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IE 5441 1
Chapter 3. Fixed Income Securities
Shuzhong Zhang
IE 5441 2
Financial instruments: bills, notes, bonds, annuities, futures contracts,
mortgages, options, ...; assortments that are not real goods but they
carry values by the promises they represent.
Securities: financial instruments that are traded in well developed
markets.
Fixed income securities: securities that promise definite cash flow
streams.
Shuzhong Zhang
IE 5441 3
The market for future cash
The only uncertainty in holding a fixed income security is that the
issuer may default. There are various forms of fixed income securities.
Savings deposits
Certificate of deposit (CD): issued in standard denominations such as
$10,000. Large CD’s can be traded in the market.
Money market instruments
Short-term (1 year or less) loans by corporations and banks.
Commercial papers: unsecured (without collateral) loans.
A banker’s acceptance: If A sells goods to B, and B promises to pay
within a fixed time. Some bank may accept the promise by promising
to pay the bill on behalf of B. A can then sell the banker’s acceptance
at a discount before expiration.
Eurodollar deposits: deposits denominated in dollars but held in a
bank outside US.
Shuzhong Zhang
IE 5441 4
US government securities
Treasury bills: issued in denominations of $10,000 or more with fixed
terms to maturity of 13, 26, and 52 weeks.
Treasury notes: maturities of 1 to 10 years, and sold in denominations
as small as $1,000. The owner of notes also receives a coupon payment
every 6 months until maturity.
Treasury bonds: maturities more than 10 years.
Treasury inflation-protected securities (TIPS): the principal value
changes with the Consumer Price Index (CPI), but the coupon rate
does not change in time.
Treasury strips: each coupon payment is sold as separate security.
Treasury strips are also known as zero-coupon bonds.
Shuzhong Zhang
IE 5441 5
Example 3.3.
Terms to learn: APR (Annual Percentage Rate); Points (the
percentage of the loan amount charged for providing the mortgage, not
including other possibly fees and expenses).
A typical mortgage broker advertisement:
Rate Pts Term Max amt APR
7.625 1.00 30 yr $203,150 7.883
7.875 0.50 30 yr $203,150 8.083
8.125 2.25 30 yr $600,000 8.399
7.000 1.00 15 yr $203,150 7.429
7.500 1.00 15 yr $600,000 7.859
Shuzhong Zhang
IE 5441 6
What does the advertisement say about the mortgage expenses?
For example, using the formula (A/P, 7.883%/12, 30× 12) we can
compute the monthly payment of a loan amount of $203,150 to be
$1,474.
Now, what is the implied expense?
If we use the annual rate of 7.625% for the monthly payment of
$1,474, then the principal would have been
$1, 474× (P/A, 7.625%/12, 30× 12) = $208, 253.
The difference, $208,253-$203,150=$5,103, is the total cost.
The loan fee itself is 1%× $203, 150 ≈ $2, 032. Therefore, the other
expense is $5,103-$2,032=$3,071.
Shuzhong Zhang
IE 5441 7
Other bonds
Municipal bonds: issued by agencies of state and local governments.
Corporate bonds: issued by corporations. Some are traded on an
exchange, many are traded over-the-counter.
Callable bonds: a feature of bonds which allows the bond issuer to
purchase back the bond at a specific price within a period.
Mortgages
Adjustable-rate mortgage: the interest rate is adjusted periodically.
Mortgage-backed securities: individual mortgages are bundled into
large packages and traded among institutions.
Shuzhong Zhang
IE 5441 8
Details of a bond
• Face value (or par value).
• Coupon payments.
• The bid price: the price the bond is sold.
• The ask price: the price the bond is bought.
• Accrued interest:
AI =# of days since last coupon
# of days in current coupon× coupon amount.
• Quality rating: Moody’s (Aaa, Ba, etc.); Standard & Poors (AAA,
BB, etc.).
Shuzhong Zhang
IE 5441 9
Example of accrued interest: Suppose we purchased on May 8 a U.S.
Treasury bond. The coupon rate is 9% per year, to be paid on
February 15 and August 15 each year. Hence,
AI =83
83 + 99× 4.5 = 2.05.
This value will be added to the quoted price. If the face value is
$1,000, then $20.50 would be added to the quoted price.
Shuzhong Zhang
IE 5441 10
Quality Ratings
Rating Classifications:
Moody’s Standard & Poor’s
High grade Aaa AAA
Aa AA
Medium grade A A
Baa BBB
Speculative grade Ba BB
B B
Default danger Caa CCC
Ca CC
C C
D
Shuzhong Zhang
IE 5441 11
Yield of a bond: Its internal rate of return (IRR).
Consider a bond with face value F , coupon payment C per annum to
be paid m times, mature in n/m years, and the purchase price P .
Then, its yield is λ, satisfying
P =F
(1 + λ/m)n+
n∑k=1
C/m
(1 + λ/m)k
=F
(1 + λ/m)n+
C
λ
{1− 1
[1 + (λ/m)]n
}.
Shuzhong Zhang
IE 5441 12
The yield is not explicitly computable in general. One exceptionally
simple case is when C = 0 (zero-coupon). In that case,
λ = m
(n
√F
P− 1
).
Another interesting case is when C/F = λ (i.e. the coupon rate is
exactly the yield). Then,
P = F
and the bond is said to be at par.
In general, the price-yield curve is convex. The ‘steepness’ of the curve
appears to be related to the length of the period.
Shuzhong Zhang
IE 5441 13
Prices of 9% coupon bonds:
5% 8% 9% 10% 11%
1 yr 103.85 100.94 100.00 99.07 94.61
5 yr 117.50 104.06 100.00 96.14 79.41
10 yr 131.18 106.80 100.00 93.77 69.42
20 yr 150.21 109.90 100.00 91.42 62.22
30 yr 161.82 111.31 100.00 90.54 60.52
where the rows are time-to-maturity, and the columns are the yields.
Clearly, as time-to-maturity increases, the price of the bond tends to
depend more sensitively to the change of yield.
Shuzhong Zhang
IE 5441 14
Duration: For a cash flow {(F1, ..., Fn) | Fi occurs at ti, i = 1, ..., n},its duration is the weighted average of the payment dates
D =
∑nk=1 PV (Fk)tk
PV
Macaulay Duration of a bond:
D =
∑nk=1
Fk
(1+λ/m)k× k
m∑nk=1
Fk
(1+λ/m)k
Explicitly, for a bond with coupon c, paid m times, and yield rate y:
D =1 + y
my− 1 + y + n(c− y)
mc[(1 + y)n − 1] +my.
Shuzhong Zhang
IE 5441 15
Example: Consider a 7% bond with 3 years to maturity. Suppose that
the yield is 8%. Then, the breakdown of the duration of its cash flows
will be:
Year Payment Discount Factor PV Weight Duration
0.5 3.5 0.962 3.365 0.035 0.017
1.0 3.5 0.925 3.236 0.033 0.033
1.5 3.5 0.889 3.111 0.032 0.048
2.0 3.5 0.855 2.992 0.031 0.061
2.5 3.5 0.855 2.992 0.031 0.074
3.0 103.5 0.790 81.798 0.840 2.520
Total 97.379 1.000 2.753
Shuzhong Zhang
IE 5441 16
Example 3.7. Consider a 10%, 30-year bond with 6-month coupons.
Suppose it is at par (yield is 10%).
We then compute from
D =1 + y
my− 1 + y + n(c− y)
mc[(1 + y)n − 1] +my
that
D =1 + y
my
[1− 1
(1 + y)n
]=
1.05
0.1
[1− 1
(1.05)60
]= 9.938.
Shuzhong Zhang
IE 5441 17
If we denote
P (λ) =n∑
k=1
Fk/(1 + λ/m)k,
then
D(λ) = −P ′(λ)
P (λ)× (1 + λ/m).
We call
DM (λ) = −P ′(λ)
P (λ)
to be the modified duration.
The duration measures the sensitivity of the price relative to the
change of interest rate.
The most sensitive bond will be zero coupon bond with a long
maturity period.
Shuzhong Zhang
IE 5441 18
Example. Consider a 30-year, 10% coupon bond, which is at par with
price $100. The duration is D = 9.94. Hence, DM = 9.94/1.05 = 9.47.
The slope of the curve at that point is dP/dλ = −947. The straight
line approximation suggests if the yield changes to 11%, then the
change in price is
∆P = −DM × 100×∆λ = −9.47× 100× 0.01 = −9.47.
Hence the estimated new price is $90.53.
On the other hand, if the bond does not carry any coupons. Then, we
have D = 30, and DM ≈ 27. If the yield changes to 11%, then the
estimated new price will be $73, which is a big change!
Shuzhong Zhang
IE 5441 19
It is important to control the risk of a portfolio with respect to the
interest rate risk.
Let us consider what happens if we hold two bonds, A and B, in a
portfolio.
We have
DA =
∑nk=0 PV A
k tkPA
and DB =
∑nk=0 PV B
k tkPB
.
Observe that the total PV is
P =n∑
k=0
(PV Ak + PV B
k ) = PA + PB.
The duration of the portfolio is
D =
∑nk=0(PV A
k + PV Bk )tk
PA + PB=
PA
PA + PB×DA +
PB
PA + PB×DB.
In other words, it is a convex combination of the two!
Shuzhong Zhang
IE 5441 20
In general, if we have m fixed income securities, each with price Pi and
duration Di, i = 1, 2, ...,m, then the portfolio will have price P and
duration D:
P = P1 + P2 + · · ·+ Pm
D = w1D1 + w2D2 + · · ·+ wmDm
where wi = Pi/(P1 + P2 + · · ·+ Pm), i = 1, 2, ...,m.
Shuzhong Zhang
IE 5441 21
Immunization: managing the interest rate risk.
Example. The X Corporate has an obligation to pay $1 Million in 10
years. It wishes to invest in some bonds in order to meet this
obligation. The following three bonds are under consideration:
Rate Maturity Price Yield
Bond 1 6% 30 yrs 69.04 9%
Bond 2 11% 10 yrs 113.01 9%
Bond 3 9% 20 yrs 100.00 9%
We calculate that D1 = 11.44, D2 = 6.54, D3 = 9.61, and
PV = 414, 643. We decide to combine Bond 1 and Bond 2, and set
PV = V1 + V2
10PV = D1V1 +D2V2
leading to V1 = $292, 788.73 and V2 = $121, 854.27.
Shuzhong Zhang
IE 5441 22
Immunization results:
9% 8% 10%
Bond 1
price 69.04 77.38 62.14
shares 4241 4241 4241
value 292,798 328,168 263,535
Bond 2
price 113.01 120.39 106.23
shares 1078 1078 1078
value 121,824 129,780 114,515
obligation value 414,642 456,386 376,889
difference -19 1,562 1,162
Shuzhong Zhang
IE 5441 23
Convexity of a bond: it is possible to improve the immunization by
using a second order approximation.
Let
C =P ′′(λ)
P (λ).
In the case of a cash flow with payments ck,
C =
n∑k=1
ck(1 + λ/m)k
× k(k + 1)
m2
P (1 + λ/m)2.
We have
∆P ≈ −DMP∆λ+CP
2(∆λ)2.
Shuzhong Zhang
IE 5441 24
It is possible to use a combination of bonds to fit the PV, the
duration, and the convexity of the obligation.
If we hold a bond portfolio (P1, · · · , Pm), then its convexity is
D = w1D1 + · · ·+ wmDm
where wi = Pi/(P1 + · · ·+ Pm), i = 1, 2, ...,m.
Back to the problem of the X Corporate, if the convexity is to be
matched as well, then we can consider the following equation
PV = V1 + V2 + V3
D × PV = D1V1 +D2V2 +D3V3
C × PV = C1V1 + C2V2 + C3V3.
One will need three bonds to do the matching.
Shuzhong Zhang