Finance And Investment Questions With Solutions

Question 1

1. This is a two period certainty model problem.

Assume that William Brown has a sole income from Bobcat Ltd in which he owns 12% of the ordinary share capital.

In its financial year 2016-17 just ended, Bobcat Ltd reported net profits after tax of $600,000, and announced its net profits after tax expectation for the next financial year, 2017-18, to be 25% higher than this year’s figure. The company operates with a dividend payout ratio of 70%, which it plans to continue, and will pay the annual dividend for 2016-17 in mid-August, 2017, and the dividend for 2017-18 in mid-August, 2018.

In mid-August, 2018, Jack wishes to spend $100,000, which will include the cost of a new car..  How much can he consume in mid-August, 2017 if the capital market offers an interest rate of 9% per year?

Answer

Dividend payable in Aug. 2017 = Net profits after tax * dividend payout ratio

= $6, 00,000 * 70%

= $4, 20,000

Dividends available to Jack = $4,20,000 * 12%

= $50,400

Net profits after tax in 2018 = $6,00,000 * 1.25

= $7,50,000

Dividend payable in Aug. 2017 = 7,50,000 * 70%

   = $5,25,000

Dividends to be available to Jack in 2018 =  $5,25,000 * 12%

= $63,000

Present value of dividend = dividend payable in 2018 / (1+rate of interest)^1

= 63000 / 1.09^1

= $57,798.2

Hence total amount available with Jack = $50,400 + $57,798.2

= $1,08,198.16

Jack needs to spend $100000 in mid August, hence the amount he can consume today will be the amount present value of amount available with Jack in 2017 less $100000.

= $1,08,198.16 – $100,000

= $8,198.16

1. This Question relates to the valuation of shares.

Big Ideas Ltd has just paid a dividend of $1.20 a share. Investors require a 12% per annum return on investments such as Big Ideas. What would a share in Big Ideas Ltd be expected to sell for today (August, 2017) if the dividend is expected to increase by 20% in August, 2018, 15% in August, 2019, 10% in August, 2020 and thereafter by 5 per cent a year forever, from August, 2021 onwards?

Answer:

Year	Dividend growth rate	Expected dividend
2017		$1.2
2018	20%	$1.44
2019	15%	$1.65
2020	10%	$1.82
2021	5%	$1.91

Price of share in year 2020 = expected dividend / (rate of return – growth rate)

= 1.91 / (0.12-0.05)

= $27.32

Price of share in 2017 = D₁ / (1+r)^1  + D₂ / (1+r)^2 +  D₃ / (1+r)^3  + P₄ / (1+r)^4

=   1.44 / 1.12^1  + 1.65 / 1.12^2 +  1.82 / 1.12^3  + 27.32 / 1.12^3

= $1.28 + $1.31 + $1.29+ $19.44

= $23.32

1. This Question relates to the time value of money and deferred perpetuities.

Colin Greenway attended Bunyip High School in the 1970s. After leaving school, Colin became a successful entrepreneur and is now very wealthy. He wishes to establish a perpetual scholarship fund which will provide $10,000 a year, payable to five high performing students at Bunyip High School each year in Year 12, that is, $50,000 a year, starting in early 2020. It is now early 2017. The High School Principal believes that the required funds can be invested at 5 per cent a year in perpetuity.

1. What is the present value in early 2017 of the whole income stream, and thus the amount which Colin must contribute to establish the fund?

Answer:

PVof perpetuity in 2020 = C₁ / r

Question 2

= 50000 / 0.05

= $10,00,000

PV of perpetuity in 2017 = PV₂₀₂₀ / (1+r)^3

= 1000000 / 1.05^3

= $ 8,63,837.6

The High School Principal, while most appreciative of Colin’s great generosity, mentions that fees at Bunyip High are rising on average by 3 per cent every year because of inflation, and that in several years, $10,000 will not be enough to keep a student in year 12 for  a whole year. Colin decides that he will increase the amount to establish the fund so as to provide for increases in the scholarship amount by 3 per cent a year in perpetuity, the first increase occurring in early 2021. How much extra (above the amount calculated in i) above, will Colin need to contribute in early 2017 so as to provide for these inflation increases forever?

 [HINT: Consider a formula similar to the dividend growth model.]

Answer:

Year	Scholarship amount	PV discounted at 5%
2017
2018	$50,000	$47,619.04
2019	$50,000	$45,351.47
2020	$50,000	$43,191.87
Total PV of cash flows		$136,162.4

PV of perpetuity in 2020 = C₂ / r-g

Here, the inflation rate is 3% per year starting from 2021. Hence we consider inflation as the growth rate.

C₂ ( cash flow in 2021) = 50000 * 1.03

= $51,500

Hence PV of perpetuity in 2020 =   51500 / (0.05 – 0.03)

    = $25,75,000

 PV of perpetuity₂₀₂₀ in 2017 = (PV₂₀₂₀ / (1+r)^3 ) + PV of cash flows from 2018 to 2020

  = (2575000 / 1.05^3) + $136,162.4

= $2,360,544.21

Extra amount to be contributed by Colin in early 2017 = $2,360,544.21 – 8,63,837.6

= $1,496,707

1. This Question relates to loan repayments and loan terms.

Ron and Robin Reid wish to borrow $540,000 to buy a home. The loan from Biggles Bank requires equal monthly repayments over 20 years, and carries.an interest rate of 7.8% per annum, compounded monthly. The first repayment is due at the end of the first month.

You are required to calculate:

1. the effective annual interest rate on the above loan.

Answer:

Effective Annual interest rate = [1+ i/n]^n  – 1

i = 7.8%

n = 12

hence, EAR = [1+ 0.078/12]^12  – 1

= 8.08%

1. the amount of the monthly repayment (consisting of interest and principal repayment components) if the same amount is to be repaid every month over the 20 year period of the loan

Answer:

PMT = Pi / (1-(1+i)^-n)

P = $540,000

I = 7.8% / 12 = 0.7%

N = 20*12 = 240

PMT = (540000*0.0065) / (1-(1.0065)^-240)

= 3510 / 0.788

= $4,449.7

Hence a monthly repayment of $4,449.7 is to be repaid every month for the 20 year loan period.

• the amount of $X, if – instead of the above – Biggles Bank agrees that Ron and Robin will repay the loan by paying the bank $3,300 per month for the first 12 months, then $3,750 a month for the next 12 months, and after that $X per month for the balance of the 20 year term.

Answer:

PV of annuity = C * [(1-(1+i)^-n) / i]

PV of $3300 to be paid for first 12 months = 3300 * [(1-1.0065^-12) / 0.0065]

= $37,976.43

PV of $3750 to be paid for next 12 months = 3750 * [(1-1.0065^-12) / 0.0065]

= $43,155.04

PV of deferred annuity = 43155.04 / (1.0065)^12

 = $39,926.96

PV of payments made in 2 years = $37,976.43 + $39,926.96

= $77,903.4

Total amount to be paid by end of 20 years = 4,449.7 * 240 = $10,67,928

Hence, balance amount to be paid = $10,67,928 – $77,903.4

     = $9,90,025

Let X be the amount to be paid per month for the rest of the period

PV of annuity for 18 years = X * [(1-1.0065^-216) / 0.0065]

= 115.88 X

PV of deferred annuity = 115.88X / (1.0065^24)

Question 3

990025 = 115.88X / 1.17

X = $9,980.86

Hence, an installment of $9980.8 will be paid after two years for a period of 18 years.

1. how long (in years and months) it would take to repay the loan if, alternatively, Ron and Robin decide to repay $2,500 per month, with the first repayment again being at the end of the first month after taking the loan, and continuing until the loan was repaid.

Answer:

The Question has given the loan installment of $2500 which is not possible because the monthly interest on the loan itself is $3,510; hence we are assuming a print mistake and taking the installment as $25,000.

25000 = (540000*0.0065) / (1-(1.0065)^-n)

25000 = 3510 / (1-(1.0065)^-n)

1-(1.0065) ^-n = 0.14

n = 24

hence number of years taken to repay the loan = 2 years

1. under option iv) above, the amount of the final repayment. [NOTE: Towards  the end of the loan repayment period, after the final full monthly instalment of $2,500 is paid, a lesser amount is likely to be outstanding. That amount, plus interest to the end of the following month, is the final loan repayment amount.] 

Answer:

The full payment has been made in the last month of the second year. Hence, there is no amount outstanding in the following month.

1. This Question relates to alternative investment choice techniques

Stanley Livingstone is considering the following cash flows for two mutually exclusive projects.

     Year        Cash Flows, Investment X ($)     Cash Flows, Investment Y ($)

         0                                -40,000                                        -40,000

         1                                 12,000                                          18,000

         2                                 18,000                                          18,000

         3                                 27,000                                          18,000

You are required to answer the following Questions:

1. If the cash flows after year 0 occur evenly over each year, what is the payback period for each project, and on this basis, which project would you prefer?

Answer:

Total cash flow for investment X = $12000+$18000+$27000

= $57,000

Even annual cash flows = $19,000

Payback period for the two investments:

Year	Investment X	Cumulative cash flow	Investment Y	Cumulative cash flow
0	-$40,000		-$40,000
1	$19,000	$-21000	$18,000	$-22000
2	$19,000	$-2000	$18,000	$-4000
3	$19,000	$17,000	$18,000	$14,000
Payback period	= 2+ (2000/19000) = 2.1 years	= 2+ (4000/18000) = 2.2 years

On the basis of the payback period of the investments, it is preferable to choose investment X as it has a lower payback period.

1. Would the payback periods then be any different to your answer in i)? If so, what would the payback periods be?

Answer:

Payback period for cash flows occurring at the end of period

Year	Investment X	Cumulative cash flow	Investment Y	Cumulative cash flow
0	-$40,000		-$40,000
1	$12,000	$-28000	$18,000	$-22000
2	$18,000	$-10000	$18,000	$-4000
3	$27,000	$17,000	$18,000	$14,000
Payback period	= 2+ (10000/27000) = 2.37 years	= 2+ (4000/18000) = 2.2 years

The payback period for investment X has increased as a result of uneven cash flows, hence now it is preferable to go with investment Y.

• Sketch freehand the net present value (NPV) profiles for each investment on the same graph. Label both axes and the NPV profile for each investment.

Answer:

The NPV profile for Investment X and Y

1. Calculate the internal rate of return (IRR) for each project and indicate them on the graph. [NOTE: It is satisfactory if the approximate IRR is calculated for Investment X by trial and error, and stated as a percentage correct to the nearer whole number. The IRR for Investment Y should be calculated as a percentage exactly, correct to 1 decimal place.]

Answer:

The IRR for X is approx. 17.3% and IRR for Y is 16.6%.

1. Calculate the exact crossover point and indicate it on the above graph.

Answer:

For crossover point, we calculate the difference in the cash flow of both the investments for each period.

Year	Difference in cash flow
0	$0
1	-$6000
2	$0
3	$9,000

Based on the above data, the equation for IRR is:

0 = -6000 / (1+r)  +  0 / (1+r)^2  +  9000 / (1+r)^3

R= 22%

Hence, the crossover rate is 22%.

1. State which of the investments you would prefer, depending on the required rate of return (i.e., depending on the discount rate).

Answer:

Depending on the discount rate, we would prefer investment X as it has a higher discount rate.

1. This Question relates to the valuation of bonds.

Bradley White, a retired school teacher, has two 6 per cent per annum $100,000 Australian Government bonds that mature on 15 August, 2020 and 15 August, 2023 respectively. At the date of the last half-yearly interest payment, viz., 15 February, 2017, both bonds were selling at par.

Since then, interest yields on bonds have risen by 2% per annum, compounded half-yearly. Bradley now intends to sell the bonds and put a deposit on a suburban townhouse.

1. Calculate the price he will receive from each bond if he sells on 15 August, 2017 at the new yield, immediately after receiving the interest payments due that day.

Answer:

Price of bond = C * F * [(1-(1+r)^-t] / r   +  F / (1+r)^t

We assume the face value of the bond to be $1000 as it is the standard price of a bond.

Yield on bond expiring in 2020 when selling at par in Feb, 2017:

1000 = 3%*1000 * [(1-(1+r)^-7] / r  +  1000 / (1+r)^7

R = 3%

Hence, the semi annual  yield on bond is 3%.

Yield on bond expiring in 2023 when selling at par in Feb, 2017:

1000 = 3%*1000 * [(1-(1+r)^-13] / r  +  1000 / (1+r)^13

R = 3%

New yield on bond expiring in 2020 = 3% + 3% = 6%

Price of 1^st bond = 30 * [(1-(1+0.06)^-7] / 0.06  +  1000 / (1+0.06)^7

= $832.52

New yield on bond expiring in 2023= 3% + 6% = 9%

Price of 2^nd bond = 30 * [(1-(1+0.09)^-13] / 0.04  +  1000 / (1+0.09)^13

= $477.16

1. Explain the relative price movements in the two bonds, as evidenced in your answer to above.

Answer:

The price of bonds maturing in 2020 and 2023 have decreased with an increase in the yield.

• Suppose that Bradley defers buying the bonds for 84 days, that is until 7 November, 2017. How much will he pay for each bond on that day? [NOTE: Between the bond interest due dates from mid-August to mid-February is 184 days, during which time interest accrues on a compound basis.]

Answer:

Accrued interest on bond = [((coupon rate * face value) / payment frequency) / days in payment period ] * days

= [((6% * 1000) / 2 ) / 184] * 84

= $13.69

Price of 1^st bond as on 7^th Nov. 2017 = price of bond on last coupon date + accrued interest

= $832.52 + $13.69

= $846.21

Price of 2^nd  bond as on 7^th Nov. 2017 = price of bond on last coupon date + accrued interest

= $477.16 + $13.69

= $490.85

Answer:

Table of cash flows for purchase of new technology

Year	0	1	2	3	4
Reduction in costs		$2,00,000	$2,00,000	$2,00,000	$2,00,000
Depreciation		-$1,42,500	-$1,42,500	-$1,42,500	-$1,42,500
Overhauling expenses			-$10,000		-$10,000
Interest on loan		-$60,000	-$60,000	-$60,000	-$60,000
Loss of rent		-$40,000	-$40,000	-$40,000	-$40,000
Operating income		-$42,500	-$52,500	-$42,500	-$52,500
[email protected]%		-$12,750	-$15,750	-$12,750	-$15,750
Income after tax		-$29,750	-$36,750	-$29,750	-$36,750
Operating cash flows		$1,12,750	$1,05,750	$1,12,750	$1,05,750
Loan repayment					-$6,00,000
After tax sale of tech.					$21,000
Recovery of working capital					$30,000
Initial investment	-$74,000
Net incremental cash flows	-$74,000	$1,12,750	$1,05,750	$1,12,750	-$4,43,250
Discount rate @10%	$1	$0.91	$0.83	$0.75	$0.68
PV of cash flows	-$74,000	$1,02,500	$87,397	$84,711	-$3,02,746

NPV = ($102,138.3)

Working notes:

Initial investment

Initial investment	$6,00,000
Lease rental	$44,000
Loan	-$6,00,000
Increase in working capital	$30,000
	$74,000

Sale of machine	$30,000
Book value	$0
profit on sale	$30,000
tax on profit	$9,000
After tax sale	$21,000

Cost	$6,00,000
Salvage value	$30,000
Depreciable value	$5,70,000
Depreciation	$1,42,500

No, the company should not go ahead with the purchase of the new technology because the NPV of the investment is negative, which means the cash outflow is more than the inflow, hence it is not preferable to undertake the investment.