Advanced Corporate Finance 2019/20

Advanced Corporate Finance

Exercise Sheet 2 with Solutions
Bond Pricing, Stock pricing, Capital Budgeting, Firm Valuation

Exercise 1 (Bond Pricing, Yield vs. Expected return, Clean and dirty price)

1. A 4 year bond with a 7.5% annual coupon is trading at a yield of 5%.

a. What is the bond’s price?

We discount « promised » cash flows at the «  yield », as is usual in financial markets:

Bond’s price = (100*0,075)/(1+0,05)^1+(100*0,075)/(1+0,05)^2+(100*0,075)/(1+0,05)^3+(100*1,075)/(1+0,05)^4

= 108.86

b. Assume that for some reason the bond price briefly drops to 98 (i.e. 98% of face
value). Will the bond’s yield be lower or higher than 7.5%?
A bond that is trading below par (=below face value) has a yield that is higher than the coupon rate.

c. Use Excel’s IRR function to determine the bond’s yield at a price of 98

Buying this bond is an investment of –98 that produces cash flows of 7.5 , 7.5, 7.5 and , 107.5 in years 1 to 4. Applying the IRR
function to this stream of cash flow we find IRR= yield = 8.11%

d. 3 months later the bond has a remaining maturity of 3.75 years. The yield is again
5%. What is the bond’s “dirty price”?
For yield of 5%:

Dirty price =(100*0,075)/(1+0,05)^0,75+(100*0,075)/(1+0,05)^1,75

+(100*0,075)/(1+0,05)^2,75+(100*1,075)/(1+0,05)^3,75
= 110.20

e. What is the bond’s “accrued interest”

Accrued interest = (days since last coupon/days between two coupons)*coupon
= (90/360)*(100*0,075)
= 1.875

f. What is the clean price?

For yield of 5%:

Clean price = Dirty price - Accrued interest

= 110.20 - 1.875
= 108.33

Prof. Michael Troege, ESCP Europe

Advanced Corporate Finance 2019/20

g. Answer the questions b) c) d) if the bond has only 9 month of remaining maturity
and is still trading at the same yield. Which one of these values converge to the
bond’s face value as maturity shortens?

For yield of 5%:

Dirty price =(100*0,075)/(1+0,05)^0,75

= 103.64

For c) Accrued interest = (days since last coupon/days between two coupons)*coupon
= (90/360)*(100*0,075)
= 1.875

For d) For yield of 5%:

Clean price = Dirty price - Accrued interest

= 103.64 - 1.875
= 101.77

The dirty price converges to Face value + coupon whereas the clean price converges to the face value of the bond as the maturity
shortens.

2. A 4 year bond with a 7.5% annual coupon is trading at a yield of 6%. Analysts think that
the bond’s early coupons are risk free, but they anticipate that with a 3% probability the
last coupon and the bond’s principal will not be reimbursed (i.e. a loss given default of
100%). Risk free rates are 2%.
a. What is the bond’s price?

Note again that we don’t need the information on default rates to evaluate the bond using the yield:

Bond price = (100*0,075)/(1+0,06)^1+(100*0,075)/(1+0,06)^2+

(100*0,075)/(1+0,06)^3+(100*1,075)/(1+0,06)^4
= 105.20

b. What is the bond’s yield spread?

Spread = Promised return – risk free rate

= 6%-2%=4%

c. What is the bond’s expected return?

The best way to answer this question is to calculate the IRR of the expected cash flow profile
-105.2, 7.5,7.5,7.5,0.97*107.5. The resulting expected return is 5.28%

d. What is the bond’s expected loss rate

Expected loss rate = Promised return – expected return

=6%-5.28%=0 .72%

e. What risk premium will you earn if you are investing in this bond?

Prof. Michael Troege, ESCP Europe

Advanced Corporate Finance 2019/20

Risk premium = Expected return – risk free rate

= 5.28%- 2%=
= 3,28%

f. The bond has a beta of 0,1, risk free interest rates are 2% and the market risk
premium is 5%. According to the CAPM what should be the required return of the
bond?

Required return (CAPM) = risk free rate + (Beta of the security*market risk premium)
= 2%+0.1*5% = 2.5%

g. If you compare the required return with the expected return, do you think that this
bond is a good investment?

As the expected risk premium is 3.28%, the investment seems to exceed the required return computed via the CAPM-model.
However see the following question for why the risk premium may be underestimated by the CAPM.

h. Why may the CAPM not be a good model for determining a bond’s required
return?

The reason lies in the partial absence or presence of symmetry in returns for each of the different bond classes – in other words,
the reasons is the beta. Stocks, which have potentially unlimited upside potential as well as significant downside potential, have
much more symmetric returns than bonds.
Corporate bonds have some upside potential, but it is limited by the fact that normal bonds can at best become default-free. Thus,
the upside potential for a AA rated bond is fairly limited. Consequently, the risk measure which needs to be taken into account
has to be a downside risk measure, which is what default risk and ratings measure. Clearly, the lower the rating of a bond, the
greater the upside potential, and thus, the greater the likelihood that we can estimate bond betas and expected returns on them.
For a junk bond, for instance, it may be possible to estimate a beta like a stock beta and get an expected return from it.

Exercise 2 (Gordon growth model, P/E ratios)

3. A company currently pays a dividend of 1,5 € per share. The company’s stock beta is 1.5,
risk free interest rates are at 4% and the market risk premium can be assumed to be 4%.

a. What is the company’s cost of equity?

4%+1.5*4%=10%

b. If the company’s dividends can be assumed to grow (forever) at a constant rate of

4% per year, what should be the company’s stock price?

Gordon growth model：1.5/ (10%-4%) =25

c. The company pays a dividend corresponding to 60% of its earnings. Its ROE is
10%. What growth rate can you expect the company to produce?

Sustainable growth formula：

Growth=b*ROE
b=plowback = reinvestment rate = 1-payout rate=40%

Prof. Michael Troege, ESCP Europe

Advanced Corporate Finance 2019/20

ROE=10%
Growth=b*ROE= 4%
Strong assumptions：
Without net investment (i.e. for Capex=D&A) the company produces a constant perpetual profit
equal to Π=ROE×BookE
Every additional dollar invested produces the same perpetual ROE as the company’s existing
book assets.

The formula is very unreliable because even without new investments existing assets can deliver
higher or lower growth. Also the marginal return on new investments is not necessarily similar
to the return of the existing investments approximated by ROE . It is very likely also not constant
but declining in the quantity of cash reinvested. However the formula can still help to assess the
credibility of assumptions about long term growth rates.

d. What should be the Price/ Earnings ratio given the assumptions from the previous
question?

1.5= D=0.6*Earnings, hence E=1.5/0.6=2.5

Gordon growth model

P/E=P/EARNING=1/E*D/(Re-g)=1/E*(1-b)E/(Re-g)=(1-b)/(Re-b*ROE)

B=40% Re=10% ROE=10%

P/E=0.6/(0.1-0.4*0.1)=10

e. Please also calculate stock price and price earnings ratio, if shareholders anticipate
a decline of the company with dividends decreasing at a rate of 2% per year.

P=D/(Re-g)=1.5/(10%+2%)=12.5
P/E= 12.5/2.5=5

f. How does the answer to the previous question change if the stock beta is not 1.5
but 0.5?

Re’=4%+0.5*0.04=6%
P’=1.5/8%=18.75
P/E= 18.75/2.5=7.5

g. The company has a very high “Book to Market Ratio”. Would this be an argument
for a higher or lower value compared to the one determined in question d)
(assuming the beta is (1.5).

We have seen in class that companies with high book market ratio deliver a higher stock market
return, because investors apply a higher risk premium, likely because of additional risk that is
not taken into account in the CAPM. This means we would need to use a higher discount rate and
therefore a lower stock value.

Prof. Michael Troege, ESCP Europe

Advanced Corporate Finance 2019/20

h. The company currently pays out only 20% of its earnings per share. Shareholders
are pressing for a higher payout, but the company estimates that by increasing its
payout to 50% it would decrease the permanent growth rate from 4% to only 2% a
year. Is increasing the payout a good idea?

Earnings are 1,5/0,2=7,5

New dividend is 0,5*7,5= 3,75.

Gordon Shapiro 3, 75/ (0,10-0,02)=46, this is higher than the initial 25 hence this is a good idea.

Exercise 3 (Capital Budgeting)

A company paying corporate taxes of 35% is trying to evaluate the Free Cash Flows of an
investment project. The project manager has made first forecast but a number of adjustments
have to be made:

a. It seems that the project requires additional working capital of 100 € in a given
year. What is the decrease in free cash flow caused by this increase in working
capital?

-100(-△BFR)

b. By how much does this increase in working capital change the project’s
accounting income?
Not at all
c. It turns out that the same investment project requires an additional energy expense
of 100 € in a given year. What is the decrease in free cash flow caused by this
increase in cost in the given year?

-100 (1-0,35)=65

d. There is also an additional machine that needs to be acquired for 100 €. How does
this affect accounting income and free cash flow if we neglect depreciation?

Income: 0

FCF:-100(-△Capex) (if we neglect depreciation and the associated tax effect)

Exercise 4 (Free Cash to the Firm, Cash to Equity)

Inherently Kristal Clean, Inc. (IKC) is a provider of parts cleaning services in the United States.
We are at the end of 2019 and the preliminary 2019 financial statements look as follows:

Prof. Michael Troege, ESCP Europe

Advanced Corporate Finance 2019/20

P&L in $ millions
Sales 120
EBITDA 10
EBIT 5
Net interest 0
Pret tax profit 5
Tax@40% 2
Net Income 3

Balance Sheet in $millions

Assets Liabilities
Net working Capital 60 Short Term Debt 0
Property Plant & Equipment 85 Long Term Debt 0
Intangibles 5 Equity 150
Total Assets 150 Total Liabilities and Equity 150

a. Please determine the Free Cash Flow to the Firm for 2019 and a forecast for 2020. In
2019 the firm has invested $ 5 million. Because of increased inventory, working
capital has increased by $ 10 million over 2019. In 2020 the level of working capital
should stay constant, but otherwise the company should deliver the same cash flow
than in 2019.
For 2019:
EBIT(1-0.4)= 3=NOP(L)AT
FCTF = NOP(L)AT+D&A-△WC-△Capex
=3+5 -10-5=-7
For 2020：
CTF2019+10=3

b. Calculate the Free Cash Flow to Equity in 2019 and a forecast for 2020, using the
same assumptions as in a), assuming that in 2019 the company has reimbursed its
entire debt of $10 million (this decrease is already included in the 2020 financial
statements) but that in 2020 and the following years the company will not change its
debt and working capital level.

Free Cash Flow to Equity：money that can be paid to shareholders after all expenses,
reinvestment and debt repayment.
2019:
=Net income +D&A-△WC-△Capex + △Debt(= New Debt - Debt Repayment)
=3+5 -10-5-10=-17

2020:

Prof. Michael Troege, ESCP Europe

Advanced Corporate Finance 2019/20

The company is not levered, hence Free Cash to Equity is Free Cash to Firm =3

Exercise 5 (Valuation of Liabilities, Valuation of Assets)

A company produces regular free cash flows to the firm (FCTF) of 20 without growth. The
company’s stock has a beta of 1,5. The risk free rate is 3% and the market risk premium is 6%.
The company is financed with a constant debt level of 100 on which it pays 5% interest. The
company pays no taxes.

a. Determine cash to equity and cash to debt

Given that there are no taxes the +FCTF is divided between FCTE and Cash to debt.
Interest is 100*5%=5/year
Debt level does not change, hence cash to debt =5/year
Cash to equity = Cash to firm – cash to debt = 15/year

b. What is the value of the company’s equity

Cost of equity = 3%+ 1,5*6 =12% Present value of cash to firm using Gordon growth formula =
15/12%=125

c. What is the company’s WACC

D=100, E=125 WACC= 100/225*5%+125/225*12% =8,88%

d. Please value the company’s assets by discounting FCTF at WACC

Present value of FCTF using Gordon growth = 20/0,0888=225

Exercise 6 (Financial View of the Firm, Issuing New Equity)

The startup company “fairtrade” has just been founded with an initial equity investment of 100
000 Euros. Currently it will produce a perpetual constant free cash flow of 20 000 Euros per year
starting at the end of this year. An additional investment of 100 000 Euros could generate
supplementary perpetual cash flow of 30 000 Euros. The cost of capital is 10%. Currently the
company has 1000 shares outstanding.

a. What is the book value of assets, assuming the company has not generated any costs or
revenues?

It will correspond to the 100K that have just been invested.

b. What is the book value per share? How many new shares will the company have to sell at
book value in order to finance the additional investment of 100k with new equity?

Prof. Michael Troege, ESCP Europe

Advanced Corporate Finance 2019/20

100Euros/share hence we have to issue 1000 new shares.

c. What is the return for new shareholders in this case? What is the NPV of buying a new
share? Is it appropriate?

New shareholders will own 50% of the company, hence receive 50% of the 50k in
FCTF/dividends for an investment of 100k. that’s a return of 25%/year, which is excessive given
that 10% is what they required

d. Please calculate the “pre money” market value of the company (i.e. before the investment
is made) as present value of future cash flows. What is the market value of one existing,
(i.e. old) share?

Future cash flows given that the investment will be made = -100k, 50k, 50k, 50k…. hence present
value = 400k and value of one share = 400 Euros

e. Decompose the market value in 1) liquidation (=book) value 2) Value created by existing
assets 3) Net present value of growth opportunities

Book value =100

NPV of existing investments = 100
NPV of investment (growth) opportunities =200

f. Assume the company sells new shares at the market value to finance the new
shareholders. How many new shares does it need to create and sell?

100k/400=25k shares at 400 Euros/Share

g. What is the ownership percentage of the new shareholders now?

25k/125k=20%

h. What is the NPV and the return for a new shareholder in this case?

Return of outsiders = 20% of 50k=10K on investment of 100k= 10%. Post money valuation =
50/0.1=500, value of one share =500/125=400, NPV=-400+400=0, hence the outside
shareholder is (reasonably) happy, he obtains exactly the required return!

Prof. Michael Troege, ESCP Europe