Calculating Bond Values and Yields

This assignment provides a step-by-step guide on calculating the price of a coupon-bearing bond, yield to maturity (YTM), and future value of an annuity. It also includes a comparison with a zero-coupon bond, demonstrating how to calculate its purchase price and market discount rate. The examples use specific values for face value, interest rates, and time periods to illustrate the calculations and provide clarity.

Corporate Finance -
Basics

Table of Contents
Question 1: ......................................................................................................................................3
Question 2. ......................................................................................................................................4
Question 3: ......................................................................................................................................5
Question 4: ......................................................................................................................................5

Question 1:
Solution:
a) Mortgage Payment can be calculated by using below formula:-
M = P r ( 1 + rn ) / ( 1 + rn ) – 1
Here, P = loan amount = £100,000
n = number of years = 20 years
r = rate of interest = 5% annually
therefore,
Mortgage payment (PMT) = P r ( 1 + r)n / ( 1 + r)n – 1
= 100000 x 0.05 (1 + 0.05)20 / (1 + 0.05)20 – 1
= £8,024.26
b) A table for each year balance of mortgage-
Mortgage Payment (PMT) = P r ( 1 + r)n / ( 1 + r)n – 1
For first year, PMT = 100000 x 0.05 (1 + 0.05)1 / (1 + 0.05)1 – 1
= 5000 (1.05) / (1.05 – 1)
= 5000 x 1.05 / 0.05
= £105,000
For second year, PMT = 100000 x 0.05 (1 + 0.05)2 / (1 + 0.05)2 – 1
= 5000 (1.05)2 / (1.05)2– 1
= 5000 x 1.0025 / 0.1025
= £53,780.49
Therefore,
Number of year Mortgage amount (£)
1 105,000
2 53,780.49
3 36,720.86
4 28,201.18
5 23,097.48
6 19,701.75
7 17,281.98

8 15,472.18
9 14,069.01
10 12,950.46
11 12,038.89
12 11,282.54
13 10,645.58
14 10,102.40
15 9634.23
16 9226.99
17 8869.91
18 8554.62
19 8274.50
20 8024.26
c)
First payment amount = 105,000
Fraction of first payment on mortgage = 105000 – 100000 / 100000
= 5000 / 100000
= 0.5% or ½ %
d)
Fraction of last payment on mortgage = 100000 – 8024.26 / 100000
= 0.9% or 9/10
Question 2.
a) Investment amount = £10,000, Rate of interest = 8% per year, T = 40 years
Amount = P (1 + R% )T
= 10000 (1 + 0.08 )40
= 10000 (1.08)40
= 2,100,000 (approximate)
b) If T = 30 years then amount of retirement money will be
Amount = P (1 + R% )T

= 10000 (1 + 0.08 )30
= 10000 (1.08)30
= £1,000,000 (approximate)
c) Annual income receives from annuity
Principal = £10,000
Duration = 15 years
Then, amount will be-
Annuity Payment (p) = P R (1 + R)N / R( 1 + R)N – 1
= 10000 x 0.08 (1 + 0.08)15 / 0.08 (1 + 0.08)15 – 1
= £855,947.87
Question 3:
a) Market discount rate = 10%, Dividend to be paid in the next year = £1.5 per share
Present Value of MVC's Share = Dividend Payment / Discount Rate
PV of MVC's Share = £1.5 / 0.10
PV of MVC's Share = £15
b) If the dividends are projected to grow at a constant rate forever and the dividend yield is
currently 7.5%, the projected dividend growth rate is:
P = D1 / (R-g)
15 = 1.5/ (0.075-g)
15*(0.075-g) = £1.5
0.075-g = £1.5/£15
0.075 – g = 0.10
g = 0.075 – 0.10
g = -0.025 or -2.5%
c)
Value of Stock = DPS / (Dividend Yield – growth)
= £1.5 / (7.5% – (-2.5%))
= £1.5 / (7.5% + 2.5%)
= £1.5/0.10
= £15
d)

Value of MVC Shares = £15/2 = £7.5
Growth Rate = -2.5%
Number of Years expected to survive at the rate of -2.5% = 7.5/2.5 = 3 years.
Question 4:
a) face value of a bond = £100
market interest rate = 5% / 2 = 2.5% semi-annually
time = 3 years for maturity of bond
semi-annual interest payment = 5% x 100 / 2
= £10
Therefore, interest payment = £20
A coupon bearing bond can be calculated by-
Here, C = periodic coupon payment
F = face value
t = time
T = total number of periods
Therefore, P = 5/ (1.05)1 + 5/ (1.05)2 +105/ (1.05)3
= 4.76 + 4.5 + 91
= 100.26 approx.
b) Yield To Maturity:
YTM = (C + (F-P)/n)
[(F+P)/2]
YTM = (0.05*(100))+((100-102.8)/2)
(100+102.8)/2
YTM = (5+ (-2.8/2))
202.8/2
YTM = (5-1.4)
101.4
YTM = (3.6)/101.4

YTM = 0.036
YTM = 3.6%
c) If Correct market discount rate for each individual payment made by the bond = yield-to-
maturity, the value of an annuity paying £100 every six months for the next three years is as
follows:
Annual Periodic Payments (P) = £100*2 = £200
Number of years (n) = 3
Market Discount Rate (r) = YTM = 3.6%
Future Value of Annuity = [P* (((1+r)n – 1))/ r)]
FV = [£200*(((1+0.036)3 – 1)/0.036)]
FV = [£200*(((1.036)3 – 1)/0.036)]
FV = [£200*((1.112– 1)/0.036)]
FV = [£200*(0.112/0.036)]
FV = [£200*(0.112/0.036)]
FV = [£200*3.11]
FV = £622.22
d) Price of the bond = £102.8
Coupon Rate = 5%
On the same day, a Zero-Coupon Bond of:
Face Value = £100
Years to Maturity = 3 years
Purchase Price = £86.23
Annual Periodic Payments (P) = £100*2 = £200
Number of years (n) = 3
YTM = (C + (F-P)/n)
[(F+P)/2]
YTM = (0)+((100-102.8)/2)
(100+102.8)/2
YTM = (0+ (-2.8/2))
202.8/2
YTM = (0-1.4)

101.4
YTM = -1.4/101.4
YTM =−0.0138
YTM = -1.38%
Market Discount Rate (r) = YTM =−0.0138
Future Value of Annuity = [P* (((1+r)n – 1))/ r)]
FV = [£200*(((1−0.0138)3 – 1)/(−0.0138)]
FV = [£200*(((0.9862– 1)/(-0.0138)]
FV = [£200*((-0.0138)/(-0.0138)])
FV = [£200*(1)]
FV = £200