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Net loan cash flows 4.0 29.0 28.0 27.0 26.0

Discount factor 1.000 0.962 0.925 0.889 0.855

Present value cash flows 4.0 27.9 25.9 24.0 22.2

Value loan (post-tax) 104.0

The answers are exactly the same as those that we got when we simply applied the law

of conservation of value. What, however, is the cause of the difference between the $106m

pre-tax value and the $104m post-tax value? Well, $106m pre-tax is the amount of money

that the bank would want to receive if it sold the loan while $104m is how much it would

expect after tax.

We can notice that the loan has gone up in value simply owing to the passage of time.

We have moved almost a full year into the future. In present value terms this $106m pre-

tax value is worth $100m today if we apply a discount rate of 6%. Similarly the $104m

post-tax value is worth $100m today if we apply the post-tax discount rate of 4%.

Part (b) is very simple. Immediately after the interest payment has been made the loan

would then be worth $100m. So we can see that the value of a loan will follow a slightly

zigzagging path owing to the exact timing of interest payments.

578 Individual work assignments: Suggested answers

Part (c) will require a spreadsheet model.

If interest rates should fall then the value of the loan would rise. This is because the loan

cash flows will be unchanged because we are dealing with a fixed rate loan. The cash flows

would, however, now be discounted at the lower rate of 5% pre-tax or 3.33% post-tax. The

calculations are being done as at the end of the first year and so the discount factor at this

point is 1.000.

The spreadsheet is shown below. Note that the initial loan is ignored because we are

trying to calculate the value of the loan after it has been made.

Lenderâ€™s perspective (end of first year just prior to interest payment) $m

Initial loan Year 1 Year 2 Year 3 Year 4 Year 5

Loan principal 0.0 25.0 25.0 25.0 25.0

Loan interest 6.0 6.0 4.5 3.0 1.5

Net loan cash flows 6.0 31.0 29.5 28.0 26.5

Discount factor 1.000 0.952 0.907 0.864 0.823

Present value cash flows 6.0 29.5 26.8 24.2 21.8

Value loan (pre-tax) 108.3

The value of the loan before tax is $108.3m.

Once again, immediately after the first interest payment of $6m had been made the

value of the loan would fall by $6m. So it would now be $102.3. The $2.3m increase in the

value of the loan compared with the calculations in part (c) is caused by the decline in

interest rates.

The post-tax valuation is as follows:

Lenderâ€™s perspective (end of first year just prior to interest payment) $m

Initial loan Year 1 Year 2 Year 3 Year 4 Year 5

Loan principal 0 25 25 25 25

Loan interest 4.0 4.0 3.0 2.0 1.0

Net loan cash flows 4.0 29.0 28.0 27.0 26.0

Discount factor 1.000 0.968 0.937 0.906 0.877

Present value cash flows 4.0 28.1 26.2 24.5 22.8

Value loan (post-tax) 105.6

This would actually have been the theoretically correct way to do the numbers because

tax represents a real charge on businesses and it should clearly therefore be incorporated

into our analysis.

In our post-tax calculations, the value of the loan immediately after the interest pay-

ment has been made will be $4m lower. This is $101.6m. The difference between the pre-

and post-tax values is equal to the tax rate times the gain in value.

8. The central bank of the hypothetical country called Lightvia aims to maintain real interest

rates of 2.5%.

a. If inflation is averaging 26% what will the nominal interest rate be?

579 Building block 2: Financial markets

b. The Lightvian currency is the dia and the current exchange rate is 235 dia/$. If the inter-

est rate on US dollars is 5.6% what would the future exchange rate be for dia/$ in one

yearâ€™s time?

The formula for part (a) is:

(1+ real interest rate) Ã— (1 + inflation rate) = (1 + nominal rate)

So the numbers are: 1.025 Ã— 1.26 = 1.2915

The nominal interest rate will be 29.15%. Compare this with the figure of 28.5% had we

simply, but incorrectly, added inflation to the real rate.

For part (b) we start with equal amounts of dia and dollars and then add one yearâ€™s local

nominal interest. The two amounts of local currency must then equal each other in value.

The exchange rate will be the ratio between the two. So the calculations are as follows:

In dia 235 Ã— 1.2915 = 304.91

In dollars 1 Ã— 1.056 = 1.056

The forward exchange rate will be 304.91 Ã· 1.056 = 288.74 dia per dollar.

9. A company is planning to raise further equity through a rights issue. It has 575m shares in

issue and the current share price is $10.87 per share. The company wishes to raise $750m of

new equity and the decision is taken to launch a 1 for 5 rights issue.

a. What price would the new shares have to be offered at?

b. What would be the anticipated share price after the rights issue if the market thought that

the cash raised would be invested in zero NPV projects?

c. How much cash would an investor who owned 1,000 shares expect to receive if they

decided to sell their rights in this situation?

d. What would this investor need to do if they wished to maintain the same amount of

money invested in the company?

e. What would be the answers to questions (a)â€“(d) if the market anticipated that each dollar

invested by the company would generate an NPV of $0.40?

a. We are told that the rights issue will be 1 for 5. Since there are currently 575m shares

in issue it is clear that 115m shares must be sold. If 115m shares are to raise $750m the

issue price must be $6.52 per share.

b. Market capitalisation before the rights issue is $575 Ã— 10.87m = $6.25bn. The issue raises

a further $0.75bn which the market believes will be invested in zero NPV projects so

the total market capitalisation after the rights issue must be $7bn. There will then be

690m shares in issue so the share price will be $10.14 per share.

c. An investor with 1,000 shares would have been awarded rights to purchase 200 shares

because the rights issue is 1 for 5. The right is to buy a share for $6.52 when the antici-

pated ex-rights price is $10.14. Hence each right must be worth $3.62. So the investorâ€™s

200 rights would be worth $724.

d. Once again, this calculation can be done in two ways. There is a long way and also a

shorter way if we invoke the principle of conservation of value. The long way is as fol-

lows. In order to maintain the same amount of money invested in the company the

investor would want their investment to remain at $10,870 worth of shares. Since the

share price is $10.14 after the rights issue the investor now needs to hold 1,072 shares.

So the investor must purchase 72 shares at a cost of $730. The shorter way is to apply

the principle of conservation of value. To maintain value the investor simply uses the

proceeds from selling their rights to purchase new shares. The investor has $724 and

580 Individual work assignments: Suggested answers

this would purchase 71 shares with a small remainder of $4. The difference between the

two methods is simply a rounding effect.

e. The rights issue strike price of $6.52 per share would be unaffected by the assumed

return from the money raised. What would change would be the ex-rights price and

all the other things that depend on this. So answer (a) is unchanged but (b)â€“(d) will be

changed. The calculation for these is as follows.

If each dollar invested was expected to generate $0.40 of NPV then the total additional

NPV would be $300m. The revised market capitalisation after the rights issue would be

$7.3bn. With 690m shares the share price would be $10.58 per share.

The 200 rights would be worth 200 x $4.06 = $812.

To maintain the investment we are no longer able to apply the law of conservation

of value because we are now assuming that the company will create value as a result of

the announcement of the rights issue and the associated value-creating investment. To

maintain the original value invested in the company we must hold shares worth $10,870.

Given the share price of $10.58 this means holding 1,027 shares. The 27 additional shares

required would cost $286.

So where, one might wonder, has the remaining $526 come from? The answer is that

it comes from the $300m value creation that is anticipated from the investments that

will be funded by the rights issue. This gain is our hypothetical shareholderâ€™s share of

this since this shareholder owns 1,000 of the 575m shares in issue. So value is actually

conserved provided one allows for this effect (once again subject to very minor round-

ing effects).

10. A company is planning to buy a high-temperature moulding machine that will manufac-

ture kitchen utensils. The machine will cost $2.5m and the company is seeking a bank loan

of $2.0m to help fund this purchase. The remaining $0.5m will come from a cash injection

from the companyâ€™s owner. How would a bank view the security on this loan and how might

it respond to the loan request in the following situations?

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