Dr. Sudhakar Raju
FN 6100
ANSWERS TO CHAPTER 7 - INTEREST RATES AND BOND VALUATION
1). FV = $1000; CIR = 7%; N = 20; YTM = 7% (When bonds are first issued they are issued at par); PB
= $1000 (Assume annual bond)
YTM = 7%
PB = $1000
YTM = 15% PB = $499.25 ← 1 Shift P/YR
1000 FV
70 PMT
20 N
7 I/YR
PV=-$1000
15 I/YR
PV= -499.25
As market interest rates rise, bond prices fall.
2.)
FV = $1000; CIR = 7%; PMT = 70; N = 10 yrs; YTM = 9%.
1 shift P/YR
1000 FV
70 PMT
10 N
9 I/YR
PV = -871.65
1
PV = $871.65
3.)
1 shift P/YR
1000 FV
100 PMT
9N
1075.25 +/- PV
I
/YR => 8.76%
4.)
1 shift P/YR
1000 FV
13 N
850 +/- PV
7.40 I/YR
PMT => $55.64
Thus, Coupon Rate = [Annual Coupon / Face Value] =
5.)
$55.64
= 5.564%
$1000
2 shift P/YR
1000 FV
10 N ←11 year bonds issued one year ago. Thus, remaining maturity is 10 years.
Shift N
43 PMT
← Annual payment is $86. Semi-annual payment is $43
7.50 I/YR
PV => $1076.43
2
6.)
2 shift P/YR
1000 FV
10 N
Shift N
39 PMT
1080 +/- PV ← 108% of par value implies that the price of the bond is 1.08 x $10000 or $1080
I
/YR => 6.69%
7.)
2 shift P/YR
850 +/- PV
1000 FV
14.50 N
Shift N
9 I/YR
PMT = $35.64 (Semi-annual payment)
Thus, the annual coupon payment is $71.28. The coupon rate is given by [Annual Coupon / Face Value]
$71.28
or
. The coupon rate is thus 7.128%
$1000
8.)
2 shift P/YR
1000 FV
50 PMT
1040 +/- PV
7N
3
Shift N
I
/YR => 9.21 %
Shift Eff % => 9.42 % (Effective Yield. The Effective Yield is similar to the notion of EAR)
AnnualCoupon
$100
=
= 9.62%. If the price of the bond is
PB
$1040
the same as the face value (par bond), then the CY and the Coupon Rate will be identical.
The Current Yield (CY) is given by: CY =
9.) Compute the YTM (Discount Rate) on the existing bond issue. This works out to 6.68%.If the
company uses the same discount rate for the new bond issue, it would need to set the coupon rate to 6.68%
so that the bond will trade at par.
2 Shift P/YR
40 PMT
1095 +/- PV
10 N
Shift N
1000 FV
I
/YR => 6.68%
10.) The problem does not give you the price of the bond. This can be figured out by using information
on the current yield (CY) of the bond. Thus:
CY =
AnnualPMT
PB
.0980 =
$120
PB
Thus, PB =
$120
= $ 1224.49
.0980
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1 Shift P/YR
1000 FV
1224.49 +/- PV
120 PMT
9 I/YR
N =>12.99 years.
11a.) The rate of return on an investment is the same as the “YTM”. The cash flows from this bond are as
follows:
Yr 0 => - $1150
1 => $90
2 => $90
.
.
.
10 => $90 + $1000
Figure out the YTM.
1 Shift P/YR
1150 +/- PV
90 PMT
1000 FV
10 N
I
/YR => 6.88 %
Thus, if the bond is held to maturity and market yields do not change over the life of the bond, you can
expect an YTM or Internal Rate of Return to be equal to 6.88%.
b.) Two years from now, the YTM has declined by 1%. Thus the new YTM is 6.88% - 1% = 5.88 %. The
remaining time to maturity is now 8 years. What is the price at which you will sell the bond? The selling
price of the bond is given by:
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1 Shift P/YR
1000 FV
90 PMT
8N
5.88 I/YR
PV => $1194.67
Thus, $1194.67 is the price you receive when you sell this bond at the end of two years.
Because the bond is not being held to maturity, the yield you receive will NOT be the same as
the YTM. The yield that you receive over the 2 year holding period is called the “holding
period yield” (HPY). The HPY can be determined by considering the cash flows from this
bond investment:
Year 0 => - 1150 (Purchase Price of Bond)
Year 1 => $90
Year 2 => $90 + $1194.67 (coupon in year 2 plus price received from selling the bond)
HPY?
1 Shift P/Yr
1150 +/- PV
90 PMT
2N
1194.67 FV (Note that the FV is not $1000 but the price you receive from selling the bond)
I/Yr = 9.68% (HPY)
12.) This is a tricky problem. Visualize the cash flows by diagramming it thus:
BOND M
$0
0
(Today)
1
$1000 every
six months
6
6.50
$1750 every six months
+ $20,000 at Yr 20
↓
14
14.50
20
Discount Rate = 12% p.a., compounded semi-annually
There are three discrete batches of cash flows: $1000 every six months from 6.50 years to 14 years, $1750
every six months from 14.50 years to 20 years; the final principal lump sum payment of $20,000. Figure
out the PV of the first two batches of cash flows.
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THE PV OF THE FIRST BATCH OF CASH FLOWS IS GIVEN BY:
2 shift P/YR
1000 PMT
8N
Shift N
12 I/YR
PV => $10,106 at Yr 6
It is important to recognize that $10,106 is the value of the cash flows at Year 6.
THE PV OF THE SECOND BATCH OF CASH FLOWS IS GIVEN BY:
2 shift P/YR
1750 PMT
6N
Shift N
12 I/YR
PV => $14,672 at year 14.
It is important to recognize that $14,672 is the value of the cash flows at Year 14.
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Now construct a diagram based on the above results. Thus:
YR 0
YR 6
↓
$10,106
YR 14
↓
$14,672
YR 20
↓
$20,000
2 shift P/YR
2 shift P/YR
2 shift P/YR
10,106 FV
14,672 FV
20,000 FV
6N
14 N
20 N
Shift N
Shift N
Shift N
12 I/YR
12 I/YR
12 I/YR
PV => $5022
PV => $2870
PV of the above CFs:
PV => $1944
Total PVs => 5022 + 2870 + 1944 =>$9836. Thus, the value of Bond M is $9836.
BOND N
Bond N is a zero coupon bond that makes no coupon payments over the life of the bond. At year 20, Bond
N makes one lump sum payment of $20,000 equal to the face value of the bond.
YR 0 YR 1
YR 19
YR 20
$20,000
Figure out the PV of $20,000, 20 years from now discounted at 12% p.a. compounded semi-annually.
Thus:
2 Shift P/YR
20,000 FV
20 N
Shift N
12 I/YR
PV => $1944
Thus, the value of Bond N is $1944.
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