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Building block 1: Economic value

1. If your CoC is 10%, what is the present value of $100 in one yearâ€™s time?

100 Ã· 1.1 = 90.9

2. By how much would this change if your CoC was (a) 5% and (b) 15%?

100 Ã· 1.05 = 95.2 â€“ an increase in value of $4.3m

100 Ã· 1.15 = 87.0 â€“ a decrease in value of $3.9m

Note that although we changed the CoC by Â±5% the change in value was not

symmetrical.

3. What is the present value of $100 in ten yearsâ€™ time if your CoC was (a) 5%, (b) 10% or (c)

15%?

Discount factors for ten yearsâ€™ time are:

5% CoC Discount factor = 0.614 so present value is 61.4

10% CoC Discount factor = 0.386 so present value is 38.6

15% CoC Discount factor = 0.247 so present value is 24.7

4. An investment offers the potential to earn $10 per annum for the next five years with the first

cash flow being in one yearâ€™s time. If your CoC is 12% what is the maximum amount you

should be prepared to pay for the investment?

Discount factors for years 1â€“5 are: 0.893 0.797 0.712 0.636 0.567

So present value of cash flows is $10m times the sum of these = 36.05

This is the maximum you should be prepared to pay unless there are any other factors to

consider that are not mentioned in the question.

5. The discount factor and annuity factor tables given in Tables 1 and 2 only cover a limited

range. Extend them to cover a wider range that can still fit on a single sheet of paper. For

example, discount rates going from 1% to 15% and a longer time horizon covering say 1â€“20

years plus also columns for 25, 30 and 40 years. Do this for both mid-year and end-year

cash flows. If you are able to, print these tables out on both sides of a sheet of paper and get

it laminated as a reference sheet.

The year-end discount factors can be checked against the tables shown in the building

block. The mid-year factors are a little more complicated. The discount factor for year

1 is calculated via the MS Excel formula 1/(1 + r)^0.5 where r is the cell reference to the

appropriate discount rate. For subsequent years the discount factor is the previous yearâ€™s

factor divided by (1 + r). As a check, if your 12 years 12% mid-year factor is 0.272 then your

table is probably correct. To check the annuity factors, make sure your mid-year 15 year

15% factor is 6.271.

These discount factor and annuity factor tables are â€˜old fashionedâ€™ but I find them very

useful and have copies to hand whenever I am at work. They save me a lot of time.

571

572 Individual work assignments: Suggested answers

6. An investment of $100 will yield you a cash flow of $15 for every year into the future with the

initial cash flow being in one yearâ€™s time. (a) If your CoC is 12%, how much value does the

investment make? (b) How many years will it take before your investment has earned back

at least its CoC?

The value of a sum to perpetuity is the number divided by the CoC.

So 15 Ã· 0.12 = 125. Deduct the investment of 100 to reach a net value creation of $25.

The investment has earned the CoC when the present value of cash inflows is $100. This

is 6.667 times the annual cash flow. Your year-end annuity factor tables should show that

payback is somewhere between years 14 and 15. Since the payments are made just once

each year the actual payback point will be when the payment is made at the end of the fif-

teenth year. Alternatively, one could simply start with the 3.605 figure from Q4 and keep

adding the discount factor until you reach 6.667.

7. Your engineering team can only cope with one further investment next year and you have

to choose from three possible projects. Project A involves a capital spend during the year

of $20m and will then generate cash flows of $8m per annum for the following four years.

Project B involves a capital spend of $7.5m and will then generate cash flows of $2m per

annum for each of the next ten years. Project C involves a capital spend of $36m and will

then generate cash flows of $11m for each of the next five years. Your CoC is 12% and it is

now the middle of the present year (so end-year discount tables can be applied). (a) Which

project would you recommend; and (b) Are there any circumstances which would make you

change this recommendation?

Project A â€“ Present value cash inflows is 8 Ã— (.797 + .712 + .636 + .567) = $21.70m

Present value cash outflow is â€“20 Ã— .893 = â€“$17.86m

So NPV is $3.84m

Project B â€“ Present value of cash inflows can either be calculated simply by adding up

the discount factors for 12% for years 2 through to 11 or through taking the 11 year annu-

ity factor of 5.938 and subtracting the one year discount factor of 0.893. The answer is

5.045. Hence the present value of cash inflows is 2 Ã— 5.045 = $10.09m. Present value cash

outflow is â€“7.5 Ã— .893 = â€“$6.70m.

So NPV is $3.39m

Project C â€“ Present value of cash inflows is 11 Ã— (4.111 â€“ 0.893) = $35.40m

Present value cash outflow is â€“36 Ã— 0.893 = â€“$32.15m

So NPV is $3.25m

So I would recommend project A as it has the highest NPV of the three options. However,

since project A has a capital cost that is $12.5m higher than project B the extra NPV

earned per unit of capital is not great. If the company was capital constrained as opposed

to engineering resources constrained I would almost certainly recommend project B.

8. You own a retail outlet that you expect to generate cash flows of $1m pa for each of the next

four years but then you expect sales to fall dramatically when a relief road for the town is

opened. Your current expectation is that you will close the site when this happens and that

your sales proceeds net of remediation costs will be $0.5m. Your CoC is 12%. It is the begin-

ning of the year and the retail site therefore has four years of economic life left. You have just

received an offer of $3.5m for the site. Should you sell? How would your answer change if

your CoC was (a) 5% or (b) 15%?

573 Building block 1: Economic value

Cash flows if we retain ownership of the outlet are projected as $1m pa for four years.

We need to use our mid-year discount factor tables. From these we find the annuity factor

for four years and a discount rate of 12% is 3.214. Hence the present value of cash inflows

is $3.21m.

Present value of residual value of $0.5m is 0.5 Ã— 0.636 = $0.32m

Note that an end-year factor has been used on the assumption that the residual value is

realised immediately on closure i.e. in four yearsâ€™ time. This may be slightly optimistic as

it might take some time to realise the residual value. So if we do assume immediate receipt

of the residual value the overall value of continued operation is $3.53m. Hence the sales

offer is a marginal call and, based on the assumptions, we would be substantially indiffer-

ent between selling or retaining ownership.

If the CoC were reduced to 5% the retain value would rise to 3.63 + 0.41 = $4.04m.

Clearly the offer would then be too low. With a 15% CoC you would be pleased to sell.

9. Build a spreadsheet model to investigate the following project. It is 1 January and your time

value of money is 8%. The capital cost of the project is $20m and this will be spent during the

current year. There will then be ten years of operation. In the first of these years the cash flow

will be $2m, in the second it will be $3m. During years 3 to 9 of operation the annual cash flow

will be $4m but in the final year this will fall to $1m. For the initial evaluation assume no clo-

sure costs. What is the NPV, IRR, discounted payback and investment efficiency? Investigate

also what capital cost increase would cause the project to have a zero NPV. Finally, identify

what closure cost incurred at the end of the project life would cause a zero NPV.

This exercise requires a very simple spreadsheet model as set out below:

Year 1 2 3 4 5 6 7 8 9 10 11

Cash inflow 0 2 3 4 4 4 4 4 4 4 1

Cash outflow âˆ’20

Net cash flow âˆ’20 2 3 4 4 4 4 4 4 4 1

Discount factor 0.962 0.891 0.825 0.764 0.707 0.655 0.606 0.561 0.520 0.481 0.446

Present value âˆ’19.25 1.78 2.47 3.06 2.83 2.62 2.43 2.25 2.08 1.93 0.45

cash flows

Cumulative PV âˆ’19.25 âˆ’17.46 âˆ’14.99 âˆ’11.93 âˆ’9.10 âˆ’6.48 âˆ’4.06 âˆ’1.81 0.27 2.19 2.64

The NPV is $2.64m, discounted payback is during year 9 and the investment efficiency

is 13.2%. The IRR can be found either by trial and error or via the IRR function (the cash

flows are all one year apart and so this function will work correctly). The IRR is 10.9%.

The maximum tolerable capital cost increase is the NPV divided by the year 1 discount

factor. This is $2.74m. (Note that we have not taken any account of tax as this was not

mentioned in the assumptions.)

The maximum closure cost is the NPV divided by the year 11 end year discount factor

of 0.429. The answer is therefore $6.15m.

APPendix

II

Building block 2: Financial

markets

1. What is the best way to invest if your primary concern is to minimise risk?

One should invest in debt rather than equity if the aim is to minimise risk. Furthermore,

one should aim only to make low- or even no-risk loans for example by investing in short

term government bills. Any long-term loans should be at floating interest rates unless you

were completely sure when you needed the loan to be repaid.

2. Which source of finance should a new company use if it was intending to go into the oil

exploration business?

The appropriate source of finance here would be equity. Oil exploration involves high risk

and the potential loss of all money spent if the so-called wildcat well is dry. Hence it would

not be appropriate to take on any debt.

3. If a 90 day US T bill was sold for $982 what (to the nearest 0.1%) would the US risk-free

rate be?

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