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```Chapter 5 :BOND PRICES AND INTEREST RATE RISK
Mr. Al Mannaei
Third Edition
What is Bond ?
• Debt instrument issued by a government or a
corporation in order to finance projects or activities.
• A form of loan.
– Issuer :
lender :
• At maturity borrower will pay lender face value + interest.
• Negotiable instruments.
2
What is Bond ?
Why issuing bonds?
• Governments : to finance infrastructure projects:
• Corporations : to finance commercial project, expand
the business, increase capital and reduce tax.
3
What is Bond ?
Mona bought a bond for \$980 issued by ALBA with 3 years
maturity , the coupon rate is 6% , paid annually ?
–
–
–
–
–
Who are the borrower & the lender?
What is the \$980 ?
How much mona will receive every year* ?
Can Mona sell the bond before maturity ?
How much mona will receive at maturity ?
Year 1
Year 2
* in other word , how much alba have to pay each year.
Year 3
(Maturity)
4
What is Bond ?
• What is the Maturity (Time) of bonds ?
Range : from
to
• What is the frequency of payment ?
• Is payment guaranteed ?
• Risk & Return ?
• Why corporation pay higher interest than government ?
• Listed vs. over the counter (OTC) ?!
5
How to price a bond ?
• The price of a bond is the present value of the future
cash flows promised, discounted at the market rate of
interest.
C1
C2
CN + F N
PB =
+
+
...
1
2
N
(1 + i) (1 + i)
(1 + i)
6
How to price a bond ?
C1
C2
CN + F N
PB =
+
+ ...
1
2
N
(1 + i) (1 + i)
(1 + i)
Where PB = price of bond or present value of promised payments;
Ct = coupon payment in period t, where t = 1, 2, 3,…, n;
Fn = par value (principal amount) due at maturity;
i = market interest rate (discount rate or market yield); and
n = number of periods to maturity.
7
Coupon rate & Market Yield
What is the difference between coupon rate & Market Yield ?!
Example :
• You buy 1 year government bond for \$1030.
• The bond pays 7% coupon annually.
• What is your profit ?! In percentage (%) ?!
2
0
1
4
2
0
1
5
C1 + F N
PB =
1
(1 + i)
8
Coupon rate & Market Yield
Example (Continue) :
if you sell the bond after 6 months for \$1050.
Calculate the market yield ?!
2
0
1
4
2
0
1
5
9
How to price a bond ?
• Example 1 : Consider 3 Years Bond with face
value of \$1000 and Coupon rate 8% , Current
market rate is 10% , Calculate the price of the
Bond ?
C1
C2
CN + F N
PB =
+
+ ...
1
2
N
(1 + i) (1 + i)
(1 + i)
10
How to price a bond ?
• Example 1 : Consider 3 Years Bond with face
value of \$1000 and Coupon rate 8% , Current
market rate is 10% , Calculate the price of the
Bond ?
11
How to price a bond ?
• Example 2 : Consider 3 Years Bond with face
value of \$1000 and Coupon rate 5% , Current
market rate is 5% , Calculate the price of the
Bond ?
C1
C2
CN + F N
PB =
+
+ ...
1
2
N
(1 + i) (1 + i)
(1 + i)
12
How to price a bond ?
• Example 2 : Consider 3 Years Bond with face
value of \$1000 and Coupon rate 5% , Current
market rate is 5% , Calculate the price of the
Bond ?
13
How to price a bond ?
Example 3: Consider a 1 year bond with a face
value of \$1000 and a coupon rate of 8%
compounded annually, current market yield is 5% ,
calculate the price of the bonds ?
14
How to price a bond ?
Example 3: Consider a 1 year bond with a face
value of \$1000 and a coupon rate of 8%
compounded annually, current market yield is 5% ,
calculate the price of the bonds ?
1.
Press (CMPD) bottom press EXE
2.
Press EXE on (Set) choose END press EXE
3.
Press EXE on (n) entre 1 press EXE
4.
Press EXE on (I) entre 5 press EXE
5.
Ignore PV ( press down )
6.
Press EXE on PMT ENTRE 80 press EXE
7.
Go back for PV press SOLVE.
15
How to price a bond ?
Example 3: Consider a 1 year bond with a face value
of \$1000 and a coupon rate of 8% compounded
annually, current market yield is 5% , calculate the
price of the bonds ?
n= 1
I=5
PV= ?
PMT=80
FV=1000
C/Y=1
16
Coupon rate & Market IR
How Does the Market IR & Coupon rate affect Bond price ?
17
Coupon rate & Market IR
How Does the Market IR & Coupon rate affect Bond price ?
If :
Coupon rate Greater Market IR >>> bond price higher than par ( issued at premium )
Coupon rate Lower Market IR >>> bond price lower than par ( issued at discount)
Coupon rate = Market IR >>> bond price equal par (issued at par)
18
Market IR & Bond Price
Negative relationship between i & Bond Price
– Increasing i ; decrease Bond Price.
– Decreasing i ; increase Bond Price.
Positive relationship between Coupon & Bond Price
– Increasing Coupon ; increase Bond Price.
– Decreasing Coupon ; decrease Bond Price.
19
Question
What is the price of a \$1000 face value with a
10% coupon if the market rate of return is 10%?
20
How to price a bond ?
• Example 4: Consider a 2 year bond with a face
value of \$1000 and a coupon rate of 5%
compounded Semi-annually, current market
yield is 6% , calculate the price of the bonds ?
21
How to price a bond ?
• Example 4: Consider a 2 year bond with a face
value of \$1000 and a coupon rate of 5%
compounded Semi-annually, current market
yield is 6% , calculate the price of the bonds ?
C1 / 2
C2 / 2
C4 / 2 + F N
PB =
+
+ ...
1
2
4
(1 + i / 2 ) (1 + i / 2 )
(1 + i)
22
Risk related with Bonds
• Credit or default risk: chance that issuer may be
unable or unwilling to pay as agreed.
23
Risk related with Bonds
• Reinvestment risk: potential effect of variability of
market interest rates on return at which payments can be
• Price risk: Inverse relationship between bond prices
and interest rates.
C1
C2
CN + F N
PB =
+
+ ...
1
2
N
(1 + i) (1 + i)
(1 + i)
24
Zero Coupon Bonds
• No periodic coupon payments.
• Issued at discount from par.
• Single payment of par value at maturity.
FV
PB =
n
(1 + i)
25
Zero Coupon Bond
• How to price Zero Coupon Bonds ?
Annual
FV
PB =
n
(1 + i)
Semi-annual, Quarterly ,
monthly, daily
FV
PB =
mn
i
(1 + )
m
PB is simply PV of FV represented by par value, discounted at market rate.
26
Example
• if you want to purchase a Company XYZ zerocoupon bond that has a \$1,000 face value and
matures in 3 years compounded annually, and
you would like to earn 10% per year on the
investment, what is the price of the bond ?
FV
PB =
n
(1 + i)
27
Example
• if you want to purchase a Company XYZ zerocoupon bond that has a \$1,000 face value and
matures in 3 years compounded semi-annually,
and you would like to earn 10% per year on the
investment, what is the price of the bond ?
FV
PB =
mn
i
(1 + )
m
28
Example
• In the previous example if the compounding
frequency increase , Bond price will :
– Decrease
– Increase
– No Change
Compounding Frequency increase
Yearly , semi-annually, monthly, daily.
Compounding Frequency decrease
29
Problem
Carol purchases a one-year discount bond with a
face value of \$1,000 for \$862.07. What is the yield
of the bond?
FV
PB =
n
(1 + i)
30
Zero bond
• Mariam bought a bond mature after 3 years for
1000 BD. The coupon rate and market yield = 8%.
• Marwa bought zero-bond mature after 3 years for
1000 BD. The market yield is 8%.
• What if both bond defaulted after year 2 ?!
Year 0
1
2
3
31
Bond Yields
• Market Yield (Interest Rate)
– Yield to Maturity
– Expected Yield
– Realized Yield
C1
C2
CN + F N
PB =
+
+ ...
1
2
N
(1 + i) (1 + i)
(1 + i)
32
Yield to Maturity (YTM)
• YTM : Investor's expected yield if bond is held to
maturity and all payments are reinvested at
same yield.
C1
C2
CN + F N
PB =
+
+ ...
1
2
N
(1 + i) (1 + i)
(1 + i)
• The longer until maturity, the less valid the
reinvestment assumption
• Example:
– Bond A mature after 30 years pay 8%.
– Bond B mature after 5 years pay 8%.
– Which bond most likely will change the coupon rate ?!
33
Yields calculation
• The Bond price , Coupon rate & maturity will be
given and the Yield (i) has to be calculated .
• The Yields can be calculated through
– Trail & error.
– Financial calculator.
C1
C2
CN + F N
PB =
+
+ ...
1
2
N
(1 + i) (1 + i)
(1 + i)
34
YTM Example
• Investor buys 5% percent coupon (semiannual
payments) bond for \$951.90; bond matures in 3
years. Solve the bond pricing equation for the
interest rate (i) such that price paid for the bond
equals PV of remaining payments due under the
bond.
C1
C2
CN + F N
PB =
1
(1 + i)
+
(1 + i)
2
+ ...
(1 + i)
N
35
YTM Example
• Investor buys 5% percent coupon (semiannual
payments) bond for \$951.90; bond matures in 3
years. Solve the bond pricing equation for the
interest rate (i) such that price paid for the bond
equals PV of remaining payments due under the
bond.
25
25
1,025
951.90 =
+
+ ...
1
2
6
(1 + (i / 2) ) (1 + (i / 2) )
(1 + (i / 2) )
36
YTM Example
• Investor buys 5% percent coupon (semiannual
payments) bond for \$951.90; bond matures in 3
years. Solve the bond pricing equation for the
interest rate (i) such that price paid for the bond
equals PV of remaining payments due under the
bond.
I:
n:
FV:
PV:
PMT:
37
Expected Yield
• Predicted yield for a given holding period
• Almost same as YTM but the expected holding
period is shorter .
C1
C2
CN + F N
PB =
+
+ ...
1
2
N
(1 + i) (1 + i)
(1 + i)
38
Realized Yield
• Realized Yield: actual rate of return, given the
cash flows actually received and their timing.
C1
C2
CN + F N
PB =
+
+ ...
1
2
N
(1 + i) (1 + i)
(1 + i)
• Differ from YTM & Expected yield , due to :
– Change in the amount of promised payments.
– Change in market interest rates.
39
Realized Yield
Investor pays \$1,000 for 10-year 8% coupon bond;
sells bond 3 years later for \$902.63.
Solve for i such that \$1,000 (the original investment)
equals PV of 2 annual payments of \$80 followed
by a 3rd annual payment of \$982.63
C1
C2
CN + F N
PB =
+
+ ...
1
2
N
(1 + i) (1 + i)
(1 + i)
40
Realized Yield
Investor pays \$1,000 for 10-year 8% coupon bond;
sells bond 3 years later for \$902.63.
Solve for i such that \$1,000 (the original investment)
equals PV of 2 annual payments of \$80 followed
by a 3rd annual payment of \$982.63
80
80
982.63
1000 =
+
+
...
1
2
3
(1 + i) (1 + i)
(1 + i)
41
Realized Yield
Investor pays \$1,000 for 10-year 8% coupon bond;
sells bond 3 years later for \$902.63.
Solve for i such that \$1,000 (the original investment)
equals PV of 2 annual payments of \$80 followed
by a 3rd annual payment of \$982.63
I:
n:
FV:
PV:
PMT:
42
Bond price volatility (price risk)
• Percentage change in price for given change in
interest rates
where %∆PB = percentage change in price
Pt = new price in period t
P t – 1 = bond’s price one period earlier
43
Bond price volatility (price risk)
44
Bond price volatility (price risk)
45
Bond price volatility (price risk)
46
Bond theorems
• Bond prices are inversely related to bond yields.
• The price volatility of a long-term bond is greater than
that of a short-term bond, holding the coupon rate
constant.
• The price volatility of a low-coupon bond is greater than
that of a high-coupon bond, holding maturity constant
47
• Price Risk
• Reinvestment Risk
• Price risk and reinvestment risk work against each other.
You bought one year bond at par, where interest rate and coupon rate= 5%.
After three months the interest rate fall to 3% ?!
48
• Price risk and reinvestment risk work against each other.
–
if interest rates fall :
• Bond prices rise but >> Coupons are reinvested at lower return.
– if interest rates rise :
• Bond prices fall but >> Coupons are reinvested at higher return.
49
Duration
• A measure of the volatility of bond price to a
change in interest rates.
• Expressed as a number of years.
• It is NOT the length of time it takes to get back
the original investment ( Payback Period ).
50
Bond : Duration will always be less than its time to maturity.
Zero-Bond : Duration is equal to its time to maturity.
Zero Bond
Regular Bond
51
Duration
• Duration
CFt * t

t
t  1 (1  i)
D  n
CFt

t
t  1 (1  i)
–
–
–
–
n
CF : Coupons payment
t: time of payment
i: interest
n: number of years
52
Duration Example
• Suppose we have a bond with a 3-year term to
maturity, an 8% coupon paid annually, and a
market yield of 10%. Duration is:
53
Duration
• If the yield increases to 15%:
What is the relationship between yield & Duration ?
54
Duration
What is the relation between coupon rates and duration ?!
Duration equals term to maturity for zero coupon securities.
55
Duration concepts
• Higher coupon rates mean shorter duration (less price volatility).
Bond A has 5 coupon rate , IR 10% , 2 years
Bond B has 8 coupon rate , IR 10% , 2 years
Which of the above bonds has lower duration ?
• Duration equals term to maturity for zero coupon securities.
What is the duration for 3 years zero bond ?
56
Duration concepts
• Longer maturities mean longer durations (greater price volatility).
Bond A has 8 coupon rate , IR 10% , 5 years
Bond B has 8 coupon rate , IR 10% , 2 years
Which of the above bonds has lower duration ?
• The higher the market rate of interest, the shorter the duration.
Bond A has 8 coupon rate , IR 20% , 2 years
Bond B has 8 coupon rate , IR 10% , 2 years
Which of the above bonds has lower duration ?
57
Duration
where: wi = proportion of bond i in portfolio and
Di = duration of bond i.
58
Duration
59
Duration
Using the 3-year, 4% coupon bond in Exhibit 5.6 (previous
slide ) , If yield increases to 12%:
 i 
%PB   D 
100

 (1  i ) 
 0.02 
 2.88

100


5
.
24
%

1
.
10


60
Thank You
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