VALUATION OF FIXED
INCOME SECURITIES
Bond: A debt instrument with periodic
payments of interest and repayment of
principal at maturity
rM rM rM rM rM
rM rM+M
|___|____|____|____|____|...…..|___ |
0 1
2
3
4
5
n-1 n
r: coupon interest rate
M: maturity (par value)
n: term to maturity
1
Bond Valuation
V= rM(PVIF)i,1+rM(PVIF)i,2 +………
rM(PVIF)i,n + M(PVIF)i,n
i: market rate of interest
Coupon payments (rM) can be regarded as
an annuity,
V= rM(PVIFA)i,n + M(PVIF)i,n
or
(1+i)n -1
1
V = rM ------------- + M -----------(1+i)n
(1+i)n
2
Bond Valuation example
n=10 years, coupon rate: 8%
M= $1,000 Market rate : 10%
$80 $80 $80 $80 $80 $80 $180
|___|____|____|____|____|...…..|___ |
0 1
2
3
4
5
9 10
V= $80x(PVIFA)10%,10 + $1,000x(PVIF)10%,10
= $877.11
If
i>r
i<r
i=r
V<M
V>M
V=M
(discount)
(premium)
(par)
Yield-to-maturity: the rate of return on a bond
In the example, the YTM is 10%.
A bond’s YTM is the market rate of interest for
that risk group and maturity.
3
Valuation Between Interest
Payment Dates
1
V
(1  i) c / g
n 1

rM
M 
rM   (1  i)t  (1  i) n 1 
t 1


V: invoice price of the bond
c: days until first payment
g: number of days between two payment periods
P= quoted price = V - accrued interest
Accrued Interest = rM (g-c)/g
4
Valuation Example
Eg. N=5 years,semiannual coupon r=8%,
i=10%, first payment 2 months from today.
9

1
40
1000 
V
40  

2/6 
t
9
(1  0.05) 
(1  0.05) 
t 1 (1  0.05)
V= Invoice Price = $953.29
Accrued Interest = 40 x (4/6)
= $26.67
Quoted price = $926.62
5
Risks Faced by a Bond
Investor
•
•
•
•
•
•
•
Default risk
Interest rate risk (price risk)
Reinvestment risk
Call risk
Inflation risk
Foreign exchange risk
Liquidity risk
6
Rating
Category
Moody’s
S&P
-----------------------------------------High Grade
Aaa
AAA
Aa
AA
------------------------------------------Investment
A
A
Grade
Baa
BBB
------------------------------------------Speculative
Ba
BB
B
B
------------------------------------------Default
Caa
CCC
Ca
CC
C
C
D
7
Interest Rate Risk
Example: Two bond issues of ABC Co.
N1=1 yr N2= 10 yrs r = 5%
Market Rate of
Interest
5%
6%
7%
8%
Bond Value
First Issue: Second Issue:
N = 1 yr
N = 10 yrs
100.00
100.00
99.06
92.64
98.13
85.95
97.22
79.87
As term to maturity increases, value of the
bond becomes more sensitive to movements
in market interest rate.
8
Bond Value and Coupon Rates
Example:Two issues of ABC Co.
n=20 yrs, r1=10%, r2=6%
Market
Interest Rate
8%
9%
10%
11%
12%
Bond 1
R=10%
119.64
109.13
100.00
92.04
85.06
Percent
change
-8.78%
-8.36%
-7.96%
-7.58%
Bond 2
R=6%
80.36
72.61
65.95
60.18
55.18
Percent
change
-9.64%
-9.17%
-8.75%
-8.31%
• Low coupon bonds are more
sensitive to changes in market
interest rates
9
Value of a Bond in Time
Example: Market rate stays at 10%, values of
two bonds with coupon rates of 8% and 12%
as the term to maturity approaches:
Maturity
Bond 1
R=8%
Bond 2
R=12%
5
4
3
2
1
0
92.42
93.66
95.03
96.53
98.18
100.00
107.58
106.34
104.97
103.47
101.82
100.00
Assuming that interest rates remain the same,
bond value approaches to par over time as
term to maturity shortens.
10
Term Structure of Interest
Rates
Relationship between yield and time to
maturity.
Example:
n=1
n=5
n=20
i=6%
i=8%
i=9%
i
Yield Curve
Maturity
11
Possible Explanations of
the Term Structure
1. Expectations Hypothesis
1 + in =[(1+ i1)(1+ 1i2)…….(1+n-1 in)]1/n
Example: i2=8%
i1=6% 1i2=?
1 + 0.08 = [(1+ 0.06)(1+ 1i2)]1/2
1i2 = 0.1004 or 10%
2. Liquidity Preference Hypothesis
Slope of the yield curve is higher than
specified in expectations hypothesis
3. Segmented Markets Hypothesis
12
Duration
Volatility in bond price is directly proportional
to term to maturity but inversely proportional
to coupon payments. Duration of a bond is a
measure that incorporates both factors that
affect volatility.
n
(t )Ct
D
V
t
t 1 (1  i )
13
Duration Example
n=5 yrs, r=8%, i=10%
(1)
Year
(2)
PMT
(3)
PVIF
(4)
(2)x(3)
1
8
0.9091
7.27
0.0787 0.0787
2
8
0.8264
6.61
0.0715 0.1430
3
8
0.7513
6.01
0.0650 0.1950
4
8
0.6830
5.46
0.0591 0.2364
5
108
0.6209
67.06
72.57
Total
(5)
(4)/V
92.41
(6)
(1)x(5)
3.6284
4.28
Bond Value = $92.41
Macaulay Duration = 4.28 years
14
Hedging Interest Rate Risk
$12 $12 $12 $12 $12 $12 $112
|___|____|____|____|____|...…..|___ |
0 1
2
3
4
5
9 10
V0=$84.94 when i=15%
After i declines to 12%, V = $100
V when term to maturity is 4 years:
V6 = $100
Future value of the first 6 coupon payments
reinvested at 12%:
12 x PVIFA 12%,6 = $97.38
Total savings = $100 + $97.38 = $197.38
$84.94 in 6 years grows to $197.38
Annual growth of 15%.
15
Immunization Example
$1,000 $2,000 $2,500 $2,000 $1650
|_____|______|______|______|______|
0
1
2
3
4
5
Total Premiums = Assets = $6,830.82
Market rate = 10% Flat yield curve
Strategy 1: Invest in 1-yr bills with 10% interest
6830.82 -> 7513.90
(1000.00)
6513.90 --> 7165.29
(2000.00)
5165.29 --> 5681.82
(2500.00)
3181.82 ->3500
(2000)
1500 ->1650
(1650)
16
Immunization Example
(Cont’d)
However, if interest rates fall, assets will
be short of liabilities
Strategy 2: Invest in 3-yr zero coupon bonds
yielding 10%
Duration of Liabilities:
1
2
3
4
5
1000
2000
2500
2000
1650
909.09
1652.89
1878.29
1366.03
1024.52
0.133
0.242
0.275
0.200
0.150
Duration = 2.99 years
0.133
0.484
0.825
0.800
0.750
2.990
17
Immunization Example
(Cont’d)
Market rate 10%, V = $6,830.82
M = $9,091.82 Duration = 3 years
If interest rates fall from 10% to 8%,
V= $9,091.82 x PVIF 8%,3 = $7,217.38
7217.38 ->7794.77
(1000.00)
6794.77 ->7338.35
(2000.00)
5338.35->5765.42
(2500.00)
3265.42->3526.66
(2000.00)
1526.66->1650
(1650)
18
Modified Duration
D
MD = ----------(1 + i)
In the example above, MD = 4.28/1.10 = 3.89
Approximate Change in V = -MD x Change in
yield
Example:
If the yield decreases from 10% to 8%
% Change in V= -4.28 x (-2) = 8.56%
In fact when i=10% V = $92.41
i=8% V = $100 increase 8.21%
19
Convexity
Price-Yield Relationship
V
Yield
The shape of the curve depends on
the coupon rate and term to maturity
High coupon + Short term -----> Linear
Low coupon + Long term ------> Convex
20
Convexity (Cont’d)
Higher convexity means that when interest
rates go up, bond value declines slowly; but
when rates decline, increase in bond price is
large
Therefore high convexity is a desirable
feature.
Factors that increase convexity:
* Low coupon
* Long term to maturity
* Low yield
21
Convexity (Cont’d)
2
dV
Convexity  2 V
di
n
Ct
d 2V
1
2

(
t
 t)
2
2 
t
di
(1  i ) t 1 (1  i )
Year
1
2
3
4
5
(1)
Ct
8
8
8
8
108
(2)
PVIF(8%,n)
0.9091
0.8264
0.7513
0.6830
0.6209
(3)
(1) x (2)
7.27
6.61
6.01
5.46
67.06
92.42
(4)
t2 + t
2
6
12
20
30
(5)
(3) x (4)
14.55
39.67
72.13
109.28
2011.79
2247.41
Convexity = [1/(1.10)2][2247.41][1/92.42]
= 20.10
Appox. Change in V = -MD x i + K x (i)2
22
Alternative Measures of
Yield
• Current Yield = rM / V
• Yield-to-maturity
– Bond is held until maturity
– All coupon and principal
repayments are made on time
– Bond is not called before maturity
– Coupon payments are reinvested
at yield-to-maturity
• Yield-to-call
• Holding period yield
Vt+1 - Vt + rM
HPY = -------------------Vt
23
Approximate yield-tomaturity
M V
rM 
n
i
V M
2
Example V= $877.11 n=3 yrs r=8% M=$1000
1000  877.11
80 
10
i
 0.0983
877.11  1000
2
24
Bond Investment Strategies
I. Passive Strategies
Investing $100 in 1925
T-bill
Deposits
Stock Market
AAA Corporate Bonds
Gold
Inflation
Passive Strategies are better when:
Interest rate risk is low, and
Inflation is low and stable
25
II. Active Strategies
• Strategies based on maturity
structure
– Maturity matching - duration
– Spreading the maturity
– Investing only in short term bills
and long term bonds
• Strategies based on forecasting
interest rate movements
– Interest rate fluctuations
• Buy when rates are high, sell when
low
• Increase duration if higher rates are
forecast, reduce duration otherwise
26
- Riding the yield curve
• Investing in bonds assuming that the
yield curve will not shift
i
A
B
Maturity
Eg. 1 year bill i=6% V1 = $943.40 B
2 year zero coupon i=8% V2 = $857.34 A
Buy the 2-year bond at $857.34, sell it next year
at $943.40
HPY = (943.40 - 857.34) / 857.34 = 10.04%
27
Strategies based on lack of
market efficiency
• Junk bonds
• Bond swaps
– Yield swap : same coupon, rating,
maturity and industry, different yield
– Exchange swap: same rating, maturity,
industry, yield, different coupon.
Exchange current yield for capital gains
– Tax swap: Selling a bond to realize a
loss, and replacing it with a similar
bond
– Swapping bonds with different tax
status: eg. AAA corporate bond vs.
municipal bond
28
Strategies based on lack of
market efficiency (cont’d)
• Possible shortcomings of bond
swaps:
–
–
–
–
time to execute the swap
taxes
transaction costs
risk level of bonds
• Portfolio rebalancing:
adjusting the bond portfolio
for the changes in market
conditions
29