Question

Suppose you purchase a​ 30-year, zero-coupon bond with a yield to maturity of 8%. You hold the bond for five years before selling it.

a. If the​ bond's yield to maturity is 8% when you sell​ it, what is the internal rate of return of your​ investment?

b. If the​ bond's yield to maturity is 9% when you sell​ it, what is the internal rate of return of your​ investment?

c. If the​ bond's yield to maturity is 7% when you sell​ it, what is the internal rate of return of your​ investment?

d. Even if a bond has no chance of​ default, is your investment risk free if you plan to sell it before it​ matures? Explain.

Assume annual compounding.

Answer 1

Solution:
Given:
Term of Bond i.e Nper	30 Year
Coupon Amount i.e PMT	0 as Zero Coupon Bond
Yield to Maturity	8%
a. If the​ bond's yield to maturity is 8% when you sell​ it, what is the internal rate of return of your​ investment?
Assuming Face value of Bond $100 i.e Future value of Bond at the end of 30th Year = $100
Given:
Term of Bond i.e Nper	5 Years	(Sold after 5 Years)
Coupon Amount i.e PMT	0
Future value at the end of 30 th Year	$100	(Assumption)
Rate	8%
We will first find the Present value of Bond i.e Purchase Price of bond
Using PV Function in excel we will calculate the PV i.e Purchase value of Bond
PV(8%,30,0,$100)
₹ -9.94
Purchase Value of Bond = $9.94
Now will will calculate Sale Price of Bond after 5 Years
Given:
Term of Bond i.e Nper	25 Years
Coupon Amount i.e PMT	0
Future value at the end of 30 th Year	$100
Rate	8%
Using PV Function in excel we will calculate the PV of Bond after 5 Years i.e sale value of Bond after 5 Years
PV(8%,25,0,$100)
₹ -14.60
Now using RATE Function in excel we will calculate the Internal rate of return when use sell the Bond
Term of Bond i.e Nper	5 Year
PV i.e Purchase Value of Bond	$9.94
FV i.e Future value of Bond	$14.60
RATE(5,0,-$9.94,$14.6)
8%
IRR = 8%
b. If the​ bond's yield to maturity is 9% when you sell​ it, what is the internal rate of return of your​ investment?
Assuming Face value of Bond $100 i.e Future value of Bond at the end of 30th Year = $100
Given:
Term of Bond i.e Nper	5 Years	(Sold after 5 Years)
Coupon Amount i.e PMT	0
Future value at the end of 30 th Year	$100	(Assumption)
Rate	8%
We will first find the Present value of Bond i.e Purchase Price of bond
Using PV Function in excel we will calculate the PV i.e Purchase value of Bond
PV(8%,30,0,$100)
₹ -9.94
Purchase Value of Bond = $9.94
Now will will calculate Sale Price of Bond after 5 Years
Given:
Term of Bond i.e Nper	25 Years
Coupon Amount i.e PMT	0
Future value at the end of 30 th Year	$100
Rate	9%
Using PV Function in excel we will calculate the PV of Bond after 5 Years i.e sale value of Bond after 5 Years
PV(9%,25,0,$100)
₹ -11.60
Now using RATE Function in excel we will calculate the Internal rate of return when use sell the Bond
Term of Bond i.e Nper	5 Year
PV i.e Purchase Value of Bond	$9.94
FV i.e Future value of Bond after 5 Years	$11.60
RATE(5,0,-$9.94,$11.6)
3.14%
IRR = 3.14%
As YTM Increases IRR is less than Initial YTM
c. If the​ bond's yield to maturity is 7% when you sell​ it, what is the internal rate of return of your​ investment?
Assuming Face value of Bond $100 i.e Future value of Bond at the end of 30th Year = $100
Given:
Term of Bond i.e Nper	5 Years	(Sold after 5 Years)
Coupon Amount i.e PMT	0
Future value at the end of 30 th Year	$100	(Assumption)
Rate	8%
We will first find the Present value of Bond i.e Purchase Price of bond
Using PV Function in excel we will calculate the PV i.e Purchase value of Bond
PV(8%,30,0,$100)
₹ -9.94
Purchase Value of Bond = $9.94
Now will will calculate Sale Price of Bond after 5 Years
Given:
Term of Bond i.e Nper	25 Years
Coupon Amount i.e PMT	0
Future value at the end of 30 th Year	$100
Rate	7%
Using PV Function in excel we will calculate the PV of Bond after 5 Years i.e sale value of Bond after 5 Years
PV(7%,25,0,$100)
₹ -18.42
Now using RATE Function in excel we will calculate the Internal rate of return when use sell the Bond
Term of Bond i.e Nper	5 Year
PV i.e Purchase Value of Bond	$9.94
FV i.e Future value of Bond after 5 Years	$18.42
RATE(5,0,-$9.94,$18.42)
13.13%
IRR = 13.13%
As YTM decreases IRR is more than Initial YTM
d)
The rate of return would change even if Bond has no chance of default