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Chapter 3. Fixed Income Securities

Shuzhong Zhang

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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.

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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.

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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.

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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

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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.

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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.

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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.).

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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.

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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

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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

}.

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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.

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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.

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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.

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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

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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.

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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.

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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!

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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!

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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.

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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.

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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

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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.

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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