Problem_Set__4_Solutions
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Jan 9, 2024
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Fin 401
Problem Set #4
Financial Instruments
Solutions
1.
Do Higgins, Chapter 5, #5.
Answer provided in the back of the book.
2.
A firm has debt with a face (book) value of 80.
The future market value of the firm’s
assets is uncertain, but the distribution of possible outcomes is shown in the table
below.
a.
What is the market value of the firm’s assets, debt, and equity?
Throughout this
question, ignore any need to discount to present value.
Payoffs to debt and equity are shown below.
Assets = 0.1*50 + 0.2*100 + 0.4*200 + 0.2*300 + 0.1*350 = 200
Debt = 0.1*50 + 0.9*80 = 77
Equity = 0.2*20 + 0.4*120 + 0.2*220 + 0.1*270 = 123
(Or, Equity = Assets –
Debt)
b.
Suppose that there is bad news about the firm.
The bad news decreases the asset
value in each case by 5%, as shown in the new table.
What is the new market value
of the firm’s assets, debt, and equity?
What was the proportional change in each of
these?
What affect would this news have on the accounting balance sheet?
Prob.
0.1
0.2
0.4
0.2
0.1
Asset Value
50
100
200
300
350
Debt payoffs
50
80
80
80
80
Equity payoffs
0
20
120
220
270
Prob.
0.1
0.2
0.4
0.2
0.1
Repeat the process from part a.
Assets = 0.1*47.5 + 0.2*95 + 0.4*190 + 0.2*285 + 0.1*332.5 = 190
Debt = 0.1*47.5 + 0.9*80 = 76.75
Equity = Assets – Debt = 113.25
Percentage change:
Assets: -5%
Debt: -0.4%
Equity: -9.2%
Equity is the residual claimant, so its value is most sensitive to new information.
Nothing happens to the accounting balance sheet as a consequence of this news.
c.
Repeat part a under the assumption that the face value of the debt was initially 150
instead of 80.
Payoffs to debt and equity are shown below.
Assets = 0.1*50 + 0.2*100 + 0.4*200 + 0.2*300 + 0.1*350 = 200
Debt = 0.1*50 + 0.2*100 + 0.7*150 = 130
Equity = Assets – Debt = 70
d.
Why is the difference between the book and market values of debt higher in part c
than in part a?
In part a, the firm had a 10% chance of default; in part c, it had a 30% chance.
The
credit risk is higher because of the higher leverage.
Asset Value
47.5
95
190
285
332.5
Debt payoffs
47.5
80
80
80
80
Equity payoffs
0
15
110
205
252.5
Prob.
0.1
0.2
0.4
0.2
0.1
Asset Value
50
100
200
300
350
Debt payoffs
50
100
150
150
150
Equity payoffs
0
0
50
150
200
2
e.
Repeat part b under the assumption that the face value of the debt was initially 150
instead of 80.
Assets = 0.1*47.5 + 0.2*95 + 0.4*190 + 0.2*285 + 0.1*332.5 = 190
Debt = 0.1*47.5 + 0.2*95 + 0.7*150 = 128.75
Equity = Assets – Debt = 61.25
Percentage change:
Assets: -5%
Debt: -1.0%
Equity: -12.5%
Nothing happens to the accounting balance sheet as a consequence of this news.
f.
Why is the proportional change in the values of debt and equity in part e different
from part b?
With more leverage, both debt and equity are more sensitive to new information.
3.
a.
A $1,000 face value zero-coupon bond that has 20 years to maturity has a yield to
maturity of 6.2%.
What is the price of the bond?
Please solve without a financial
calculator.
Price = 1000 / (1.062)^20 = 300.27
b.
What is the price of a $1,000 face value bond that has 4 years to maturity and that
pays annual coupon payments at a 4.4% rate, if the current yield to maturity is 7%?
What would be the new price if yields fell to 6%?
Please solve without a financial
calculator.
Price = 44/(1.07) + 44/(1.07)^2 + 44/(1.07)^3 + 1044/(1.07)^4
= 41.12 + 38.43 + 35.92 + 796.46
= 911.93
Prob.
0.1
0.2
0.4
0.2
0.1
Asset Value
47.5
95
190
285
332.5
Debt payoffs
47.5
95
150
150
150
Equity payoffs
0
0
40
135
182.5
3
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For the new price, replace “1.07” above with “1.06.”
Price = 944.56
4.
a.
What is the price of a $1,000 face value 6% coupon bond with annual interest
payments that has 3 years to maturity if the interest rate is 5.6%?
Please solve
without a financial calculator.
Price = 60/(1.056) + 60/(1.056)^2 + 1060/(1.056)^3
= 56.82 + 53.81 + 900.15
= 1010.78
b.
Now find the present value assuming that coupon payments are made semiannually.
(This is the more typical case in the U.S.)
In order to do this, you must remember
some conventions.
In the U.S. if a bond with semiannual coupon payments is said
to have a coupon rate of 6%, this usually means that 3% ($30 in this case) will be
paid every six months.
Likewise, if the bond’s interest rate is stated as 5.6%, this
usually means that the 6-month rate is 5.6/2=2.8%.
That is, you can think of the
stated rate as the APR, not the effective annual rate.
Price = 30/(1.028) + 30/(1.028)^2 + 30/(1.028)^3 + 30/(1.028)^4 + 30/(1.082)^5 +
1030/(1.082)^6
= 29.18 + 28.39 + 27.61 + 26.86 + 26.13 + 872.73
= 1010.90
c.
Now replicate your answers from (a) and (b) using Excel.
To do so, enter the data
from (a) in a worksheet that looks like this:
Bond Valuation
Settlement Date
1/25/2017
Maturity Date
1/25/2020
Rate (Coupon)
6.00%
Yield
5.60%
Redemption
100
Frequency
1
Basis
1
Value
4
Explanation of terms
Settlement: Date when bond would be sold
Maturity: Date when bond matures
Rate:
Annual coupon rate
Yield:
Yield to maturity
Redemption:
Amount received at maturity
as a % of par
Frequency:
Number of payments per year
Basis:
Code for date convention (Just set this to 1).
Note that here the dates have been set somewhat arbitrarily just to be 3 years apart.
(Any dates 3 years apart would give the same answer.)
One of the nice things
about bond valuation in Excel is the flexibility allowed in setting these dates.
Now solve for the value using the PRICE function and the inputs as listed on the
worksheet.
(Note: if you don’t have PRICE available you need Analysis ToolPak
add-in.
Go to Tools | Add-ins.)
You should get the same answer as in (a) above,
but it will be as a % of par.
Multiply the answer by 10 to get the price for a
standard $1,000 face value bond.
Now switch the frequency to 2 and you should
get the same answer as in (b) above.
d.
Use the YIELD function in Excel, the inputs above, and the price found in (c) to
verify that the yield to maturity is 5.6% when the price is as found in (c).
e.
What is the value of a $1,000 face value bond to be sold on Feb. 22, 2016, that
matures on November 15, 2028, that has semiannual coupon payments and an
annual coupon rate of 7.25%, if the yield to maturity is 8.43%?
Plugging these values into Excel, and multiplying the answer by 10, gives $908.77.
5
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