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The interest rate over the 90 day period is 1,000 Ã· 982 = 1.833%

If we work on the basis of there being four 90 day periods in a year (justified because we

are only seeking an answer to the nearest 0.1%) then the annual interest rate is:

(1 + 0.01833)4 âˆ’ 1 = 7.5%

4. If the dividend yield on a share is currently 2% and the market consensus is that this

dividend will grow at 5Â½%, what cost of equity does the market appear to be using for this

company?

The relevant equation is cost of equity = yield + growth

So the implied cost of equity is 7Â½%

5. A company currently pays a dividend of $3.75 per share. It appears to be subject to the same

degree of risk as a typical large US company and so we decide to apply the time value of

money that was calculated in the â€˜Estimating the cost of capitalâ€™ section above. What would

the share price be in the following situations?

a. If the dividend was expected to grow at a steady rate of 4% into the future;

b. If the dividend was expected to decline by 10% pa into the future;

c. If the dividend was expected to grow by 5% a year for the next ten years but then growth

would decline to 2.5% pa; and

d. If the dividend was expected to grow by 20% for each of the next five years and then

growth was expected to fall to 5% pa.

574

575 Building block 2: Financial markets

We must start by determining the appropriate time value of money. In this instance we

are dealing with shares and so we need to apply the cost of equity and not the CoC. Our

rough analysis on page 58 suggested a cost of equity for the US market of 10.2%.

The valuation is then obtained by multiplying the dividend by one plus growth (to deter-

mine the dividend in one yearâ€™s time) and dividing this by the cost of equity less growth.

So the answer to (a) is $3.75 Ã—1.04 Ã· 0.062 = $62.90

The calculation for (b) is similar except that growth is negative. The numbers are:

$3.75 Ã— 0.90 Ã· 0.202 = $16.71

Part (c) can be computed in two different ways. I will use the more sophisticated two-

stage growth model first. We can also apply what I call the longhand approach which is

demonstrated below when I deal with part (d).

We are helped in answering this by the additivity of value. The value of a company grow-

ing for ever will be greater than the value of a company growing in this two stage manner.

We can determine the difference in value at the point when growth changes and then dis-

count this difference back to the present. This number is then subtracted from the value of

the company had it grown at the high rate for ever. The numbers work like this:

Value with 5% growth to perpetuity:

= $3.75 Ã— 1.05 Ã· 0.052

= $3.75 Ã— 20.19

= $75.72.

After five years the dividend will have grown to:

3.75 Ã— 1.055 = $4.79

The values with 5% and 2.5% growth will then be:

5% growth: $4.79 Ã— 1.05 Ã· 0.052 = $96.64

2.5% growth $4.79 Ã— 1.025 Ã· 0.077 = $63.71

So the decline in growth prospects will lower value at a point in five yearsâ€™ time by $32.93.

The present value of this is:

$32.93 Ã· 1.1025

= $32.93 Ã· 1.625

= $20.26.

So the share price will be this much lower than $75.72

i.e. $55.46.

The advantage of this approach is that it can be quite simply programmed into a spread-

sheet model with the number of years of higher growth treated as an input variable. The

alternative approach which I will show as the answer to part (d) is harder to programme

if the number of years of higher growth is treated as a variable.

In part (d) we cannot simply replicate the approach used above. This is because the

growth rate in the early years is above the cost of equity which we are using. The perpetu-

ity valuation formula does not apply for growth rates that exceed the discount rate. We are

forced to adopt what I think of as the longhand approach. We need estimates of annual

dividends which are then discounted to the present. We can switch to the perpetuity for-

mula once the sustainable growth rate is below the cost of equity.

576 Individual work assignments: Suggested answers

The numbers will be as follows. The dividend grows at 20% and the terminal value

factor is 20.19 (for calculation see part (c) above) times the year 5 dividend. The discount

factors are based on the 10.2% cost of equity and year-end cash flows.

Year 1 2 3 4 5 Terminal value

Dividend 4.50 5.40 6.48 7.78 9.33 188.37

Discount factor 0.907 0.823 0.747 0.678 0.615 0.615

Present value 4.08 4.44 4.84 5.27 5.74 115.91

Cumulative

present value 4.08 8.52 13.36 18.63 24.37 140.28

Hence we conclude that the share price in this situation should be $140.28.

This approach is nice and easy to follow but requires quite sophisticated spreadsheet

skills if one wants to allow the number of years of high growth to be treated as a variable.

6. A company takes out a loan of $100m at a fixed interest rate of 6%. It will repay the loan in

four equal annual instalments starting at the end of the second year and interest is paid at

the end of each year. (a) What will the pre-tax cash flows associated with the loan be? (b) If

the corporate tax rate is 33.3%, how will this affect the after-tax cash flows?

The loan repayments will each be $25m. Thanks to the repayments occurring at the end of

each year we can calculate the annual interest charge quite easily. The numbers will look

as follows:

Borrowerâ€™s perspective $m

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

Loan principal 100.0 0.0 âˆ’25.0 âˆ’25.0 âˆ’25.0 âˆ’25.0

Loan interest âˆ’6.0 âˆ’6.0 âˆ’4.5 âˆ’3.0 âˆ’1.5

Net loan cash flows 100.0 âˆ’6.0 âˆ’31.0 âˆ’29.5 âˆ’28.0 âˆ’26.5

The loan interest will be allowable for tax relief and so will result in a saving of one

third of the interest charge. There may well be a timing difference between the companyâ€™s

having to pay the interest and receiving back the benefit of the associated tax relief but

we will ignore this for the purpose of this calculation. So the negative outflows associated

with the loan would be reduced by $2m in each of the first two years then $1.5m in year 3,

$1.0m in year 4 and $0.5m in year 5.

7. Now consider the loan described in question 6 from the perspective of the lender.

a. If interest rates remain unchanged what would the loan be worth to the lender just before

the end of the first year (i.e. just before the first interest payment has been made)?

b. What would the loan be worth immediately after the first interest payment has been made?

c. If interest rates fell by 1% what impact would this have on the answers to questions (a)

and (b)?

We can answer question (a) in two ways. The simplest approach is to apply the principle of

additivity of value. When the interest is paid we have in place what is exactly the same as a

577 Building block 2: Financial markets

loan of $100m that will pay interest at the ruling rate in one yearâ€™s time. By definition this

must be worth $100m. The interest which we are about to receive is $6m. So if we ignore

tax, the value of the loan just before this first interest payment is made must be $100m +

$6m = $106m.

If we allow for tax (which we should always do!) the value would be $104m.

The other way to approach the valuation is simply to calculate the annual cash flows

and discount them to the present. The pre-tax answer would look like this. Note that the

loan repayments are shown as positive numbers because they represent cash inflows to

the lender:

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 0.943 0.890 0.840 0.792

Present value cash flows 6.0 29.2 26.3 23.5 21.0

Value loan (pre-tax) 106.0

After tax the numbers look like this (in this situation we use a discount rate of 4% which

is the interest rate net of tax):

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

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