Financial Questions On Annuities, Loans, Capital Budgeting And Valuation

Question 1: Time value of money and deferred annuities

This is a two period certainty model problem.

Assume that Bradley Lane has a sole income from Fisher Ltd in which he owns 12% of the ordinary share capital. Currently, Bradley has no savings.

In August, 2018, Fisher Ltd reported net profits after tax of $800,000 for the last financial year, 2017-18 (1 July, 2017 to 30 June, 2018), and announced it expects net profits after tax for the current financial year, 2018-19, to be 20% higher than last financial year’s figure. The company has a dividend payout ratio of 70%, which it plans to continue, and will pay the annual dividend for 2017-18 in late-September, 2018, and the dividend for 2018-19 in late-September, 2019.

In late-September, 2019, Bradley wishes to spend $95,000, which will include the cost of a new car. How much can he consume in late-September, 2018 if the capital market offers an interest rate of 10% per year?

Solution

	PAT Save Time On Research and Writing Hire a Pro to Write You a 100% Plagiarism-Free Paper. Get My Paper	Dividend Payout Percentage	Dividend paid	Dividend Income of Bradley	Interest Income	Closing Income
2017-18	800000	70%	560000	67200	6720	73920
2018-19	960000	70%	672000	80640	15456	96096

Amount required at the end of September, 2019 = $95000

Amount that will be available = $ 96096

Amount that can be used = $ 1096

QUESTION 1 continued

This is an annual equivalent costs (AEC) problem.

Speedy Delivery Ltd, which operates a courier service, requires a new van. It has received two quotes. Van A will cost $70,000 now, has a three year life and will cost $7,000 a year to operate. Van B will cost $90,000 now, has a four year life and will cost $9,000 a year to operate. The relevant discount rate is 6 per cent per annum. Ignoring depreciation and taxes, calculate the AEC for each. Which van do you recommend that Speedy Delivery Ltd buy, and state why?

Solution

AEC= NPV/ Project Life

Van A
	0	1	2	3
Cost	-$ 70,000.00
Inflows		$ 7,000.00	$ 7,000.00	$ 7,000.00
DCF	1.000	0.943	0.890	0.840
PV of Cash Flows	-$ 70,000.00	$ 6,603.77	$ 6,229.98	$ 5,877.33
NPV	-$ 51,288.92
EAC	-$ 17,096.31
EAC=	-$ 51,288.92/3	=-$17,096.31

Van B
	0	1	2	3	4
Cost	-$ 90,000.00
Inflows		$ 9,000.00	$ 9,000.00	$ 9,000.00	$ 9,000.00
DCF	1.000	0.943	0.890	0.840	0.792
PV of Cash Flows	-$ 90,000.00	$ 8,490.57	$ 8,009.97	$ 7,556.57	$ 7,128.84
NPV	-$ 58,814.05
EAC	-$ 14,703.51

EAC=

-$ 58,814.05/4

=-$14,703.51

Van B will be selected as it has less negative NPV

QUESTION 1 continued

This question relates to the valuation of interest-bearing securities.

Because of the drought, Farmers Bank Ltd has experienced large losses on its rural loan portfolio and is unable to meet its next two annual interest payments on its recent issue of unsecured notes. The notes are of $1,000 face value each, mature in September, 2023 and bear a yearly interest coupon payment of 13%.

The Bank paid the interest due this month (September, 2018), and following a meeting of creditors, arranged to defer payment of the next two interest coupons due in September, 2019 and September, 2020 respectively. Under the arrangement with creditors, the Bank will pay the remaining interest coupons (due in September, 2021, September, 2022 and September, 2023) on their due dates, and pay the two deferred coupons (without interest) along with the normal final interest payment and face value of the notes on the maturity date. Farmers Bank Ltd’s notes are now seen as risky, and require a 19% per annum return.

Question 2: Loan repayments and loan terms

REQUIRED: Calculate the current value of each Farmers Bank unsecured note.

Solution:

Face Value	1000
Interest Rate	13%
Required Rate	19%
	Year	Interest	PVF @ 19%	PV
1	2019		0.840	$ –
2	2020		0.706	$ –
3	2021	$ 130.00	0.593	$ 77.14
4	2022	$ 130.00	0.499	$ 64.83
5	2023	$ 390.00	0.419	$ 163.43
5	2023	$ 1,000.00	0.419	$ 419.05
				$ 724.45

Current Market Price of Unsecured Note		$ 724.45

QUESTION 2

This question relates to the time value of money and deferred annuities.

Ruth Bray is age 42 today and plans to retire on her 63^rd birthday. With future inflation, Ruth estimates that she will require around $1,600,000 at age 63 to ensure that she will have a comfortable life in retirement. She is a single professional and believes that she can contribute $3,700 at the end of each month, starting in one month’s time and finishing on her 63^rd birthday.

If the fund to which she contributes earns 4.8% per annum, compounded monthly (after tax), how much will he have at age 63? Will she have achieved her targeted sum? What is the surplus or the shortfall?

At the age of Bray will have $ 1,604,515.59 (Refer Appendix: Table 1).

There is a surplus of $ 4,515.59 as the required amount of funds at the age of 63 years is for only $ 1600000.

Using the entire fund balance, Ruth then wishes to commence a monthly pension payable by the fund starting one month after her 63^rdbirthday, and ending on her 87^th birthday, after which she expects that the fund will be fully expended. If the fund continues to earn the above return of 4.8% per annum, compounded monthly, how much monthly pension will Ruth receive, if the fund balance reduces to zero as planned after the last pension payment on her 87^th birthday?

Monthly pension that Bray will receive will be = $ 26.44

Monthly Pension= Surplus/ Cumulative Factor of 288 terms i.e. 0.4%

= $4515.59/170.817

= $26.44

Workings:

Calculation of Number of terms=

No. of years = 87-63

= 24

No. of Months in each year = 24*12 = 288 Month

Calculation of monthly rate= 4.8/12

QUESTION 2 continued.

This question relates to loan repayments and loan terms.

James and Mary Hall wish to borrow $750,000 to buy a home. The loan from the Federal Bank requires equal monthly repayments over 25 years, and carries an interest rate of 4.5% per annum, compounded monthly. The first repayment is due at the end of one month after the loan proceeds are received.

You are required to calculate:

The effective annual interest rate on the above loan (show as a percentage, correct to 3 decimal places).

Loan Amount		750000
Loan Terms		25
Interest p.a.		4.50%
Compounding		Monthly

Effective Annual Rate		i=(1+(r/m))^m−1

		((1+(4.50/12))^12)-1

EAR=		4.594%

The amount of the monthly repayment (consisting of interest and principal repayment components) if the same amount is to be paid every month over the 25 year period of the loan.

Refer Table 3

EMR= EAC/12	=4.594/12	0.383%
Amount of Loan		$ 750,000.00
Cumulative PVAF @ 0.383% for 300 terms (25 years * 12 Months	300 Months	178.196
Installment= Loan Amount/ PVAF	$ 750000/4208.85	$ 4,208.85

Therefore, the amount of installment is $ 4208.85

QUESTION 2 continued.

The amount of $Y, if – instead of the above – the Federal Bank agrees that James and Mary will repay the loan by paying the bank $3,000 per month for the first 12 months, then $3,500 a month for the next 12 months, and after that $Y per month for the balance of the 25 year term.

Solution:

Refer Table 4

Opening Balance	$ 750,000.00
Installment (1-12th )	$ 3,000.00
Installment (12-24th )	$ 3,500.00
25-300th	$ Y
Cumulative Factor @ 0383% from 25^th to 300 term, taking 25^thmonth is the first term.	170.223
Therefore Y=	$4,351.01

How long (in years and months) would it take to repay the loan if, alternatively, James and Mary decide to repay $4,400 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. [HINT: The final repayment is likely to be less than $4,400, and will be paid one month after the final full installment of $4,400 is paid.)

Solution:

Time take to repay the loan in installments of $ 4400 will be 23 years and 1 month (Refer Table 5)

23.0833 years

Therefore in terms of months and years, it will take 23 years and around 1 month.The answer is arrived by preparing the table at the discounting rate of 0.383%

At 277^th month, the balance became zero.

Therefore, the total number of years= 277/12

= 23 years 1 mon

QUESTION 3

This question relates to alternative investment choice techniques

William Slater is considering the following cash flows for two mutually exclusive projects.

Year Cash Flows, Investment P ($) Cash Flows, Investment Q ($)

0 -60,000 -60,000

1 20,000 30,000

2 30,000 30,000

3 44,000 30,000

You are required to answer the following questions:

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?

Total Cash Flows	$ 94,000.00
Annual Cash Flows (20000+30000+44000)/3	$ 31,333.33

Project P
Years	Cash Flows
0	-$ 60,000.00
1	$ 31,333.33
2	$ 31,333.33
3	$ 31,333.33

Years	1.91 years

Project Q
Years	Cash Flows
0	-$ 60,000.00
1	$ 30,000.00
2	$ 30,000.00
3	$ 30,000.00

Years	2 Years

Decision:

On the basis of payback period Project P will be selected under the assumption that project cash flows will occur evenly from year 1 to year 3.

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

Project P
Years	Cash Flows	Cumulative CF
0	-$ 60,000.00	-$ 60,000.00
1	$ 20,000.00	-$ 40,000.00
2	$ 30,000.00	-$ 10,000.00
3	$ 44,000.00	$ 34,000.00

Years		2.23

Project Q
Years	Cash Flows
0	-$ 60,000.00
1	$ 30,000.00
2	$ 30,000.00
3	$ 30,000.00

Years	2.00

Decision:

On the basis of payback period Project Q will be selected under the assumption that project cash flows will occur at the end of each year from 1 to year 3.

QUESTION 3 continued.

If the required return is 8% per annum, what are:

– The net present values of each project?

Project P
Years	Cash Flows	PVF @ 8%	NPV
0	-$ 60,000.00	1.000	-$ 60,000.00
1	$ 20,000.00	0.926	$ 18,518.52
2	$ 30,000.00	0.857	$ 25,720.16
3	$ 44,000.00	0.794	$ 34,928.62

			$ 19,167.30


Project Q
Years	Cash Flows	PVF @ 8%	NPV
0	-$ 60,000.00	1.000	-$ 60,000.00
1	$ 30,000.00	0.926	$ 27,777.78
2	$ 30,000.00	0.857	$ 25,720.16
3	$ 30,000.00	0.794	$ 23,814.97

			$ 17,312.91

Decision:	On the basis of Net Present Value, Project P will be selected.

– The present value (or profitability) indexes of each project?

Profitability Index=

(NPV + Initial Investment)/ Initial Investment

Project P
Years	Cash Flows	PVF @ 8%	NPV
0	-$ 60,000.00	1.000	-$ 60,000.00
1	$ 20,000.00	0.926	$ 18,518.52
2	$ 30,000.00	0.857	$ 25,720.16
3	$ 44,000.00	0.794	$ 34,928.62

	NPV		$ 19,167.30

Profitability Index= (19167.3+60000)/60000

= 1.32

Project Q
Years	Cash Flows	PVF @ 8%	NPV
0	-$ 60,000.00	1.000	-$ 60,000.00
1	$ 30,000.00	0.926	$ 27,777.78
2	$ 30,000.00	0.857	$ 25,720.16
3	$ 30,000.00	0.794	$ 23,814.97

	NPV		$ 17,312.91

Profitability Index = (17312.91+60000)/60000

= 1.29

Decision:

On the basis of Profitability Index, Project P will be selected.

QUESTION 3 continued.

Calculate the internal rate of return (IRR)for each project.

[NOTE: It is satisfactory if the approximate IRR is calculated for Investment P by trial and error, and stated as a percentage correct to the nearer whole number. The IRR for Investment Y, where the positive cash flows form an ordinary annuity, should be calculated as a percentage exactly, correct to 1 decimal place.]

Project P
Years	Cash Flows
0	-$ 60,000.00
1	$ 20,000.00
2	$ 30,000.00
3	$ 44,000.00

IRR	22.74%


Project Q
Years	Cash Flows
0	-$ 60,000.00
1	$ 30,000.00
2	$ 30,000.00
3	$ 30,000.00

IRR	23.4%


Decision:	On the basis of IRR, Project Q will be selected.

Project P

Year	Cash Flows	PVF @ 22%	PV	PVF @ 23%	PV	PVF @ 22.74%	PV
0	-60000	1.000	-60000.00	1.000	-60000.000	1.000	-60000.00
1	20000	0.820	16393.44	0.813	16260.163	0.815	16294.61
2	30000	0.672	20155.87	0.661	19829.467	0.664	19913.57
3	44000	0.551	24231.10	0.537	23644.892	0.541	23795.47

NPV			600.962		-265.478		0.00

Question 3: Capital budgeting

IRR=	LDR+	NPV at LDR	x (UDR-LDR)
		NPV at LDR- NPV at UDR

	0.22 +	600.962	*(.23-.22)
		335.484

	0.22 +	1.79132906	* 0.01

IRR = 23.79%

Project Q
Year	Cash Flows	PVF @ 24%	PV	PVF @ 23%	PV	PVF @ 23.375%	PV
0	-60000	1.000	-60000.00	1.000	-60000.000	1.000	-60000.00
1	30000	0.806	24193.55	0.813	24390.244	0.811	24316.11
2	30000	0.650	19510.93	0.661	19829.467	0.657	19709.11
3	30000	0.524	15734.62	0.537	16121.518	0.532	15974.96

NPV			-560.908		341.228		0.175

LDR+	NPV at LDR	x (UDR-LDR)
	NPV at LDR- NPV at UDR

0.23 +	560.908	*(.24-.23)
	902.136

0.23 +	0.6217551	* 0.01

IRR = 23.62%

Calculate the exact crossover point(an interest rate, expressed as a percentage correct to two places of decimals) of the respective net present values (NPVs) for the above projects.

CROSSOVER POINT

Project P
IRR	23%	0%	18.32%
NPV	0.00	34000	4893.17
Project Q
IRR	23%	0%	18.32%
NPV	0.00	30000	4893.23

Cross over point = 18.32%

Graphical presentation of cross over point

Having regard to the above calculations, state – with reasons – which of investments P and Q you would prefer.

Solution:

The evaluation of two mutually exclusive projects i.e. Project P and Project Q, which are proposed to be undertaken by William Slater is done using various techniques of capital budgeting. Capital budgeting decisions require significant evaluation as they involve investment of large sums of funds and also they take some period of time to generate returns (Bennouna, Meredith & Marchant, 2010). The basic feature of projects that are mutually exclusive in nature is that only one of such projects can only be undertaken and not all the projects at a time due to limited availability of funds (Finance Management, 2018). In order to select the best suitable project, application of following capital budgeting techniques has been made:

Payback period: It is the length of time that will be taken by a project to recover its cost of initial outlay. When it is assumed that the cash flows in respect of all the projects will occur evenly, it is observed that project P is better than Q because P has a shorter payback period than Q. However, with the change in assumption that cash flows will occur only at the end of each year, the decision on the payback period basis changed. Now, project Q is found to be better than P due to Q’s shorter length of payback term.

Net Present Value: On the basis of NPV, Project P must be selected as it has higher NPV than that of Project Q as higher NPV shows higher profitability of the project (Management Study Guide, 2018).

Profitability Index: On the basis of Profitability Index, Project P will be selected because it has higher PI.

Internal Rate of Return: On the basis of IRR, Project Q will be selected as it has higher IRR and therefore it has the capacity to offer higher returns to the project owner (Management Study Guide, 2018).

QUESTION 4

This question relates to capital budgeting.

Bayside Ferries Ltd is considering the purchase in September, 2018 of two new small hydrofoils costing $480,000 each, which it will fully finance with a fixed interest loan of 9% per annum, with interest paid monthly and the principal repaid at the end of 4 years. The hydrofoils will be used in the company’s passenger transport business.

The two new hydrofoils will replace three existing steam ferries and will permit the company to reduce its energy and labour costs by a total of $160,000 a year, over the next 4 years. [Assume these savings are realized at the end of each year.]

The new hydrofoils may be depreciated for tax purposes by the straight-line method to zero over the next 4 years. The company thinks that it can sell the hydrofoils at the end of 4 years for $75,000 each.

The three old ferries being replaced were bought two years ago for $300,000 each, with a then life expectancy of 6 years, and are being depreciated by the straight-line method to zero over 6 years. If the company proceeds with the above purchase, the old ferries will be sold in September, 2018 for $170,000 each.

This is not the first time that the company has considered this purchase and replacement. Twelve months ago, the company engaged Sea Travel Consultants, at a fee of $25,000 paid in advance, to conduct a feasibility study on savings strategies and Sea Travel made the above recommendations. At the time, Bayside Ferries Ltd did not proceed with the recommended strategy, but is now reconsidering the proposal.

Bayside Ferries Ltd further estimates that it will have to spend tax-deductible amounts of $30,000 in 2 years’ time and $40,000 in 3 years’ time overhauling the hydrofoils.

It will also require additions to current assets of $30,000 at the start of the hydrofoils project, which will be fully recoverable at the end of the fourth year after purchase.

Bayside Ferries Ltd’s cost of capital is 11%. The tax rate is 30%. Tax is paid in the year in which earnings are received.

Calculate the net present value (NPV), that is, the net benefit or net loss in present value terms of the proposed purchase costs and the resultant incremental cash flows.

[HINT: As shown in the text-book, it is recommended that for each year you calculate the tax effect first, then identify the cash flows, then calculate the overall net present value. Finally, make your recommendation.]

Opening Balance	Interest @ 9% p.a.	Total Interest	Interest Per year
480000	3600	172800	43200
480000	3600	172800	43200
		345600	86400

Reduction in labor cost	Per Year	160000
				Total
Depreciation on New Machine	Cost	480000	480000
	Less Residual Value	75000	75000
		405000	405000
	Useful life	4	4
	Depreciation per annum	101250	101250	202500

Old Berries Cost	300000	300000	300000	900000
Useful life	6	6	6
Depreciation per year	50000	50000	50000	150000
Depreciation for 2 years	100000	100000	100000	300000
WDV at the end of 2nd year	200000	200000	200000	600000

Initial Investment	$ 960,000.00
Working Capital	$ 30,000.00
Total Initial Outlay	$ 990,000.00

Years	0	1	2	3	4
Total Initial Outlay	-$ 990,000.00
Sale of Old Berries	$ 483,000.00
Decrease In Labor Cost		$ 160,000.00	$ 160,000.00	$ 160,000.00	$ 160,000.00
Incremental Depreciation		-$ 52,500.00	-$ 52,500.00	-$ 52,500.00	-$ 52,500.00
Interest on loan		-$ 86,400.00	-$ 86,400.00	-$ 86,400.00	-$ 86,400.00
Overhauling Expenditure		$ –	-$ 30,000.00	-$ 40,000.00	$ –
Net Cash Flows		$ 21,100.00	-$ 8,900.00	-$ 18,900.00	$ 21,100.00
Add Salvage Value					$ 150,000.00
Less Tax		-$ 6,330.00	$ 2,670.00	$ 5,670.00	-$ 51,330.00
Cash Flows After Tax		$ 14,770.00	-$ 6,230.00	-$ 13,230.00	$ 119,770.00
Add Incremental Depreciation		$ 52,500.00	$ 52,500.00	$ 52,500.00	$ 52,500.00
CFAT		$ 67,270.00	$ 46,270.00	$ 39,270.00	$ 172,270.00
Less Working Capital Recovered					-$ 30,000.00
Total Cash Flows	-$ 507,000.00	$ 67,270.00	$ 46,270.00	$ 39,270.00	$ 142,270.00
PVF @ 11%	1.000	0.901	0.812	0.731	0.659
PV	-$ 507,000.00	$ 60,603.60	$ 37,553.77	$ 28,713.89	$ 93,717.66
NPV	-$ 286,411.09

References:

Bennouna, K., Meredith, G. G., & Marchant, T. (2010). Improved capital budgeting decision making: evidence from Canada. Management decision, 48(2), 225-247.

Finance Management. (2018). Why Net Present Value is the Best Measure for Investment Appraisal? Retrieved from: https://efinancemanagement.com/investment-decisions/why-net-present-value-is-the-best-measure-for-investment-appraisal

Management Study Guide. (2018). Problems With Using Internal Rate of Return (IRR) for Investment Decision Making. Retrieved from: https://www.managementstudyguide.com/problems-with-using-internal-rate-of-return.htm

Management Study Guide. (2018). What is Internal Rate of Return (IRR) ? Retrieved from: https://www.managementstudyguide.com/internal-rate-of-return.htm

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