1. Calculating Duration on a $1,000 10-Year 10% Coupon Bond when Its Interest Rate Is 9%.

If a pension fund manager is ho

bond in the fund’s portfolio, and the interest rate is currently 9%, what loss would the fund be exposed to if the interest rat

FV= $1,000.00 Year Cashflow DCF Weight(Wt) Wt*YR

Coupon Rate 10.0% 1 $100.00 $92 0.0862 0.086
Coupon Pmt $ 100.00 2 $100.00 $84 0.0791 0.158
Interest Rate (i1)= 9.0% 3 $100.00 $77 0.0726 0.218
N 10 Years 4 $100.00 $71 0.0666 0.266
PV= $1,064 5 $100.00 $65 0.0611 0.305
6 $100.00 $60 0.0560 0.336
7 $100.00 $55 0.0514 0.360
8 $100.00 $50 0.0472 0.377
9 $100.00 $46 0.0433 0.389
10 $100.00 $42 0.0397 0.397
10 $1,000.00 $422 0.3969 3.969
$1,064 6.863 Years
Duration
FV= $1,000.00
Coupon Rate 10%
Coupon Pmt $ 100.00
Interest Rate (i2)= 10%
N 10 Years
PV= $1,000 Check
 sion fund manager is holding a 10-year 10% coupon
osed to if the interest rate rises to 10% tomorrow?

i1= 9.0%
i2= 10.0%
i2-i1= 1.0%
% change in price= -6.30%
P1= $1,064
P2= $997
1. A 3-year, $1,000 par, zero-coupon corporate bond has a hazard rate of 3% per year. Its recovery rate is 70% and the benchm
5%. Calculate the expected exposure, probability of default, loss given default, CVA, and the credit spread on th

Rate= 5% Year Exposure LGD PD Expt.Loss

Hazard rate= 3% 1 $907 $272 3.0% $8.2
Recovery rate= 70% 2 $952 $286 2.9% $8.3
1-RR= 30% 3 $1,000 $300 2.8% $8.5
Par value= $1,000
N= 3 Years
PV of risk-free bond= $864
Pv of credit risky bond= $841

YTM of risk-free bond= 4.87% 5.00%

YTM of risky bond= 5.75% 5.93%
Credit spread= 0.88% 0.93%
 te is 70% and the benchmark rate curve is flat at
nd the credit spread on the bond.

PV of Expt.Loss
$7.77
$7.54
$7.32
$22.63
 Given the following zero-coupon yields, compare the yield to maturity for a three-year zero-coupon bond, a three-year coupo
coupon bond with 9% annual coupons. All of these bonds are default free

Maturity Zero-Coupon
(years) YTM Year CashFlow DCF Year CashFlow DCF
1 4% 1 $ - $ - 1 $ 50 $ 48
2 4.50% 2 $ - $ - 2 $ 50 $ 46
3 5% 3 $ 1,000 $ 864 3 $ 1,050 $ 907
4 5.50% $ 864 $ 1,001
5 6%
FV $ 1,000 FV $ 1,000
Cpn Pmt $ - Cpn Pmt $ 50.00
PV(Price) $ (864) PV(Price) $ (1,001)
N 3 N 3
Rate(YTM) 5.00% Rate(YTM) 4.97%

Approx.YTM 4.87% Approx.YTM 4.97%

n bond, a three-year coupon bond with 5% annual coupons, and a three-year
ese bonds are default free.

Year CashFlow DCF

1 $ 90 $ 87
2 $ 90 $ 82
3 $ 1,090 $ 942
$ 1,111

FV $ 1,000
Cpn Pmt $ 90.00
PV(Price) $ (1,111)
N 3
Rate(YTM) 4.95%

Approx.YTM 5.04%
 You have just turned 22 years old, received your bachelor’s degree, and accepted your first job. Now you must decide how m
The plan works as follows: Every dollar in the plan earns 6.7% per year. You cannot make withdrawals until you retire on yo
withdrawals as you see fit. You decide that you will plan to live to 100 and work until you turn 65. You estimate that to liv
$115,000 per year, starting at the end of the first year of retirement and ending on your 100th birthday. You will contribute
every year that you work. How much do you need to contribute each year to fund your re
PVA= ?
Pmt= $ 115,000 PVA= $1,539,053
i= 6.70%
n=(100-65) 35 Years

FVA= $1,539,053 Pmt= $6,758

Pmt= ?
i= 6.70%
n=(65-22) 43 Years
 Now you must decide how much money to put into your retirement plan.
hdrawals until you retire on your 65th birthday. After that, you can make
n 65. You estimate that to live comfortably in retirement, you will need
h birthday. You will contribute the same amount to the plan at the end of
ute each year to fund your retirement?
 1. You are thinking of making an investment in a new factory. The factory will generate revenues of $1.71 million per year for a
costs will start at $97,470 per year and will increase 5% per year thereafter. Assume that all revenue and maintenance costs o
long as it continues to make a positive cash flow (as long as the cash generated by the plant exceeds the maintenance c
immediately and the interest rate is 6% per year. What is the present value of the revenues? What is the present value of the
should you invest in the factory?

Rate of Increase of Cost 5%

Int Rate 6%
Year Revenue Cost Net Cashflow PV of Revenue PV of Cost PV of Revenue
1 $1.71 $ 0.975 $0.74 $1.61 $ 0.92 PV of Cost
2 $1.71 $ 1.02 $0.69 $1.52 $ 0.91 Initial Investment
3 $1.71 $ 1.07 $0.64 $1.44 $ 0.90 Net PV
4 $1.71 $ 1.13 $0.58 $1.35 $ 0.89
5 $1.71 $ 1.18 $0.53 $1.28 $ 0.89 As the NPV is Negetive the invest
6 $1.71 $ 1.24 $0.47 $1.21 $ 0.88
7 $1.71 $ 1.31 $0.40 $1.14 $ 0.87
8 $1.71 $ 1.37 $0.34 $1.07 $ 0.86
9 $1.71 $ 1.44 $0.27 $1.01 $ 0.85
10 $1.71 $ 1.51 $0.20 $0.95 $ 0.84
11 $1.71 $ 1.59 $0.12 $0.90 $ 0.84
12 $1.71 $ 1.67 $0.04 $13.49 $ 9.65
13 $1.71 $ 1.75 -$0.04 Sum

Based on the criteria , we will run the company for 12 years after which the cashflow will be negetive
 $1.71 million per year for as long as you maintain it. You expect that the maintenance
e and maintenance costs occur at the end of the year. You intend to run the factory as
xceeds the maintenance costs). The factory can be built and become operational
is the present value of the maintenance costs? If the plant costs $17.1 million to build,
e factory?

$13.49
$ 9.65
$ 17.10
-$13.26

NPV is Negetive the investment should not be made.

You are thinking of purchasing a house. The house costs $350,000. You have $50,000 in cash that you can use as a down pa
rest of the purchase price. The bank is offering a 30-year mortgage that requires annual payments and has an interest rate of 7
you sign this mortgage?
You would like to buy the house and take the mortgage described above. You can afford to pay only $23,500 per year. The ba
year, yet still borrow $300,000. At the end of the mortgage (in 30 years), you must make a balloon payment; that is, you must
How much will this balloon payment be?

N= 30 Years Pmt(Required) $ 24,176

i= 7%
PVA= $ 300,000
Pmt(Required) ?
Pmt( Can Afford) $ 23,500
Ballon Payment = ? Ballon Payment = $ 63,848
t you can use as a down payment on the house, but you need to borrow the
and has an interest rate of 7% per year. What will your annual payment be if
?
ly $23,500 per year. The bank agrees to allow you to pay this amount each
payment; that is, you must repay the remaining balance on the mortgage.
ment be?
 1. You are saving for retirement. To live comfortably, you decide you will need to save $2 million by the time you are 65. Tod
decide, starting today and continuing on every birthday up to and including your 65th birthday, that you will put the same amo
interest rate is 9%, how much must you set aside each year to ensure that you will have $2 million in the account

FVA(Due) (2*1.09) = $ 2.18

Pmt= ?
i= 9%
N=65-22+1 44 Years

Pmt= $ 0.0041
on by the time you are 65. Today is your 22nd birthday, and you
that you will put the same amount into a savings account. If the
have $2 million in the account on your 65th birthday?
 1. You work for a pharmaceutical company that has developed a new drug. The patent on the drug will last 17 years. You ex
profits will be $2 million in its first year and that this amount will grow at a rate of 6% per year for the next 17 years. Once the p
pharmaceutical companies will be able to produce the same drug and competition will likely drive profits to zero. What is the p
new drug if the interest rate is 9% per year?

Growth Rate= 6%
Int. Rate= 9%

Year Cashflow DCF

1 2 $ 1.83
2 2.12 $ 1.78
3 2.25 $ 1.74
4 2.38 $ 1.69
5 2.52 $ 1.64
6 2.68 $ 1.60
7 2.84 $ 1.55
8 3.01 $ 1.51
9 3.19 $ 1.47
10 3.38 $ 1.43
11 3.58 $ 1.39
12 3.80 $ 1.35
13 4.02 $ 1.31
14 4.27 $ 1.28
15 4.52 $ 1.24
16 4.79 $ 1.21
17 5.08 $ 1.17
$ 25.18
he drug will last 17 years. You expect that the drug’s
r for the next 17 years. Once the patent expires, other
drive profits to zero. What is the present value of the
ar?
You are trying to decide how much to save for retirement. Assume you plan to save $5000 per year with the first investment
earn 10% per year on your investments and you plan to retire in 43 years, immediately after making your
A.How much will you have in your retirement account on the day you retire?
B.If, instead of investing $5000 per year, you wanted to make one lump-sum investment today for your retirement that will
much would that lump sum need to be?
C.If you hope to live for 20 years in retirement, how much can you withdraw every year in retirement (starting one year after r
savings with the 20th withdrawal (assume your savings will continue to earn 10% in retire
D.If, instead, you decide to withdraw $300,000 per year in retirement (again with the first withdrawal one year after retiri
exhaust your savings?
E.Assuming the most you can afford to save is $1000 per year, but you want to retire with $1 million in your investment accou
on your investments?

Pmt= $ 5,000
N 43 Years
10%
FVA= $ 2,962,003 Answer A.

PV= $49,170 Answer B.

N= 20 Years
PVA= $ 2,962,003
Pmt= ? Pmt= $ 347,916 Answer C.

Pmt= $ 300,000
N= ? N= 45.8 Years Answer D.
PVA= $ 2,962,003

Pmt= 1000
N= 43 Years i= 12% Answer E
i= ?
FVA= 1000000
 r with the first investment made one year from now. You think you can
mediately after making your last $5000 investment.
nt on the day you retire?
or your retirement that will result in the same retirement saving, how
to be?
ent (starting one year after retirement) so that you will just exhaust your
ntinue to earn 10% in retirement)?
drawal one year after retiring), how many years will it take until you

n in your investment account, how high of a return do you need to earn

 When Alex Rodriguez moved to the Texas Rangers in 2001, he received a lot of attention for his “$252 million” contract (the
moved to the Yankees, but assume the following in order to determine the value of his con
Rodriguez earns $16 million in the first year, $17 million in years 2 through 4, $19 million in years 5 and 6, $23 million in year 7
$10 million signing bonus spread equally over the first 5 years ($2 million per year). His deferred payments begin in 2011. T
million, then $4 million, then eight amounts of $3 million (ending in 2020). However, the actual payouts will be different. All of
paid. For example, the $5 million is deferred from 2001 to 2011, or 10 years, meaning that it will actually be $6.7196 million whe
is deferred from 2002 (each payment is deferred 10 years). The contract is a 10-year contract, but each year has a deferred c
years. The contractual payments, signing bonus, and deferred components are given below. Note that, by contract, the deferr
instead are paid (plus interest) 10 years later.
Assume that an appropriate discount rate for A-Rod to apply to the contract payments
Compare the present value of the contract to the quoted value of $252 million. What explains the diffe

Deferred
Rate 7% 3%
Rate
year CF1 CF2 CF3(Deferred) CF3(Actual) Total CF DCF
1 $ 16 $ 2 $ 5 $ 18.00 $ 16.82
2 $ 17 $ 2 $ 4 $ 19.00 $ 16.60
3 $ 17 $ 2 $ 3 $ 19.00 $ 15.51
4 $ 17 $ 2 $ 3 $ 19.00 $ 14.50
5 $ 19 $ 2 $ 3 $ 21.00 $ 14.97
6 $ 19 $ 3 $ 19.00 $ 12.66
7 $ 23 $ 3 $ 23.00 $ 14.32
8 $ 27 $ 3 $ 27.00 $ 15.71
9 $ 27 $ 3 $ 27.00 $ 14.69
10 $ 27 $ 3 $ 27.00 $ 13.73
11 $ 6.72 $ 6.72 $ 3.19
12 $ 5.38 $ 5.38 $ 2.39
13 $ 4.03 $ 4.03 $ 1.67
14 $ 4.03 $ 4.03 $ 1.56
15 $ 4.03 $ 4.03 $ 1.46
16 $ 4.03 $ 4.03 $ 1.37
17 $ 4.03 $ 4.03 $ 1.28
18 $ 4.03 $ 4.03 $ 1.19
19 $ 4.03 $ 4.03 $ 1.11
20 $ 4.03 $ 4.03 $ 1.04
$ 165.77
 r his “$252 million” contract (the total of the payments promised was $252 million). He later
to determine the value of his contract when he signed it:
ars 5 and 6, $23 million in year 7, and $27 million in years 8 through 10. He also receives his
erred payments begin in 2011. The deferred payment amounts total $33 million and are $5
al payouts will be different. All of the deferred payments will earn 3% per year until they are
ll actually be $6.7196 million when paid. Assume that the $4 million payment deferred to 2012
t, but each year has a deferred component so that cash flows are paid out over a total of 20
Note that, by contract, the deferred components are not paid in the year they are earned, but

apply to the contract payments is 7% per year. Calculate the present value of the contract.
2 million. What explains the difference?