11.11 Principal FV @ 16% for 8 years Future value
= R 7,500 = 3.278 = R 24,588.11
11.12 Future value of an annuity FVA @ 16% for 5 years Annual instalments
R 70,000 = 6.877 = R 10,178.66
11.13 Future value . PV @ 10% for 15 years Amount to invest now
= R 400,000 = 0.239 = R 95,756.82
Annuity instalment FVA @ 12% for 5 years Future value of the annuity
= R 12,000 = 6.353 = R 76,234.17
11.14
11.15 Annuity instalment for retirement FVA @ 12% for 20 years Future value of the annuity
= R 24,000 = 72.052 = R 1,729,258.62
Future value to receive PV @ 14% for 6 years Amount to invest now
= R 12,000 = 0.456 = R 5,467.04
11.16
11.17 Annual annuity required PVA @10% for 10 years Amount to invest now If first investment at end of year PVA @10% for 9 years Plus the first instalment immediately withdrawn Factor to use Amount to invest at the end of the first year
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= R 8,000 = 6.145 = R 49,156.54
[if invested at the beginning of 1st year]
5.759 1.000 6.759 R 54,072.19
[if invested at the end of 1st year]
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11.18 Annual annuity required PVA @12% for 7 years Amount to invest now If first investment at end of year PVA @ 12% for 6 years Plus the first instalment immediately withdrawn Factor to use Amount to invest at the end of the first year
= R 17,000 = 4.564 = R 77,583.86
[if invested at the beginning of 1st year]
4.111 1.000 5.111 R 86,893.92
[if invested at the end of 1st year]
11.19 11.20 Annual annuity to be paid to Mortgagor PVA @ 16% for 20 years Maximum amount of Mortgage Loan
= R 30,000 = 5.929 = R 177,865
Note the amount which will be repaid
= R 600,000 = 3.4
[R30 000 x 20 years] times the amount borrowed!!
Annual amount able to be repaid PVA @ 15% 6 years Instalment to befor paid each year [5 years] Maximum amount of Loan PVA @ 10% for 5 years
= R 24,000 == R3.784 5,000 R 90,828 == 3.791
Amount needed in 7 years time Note the amount which will be repaid
= R 18,953.93 Note - this is invested on her 17th birthday = R 144,000 and used [R30 for 000 20 years] thex first annuity payment on = 1.6 times the amount borrowed her 18th birthday = 0.513 = R 9,726.37
PV @ 10% for 7 years Amount to be invested today
on 18th to 22nd birthday
11.21 11.22
Required rate of return Risk free rate Beta Market return Required rate of return
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= = = = =
Risk free Rate + Beta (Risk Premium) 12.00% 1.4 22.00% 26.00% 12% + 1.4(22%-12%)
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11.23 Annual income required PVA @ 15% for 8 years Investment required now
= R 15,000 = 4.487 = R 67,310
11.24 Year Expected Dividends Expected selling price Cash Flow [cents per share] PV factor @ 15% Present Value Highest price to pay [cents]
NOW
1 2 3 65 cents 72 cents 80 cents 65 0.8696 56.52
72 0.7561 54.44
4 95 cents 1 150 cents 80 1245 0.6575 0.5718 52.60 711.83
875.40
11.25 Year Expected Dividends Expected selling price Cash Flow [cents per share] PV factor @ 18% Present Value Highest price to pay [cents]
NOW
1 2 22 cents 24 cents 22 0.8475 18.64
24 0.7182 17.24
3 30 cents 450 cents 480 0.6086 292.14
328.02
11.26 SCORES OBTAINED BY CLASS OF 30 STUDENTS 71 66 62 51 70 66 60 50 69 65 57 48 68 63 55 46 66 63 54 46
87 83 76 76 72
394
344
323
Aggregate [Sum of all 30 scores] Mean
288
241
44 43 42 41 40 210
= =
1800 60
=
62.5
=
66
[Sum of allnumbers divided by number of students]
Median [Number in the middle ie between 15th &16th number]
Mode [Number which recurs most frequently]
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11.27 YEAR 20.30 20.29 20.28 20.27 20.26
RETURN 18% 12% -2% 8% 14%
YEAR 20.25 20.24 20.23 20.22 20.21
50% 10%
RETURN 8% 10% 12% 7% 8%
YEAR 20.20 20.19 20.18 20.17 20.16
45% 9%
RETURN 18% 21% 15% -4% -3%
YEAR 20.15 20.14 20.13 20.12 20.11
RETURN 7% 13% 19% 23% 16%
47% 9%
Aggregate [Sum of all 20 returns] Mean [Sum of allnumbers divided by number of students] Median [Number in the middle ie between 10th &11th number] Mode
= =
220% 11%
=
12%
=
8%
78% 16%
[Number which recurs most frequently]
11.28 (a) Quarter QI Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Sum Mean Variance Std Dev
Data 8 6 5 4 7 0 -2 5 9 7 6 5 60 5
Deviation Deviation2 3 9 1 1 0 0 -1 1 2 4 -5 25 -7 49 0 0 4 16 2 4 1 1 0 0 110 9.17 3.03
(110/12) Square root of 9.17
(b) Probability of one standard deviaiton above the mean is (100 - 68)/2 = 16% (c ) To calculate these prbabilities, we must use the Z score Below (%) 5 = (5-5)/3.03 Z score = 0 Probability of earning below (or above) is exactly 50% (because 5% is the Mean Above (%) 10 = (10-5)/3.03 Z Score = 1.65 Table reading = 0.4505 Interpretation = .5000 - .4505 Probability of above 10% = 4.95%
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11.29 (a) Quarter QI Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Sum Mean (b) Variance Std Dev
Data
Deviation
Deviation2
4 -2 5 4 5 4 7 8 6 3 7 -3
0 -6 1 0 1 0 3 4 2 -1 3 -7
0 36 1 0 1 0 9 16 4 1 9 49
48 4
126 10.50 3.24
(126/12) Square root of 10.5
(d) To calculate the probability, the Z score must be used Higher (%) 4 = (4-4)/3.24 Z score = 0 Probability of earning below (or above) is exactly 50% (because 4% is the Mean Below (%) 0 = (0-4)/3.24 Z Score = -1.24 Table reading = 0.3925 Interpretation = .5000 - .3925 Probability of below 0% = 10.75% (This is also the probability of capital invested being eroded)
11.30 Required rate of return Risk free rate Beta Market return Required rate of return
= Risk free Rate + Beta (Risk Premium) = 10.00% = 0.8 = 18.00% = 16.40% 10% + 0.8(18%-10%)
11.31 (a) Required [Expected] rate of return Risk free rate [Govt Bond rate] Beta of Stable Ltd Market return Expected rate of return from Stabler Ltd
= = = = =
(b) Expected Portfolio return Portfolio (50% in each) Beta
= 14.05% = 1.1/2 = 0.55
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Risk free Rate + Beta (Risk Premium) 8.00% 1.1 19.00% 20.10% 8% + 1.1(19%-8%) (.5x8%)+(.5x20.1%)
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11.32 COMPARISON OF 2 SHARES
SUM MEAN VARIANCE STD DEV
YEAR
Rp
Rq
1 2 3 4 5
15 18 20 17 20
17 20 18 23 22
90 18
100 20
dp
dq
dp2
dq2
3 0 -2 1 -2
3 0 2 -3 -2
9 0 4 1 4
9 0 4 9 4
0
0
18
26
3.6 1.90
5.2 2.28
(c )
Consistent with market forces, share P has a lower expected return with lower risk than Share Q. Selection between P or Q depends entirely upon the risk preference of the investor. Some would prefer expecting 20% (share Q) with a lower probability of achieving it than expecting 18% with a higher chance of that being the return.
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