Problem Set 1a: Present Valuing Bonds, Loans, and other Cash-Flows
General Instructions:
Write out a time line showing the Cash-flows (CF’s) and the algebraic equations for each problem. Solve
the problems algebraically, before using your calculator. Do not use the TVM or Cash-flow functions
(except to check your solutions).
1) Suppose that you are going to receive 300$ at the end of each year for the next 3 years, and $200
for the subsequent five years. What is the PV of these CF’s if the current discount rate is 9%?
What amount would you be willing to pay in year 3 for these same CF’s? In year 8? In year 10?
0
1
300
2
300
3
300
3
300
1
PV  

t
1.09 3
t 1 1.09
4
200
5
200
5
200
 1,360.09

t
t 1 1.09
6
200
7
200
8
200
9
10
N=3
N=5
I/YR = 9
I/YR = 9
PMT = 300 + PMT = 200
FV = 0
FV = 0
 PV
 PV
× 1.09-3
Alternatively this can be computed as:
3
8
100
200

 1,360.09
t
t
t 1 1.09
t 1 1.09
PV  
Amount in Year 3:
1,360.09  1.09 3  1,761.36
Amount in Year 8:
1,360.09  1.098  2,710.07
Amount in Year 10:
1,360.09  1.0910  3,219.83
N=3
N=8
I/YR = 9
I/YR = 9
PMT = 100 + PMT = 200
FV = 0
FV = 0
 PV
 PV
2) Suppose that you are going to receive two payments of $500, the first at the end of year 3, and the
other at the end of year 6. How much are you willing to pay for those CF’s today if the current
discount rate is 5%? How about 6%? How much would you be willing to pay in at the end of this
year (year 1) if the discount rate is 5%? At the end of year 2? At the end of year 5? Year 10?
PV (r ) 
500
1  r 
3

500
1  r 6
; PV(5%) = 805.03 PV(6%) = 772.29
To compute values at the end of 1, 2, and 5 years, merely multiply by 1.05 t
FV(5%, year 1) = 845.28 FV(5%, year 2) = 887.55 FV(5%, year 5) = 1,027.44
3) Suppose that you take out a loan of $10,000 at 14%, and must repay the loan in a single lump sum
at the end of 4 years. How much must you repay? What is the principle and what is the interest?
FV  $10,0001.14  16,889.60
4
Principle: $10,000
Interest: $6,889.60
N=4
I/YR = 14
PMT = 0
PV = 10,000
 FV
4) Suppose that you purchase a $10,000 zero coupon bond that matures 3 years from now and is
priced to yield 11%, how much will you pay? How about if it is priced to yield 12%, will you pay
more or less (think first—write out the equation)? How much will you pay? If you paid $8500 for
the bond, what will be the yield if you hold it to maturity?
PV (r ) 
10,000
1  r 
3
N=3
I/YR = 11, 12
PMT = 0
FV = 10,000
 PV
;
PV(11%) = 7,311.91
PV(12%) = 7,117.80
8,500 
5)
N=3
PV = 8,500
PMT = 0
FV = 10,000
 I/YR
10,000
1  YTM 3
Suppose that you purchase a $5000 coupon bond with an annual coupon of 8%, and a maturity of
5 years. If the bond is sold at Par value (that is to yield the coupon rate) what does it cost? How
about if the bond is priced to yield 9%, will it cost more or less than par value? How much will it
cost? How about if it is priced to yield 7%?
N=5
I/YR = 7, 9
PMT = 400
FV = 5,000
 PV
At Par, the price of the bond is face value, or $5,000.
0
1
400
2
400
5
PV (r )  
t 1
400
1  r 
t
3
400

4
400
5000
1  r 5
;
5
5400
PV(9%) = 4,805.52
PV(7%) = 5,205.01
6) Suppose that you take out a $20,000 fixed payment loan at 10% that must be paid off in 4 years.
Assume that you must make annual payments. How much will you pay each year? How much
principle and interest is paid each year? (This is called an amortization table.) Now assume that
this is a car loan, and you must make monthly installments at the end of each month for 48
months. How much do you pay each month? How about if it was at the beginning of a month?
4
$20,000  
t 1
Principle
Interest
Payment
Principle
Paid
PMT
1.10t
1
$20,000
2,000
($6,309.42)
2
$15,690.58
1,569.06
($6,309.42)
3
$10,950.22
1,095.02
($6,309.42)
4
$5,735.83
573.58
($6,309.42)
$4,309.42
$4,740.36
$5,214.39
$5,735.83
N=4
I/YR = 10
PV = 20,000
FV = 0
 PMT
If this is a 48 month car loan, and the interest rate is an APR rate, we must first determine the
monthly rate: 10%/12 = .8333% per month, then compute the PV for 48 months. Alternatively,
setting payments per year (P/Yr) to 12, you can enter an interest rate of 10%.
48
$20,000  
t 1
PMT
1.0833t
 507.25
P/Yr = 1
or:
N = 48
I/YR = .8333
PV = 20,000
FV = 0
 PMT
P/Yr = 12
N = 48
I/YR = 10
PV = 20,000
FV = 0
 PMT
To determine payments for the beginning of the month, set the BEG mode:
47
PMT
$20,000  
 503.06
t
t 0 1.0833
Problem Set 1b: Amortization Schedules, Yields, Sensitivity, and EAR
General Instructions:
Write out a time line showing the Cash-flows (CF’s) and the algebraic equations for each
problem. Solve the problems algebraically, before using your calculator. Do not use the TVM or Cashflow functions, unless otherwise directed.
7) Suppose that the current price of a zero $1000 coupon bond is $700, and the maturity is 8 years.
Compute the yield to maturity—show the algebra. It is now 2 years later and the price is $725.
Has the yield increased or decreased? Have you incurred a capital gain or loss? What must have
happened to interest rates? If interest rates haven’t changed, what has happened to the credit of
the firm issuing the bond? Finally, check your work using the TVM functions on your calculator.
$700  $1000
$725  $1000
1  r 8
1  r 6

700


725
 r  1000
 r  1000
  1  4.55932%
1
8
1
6
N=8
PV = -700
PMT = 0
FV = 1000
 I/YR
N=6
PV = -725
PMT = 0
FV = 1000
 I/YR
 1  5.50596%
Price has increased $25  Capital Gain
Because the yield to maturity has increased (and the maturity has fallen) in general, interest rates
have increased; otherwise, the firm’s credit rating has declined.
8) Graphically solve for the yield to maturity (YTM) for a 7% annual coupon bond with 5-years left
until maturity and $5000 face value, given that it has a current price of $4400. Start by computing
the PV for interest rates of 7%, 9%, and 12% and plot those points. Suppose that the price were
$5400, would you expect the YTM to be higher or lower? Again, graphically solve for the YTM.
Compare the solution with the YTM using the TVM function on your calculator.
0
1
350
2
350
3
350
4
350
5
5350
N=5
PV = -4400
PMT = 350
FV = 5000
 I/YR
$5000
YTM = 10.18%
$4700
$4400
5
7.5
10
12.5
15
$4100
If the price is higher, the YTM should be lower, e.g. YTM($5400) = 5.14480%
9) Suppose that you purchased an eight-year, 10% annual coupon bond, with a face of $1000, at par
value, two years ago, and you received an interest payment yesterday. (How many years are left
till maturity?) You look in the paper today and the yield for similar bonds being issued today is
8.5%. If you sold your bond today, would you have a capital gain or loss on sale? How much
would you sell it for? Suppose that your bond is listed as selling for $925, has the yield increased
or decreased from when you bought it? What is the yield? Check your work using the TVM and
CFLO functions on your calculator.
Effectively, this is a 6 year bond and should be treated as such. The bond should sell for:
6
$100
$1000
PV  

 $1,068.30
t
1.085 6
t 1 1.085
N=6
I/YR = 8.5
PMT = 100
FV = 1000
 PV
Since you purchased the bond at par value (face value), $1,000, you have made a capital gain of
$68.30. Alternatively, if the bond is selling for $925, the yield has increased substantially.
6
$925  
t 1
$100

$1000
1  YTM t 1  YTM 6
 YTM  11.81%
N=6
PV = -925
PMT = 100
FV = 1000
 I/YR
10) Referring to the bond in question 9, would it be more or less sensitive to a change in the interest
rates then a zero coupon bond with the same maturity? Why? (Write out the algebraic equations
for both bonds and compare them.) Show why using a numerical example. Would the bond in
question 9 be more or less sensitive than a bond with a 15% annual coupon, same face value and
maturity, to a 1% change in the respective yields? (Again, write out the equations of the 10% and
15% coupon bonds and compare them.) Show which bond would be more sensitive to changes in
the interest rate using a numerical example. How would the bond in question 9 compare with a
longer maturity bond with the same coupon?
Less sensitive than a zero coupon bond; the reason is that the entire value of the zero is tied up in
the final year 6 payment, therefore its entire value is subject to a 6 th power discounting factor. In
contrast, the coupon bond has more of its value subject to lower discounting factors.
PV 
PV 
PV 
100
100
100
100
100
1100





2
3
4
5
1  r  1  r  1  r  1  r  1  r  1  r 6
1000
1  r 6
150
150
150
150
150
1150





2
3
4
5
1  r  1  r  1  r  1  r  1  r  1  r 6
The bond in question 9 would be more sensitive than a bond with a 15% coupon, because the
higher coupon means a greater value of the bond is subject to lower discounting factors.
11) Suppose that you take out a loan with a 12% APR that is compounded quarterly, what is the
Effective Annual Rate (EAR)? How about if the loan is compounded annually—what is the EAR?
Suppose the bond in question 5 is a semi-annual coupon bond (instead of being an annual bond).
What do its cash-flows look like? (Draw a timeline and plot them.) If similar bonds are being
sold today with coupons of 8.5%, what should the price of that bond be?
4
 .012 
EAR  1 
  1  12.5509%
4 

If compounded annually, the EAR = APR = 12%.
If the $5000 bond is semi-annual:
0
1
2
3
4
5
200 200 200 200 200 200 200 200 200 5200
10
PV (rSemi )  
t 1
200

1  rSemi 
t
5000
1  rSemi 10
Because comparable bonds are being sold with 8.5% coupons, rSemi  .085
10
PV (.0425)  
t 1
200
1.0425
t

5000
1.042510
 4,899.86
P/YR = 1 or
N = 10
I/YR = 4.25
PMT = 200
FV = 5000
 PV
2
P/YR = 2
N = 10
I/YR = 8.5
PMT = 200
FV = 5000
 PV