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Fall 2011 Bond and Interest Rate Basics Prof. Page

BUSM 411: Derivatives and Fixed Income

4. Bond and Interest Rate Basics

4.1. Discount Factors

• Receiving a dollar today is not the same as receiving it in a month or in a year.

– Money today can be put in a safe place (a bank, under the mattress), but the

opposite is not easily doable

– Money in hand gives the holder the option to use it however he/she desires,

including transferring it to the future through deposit or investment

• How much is $1 in the future worth today? This value is called the discount factor

– The notion of discount factors is at the the heart of fixed income securities (and

of any other security)

• As a concrete example, the US government needs to borrow money from investors to

finance its expenses. It does so by issuing a number of securities, such as Treasury

bills, notes, and bonds, to investors, receiving money today in exchange for money in

the future. The US Treasury is extremely unlikely to default on its obligations (why?),

so the relation between the purchase price and the payoff of Treasury securities reveals

the market time value of money–that is, the exchange rate between money today and

money in the future

• Example:

– On August 10, 2006, the Treasury issued 182-day (six month) Treasury bills. The

market price was $97.477 for $100 of face value. This bill would not make any

other payment between the two dates.

– Thus, the ratio between purchase price and payoff, 0.97477 = $97.477/$100, can

be considered the market-wide discount factor between the two dates August 10,

2006 and February 8, 2007.

– That is, market participants were willing to exchange 0.97477 dollars on the first

date for 1 dollar six months later.

• Notation:

– The discount factor between dates t1 and t2, that is, that price on date t1 to

receive $1 on date t2, can be denoted by Z(t1, t2).

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– If you have a future cash flow at time t, Ct, you can multiply it by the discount

factor Z(0, t) to obtain the present value of the cash flow.

4.1.1 Discount factors across maturities

• The example and notation above highlight that the discount factor at some date t1

depends on its maturity t2. If we vary the maturity, making it longer or shorter, the

discount rate varies as well.

• For the same reason investors value $1 today more than $1 in six months, they also

value $1 in three months more than $1 in six months.

• Example:

– On August 10, 2006, the Treasury also issued 91-day (three month) Treasury bills.

The market price was $98.739 for $100 of face value.

– If we denote t1 = August 10, 2006, t2 = November 9, 2006, and t3 = February 8,

2007, we find that the discount factor Z(t1, t2) = 0.98739, which is higher than

Z(t1, t3) = 0.97477

• FACT: At any given time t1, the discount factor is lower the longer the maturity t2.

That is, given two dates t2 and t3 with t2 < t3, it is always the case that

Z(t1, t2) ≥ Z(t1, t3)

4.1.2 Discount factors over time

• A second important characteristic of discount factors is that they care not constant

over time, even while keeping constant the time-to-maturity t2 − t1.

• As time goes by, the time value of money changes. For example, the Treasury issued a

182-day T-bill on August 26, 2004, for a price $99.115, which implies a discount factor

much higher than that for the same time-to-maturity two years later.

• Why do discount factors vary over time?

– Most obvious and intuitive reason is that expected inflation varies over time.

– Other explanations are related to the behavior of the US economy, its budget

deficit, and the actions of the Federal Reserve, as well as investors’ change appetite

for risk.

– These macroeconomic conditions affect the relative supply and demand of Trea-

sury securities and thus their prices.

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4.2. Interest rates

• Grasping the concept of a rate of interest is both easier and more complicated than

the concept of a discount factor

– Easier because interest is closer to our everyday notion of a return on an invest-

ment, or the cost of a loan. For instance, if you invest $100 for one year at an

annual interest rate of 5%, you receive in one year $105. The discount factor is

$100/$105 = 0.9524. It carries the same information as the interest rate, but

perhaps less intuitively descries the return on investment.

– More complicated because it depends on the compounding frequency of the inter-

est paid on the initial investment.

• The compounding frequency of interest accruals refers to the number of times

per year in which interest is paid and reinvested on the invested capital

– Mentioning only an interest rate is an incomplete description of the rate of return

of an investment–the compounding frequency is a crucial element that must be

attached to the interest rate figure

– In the above example, we implicitly assumed that the 5% rate of interest is applied

to the original capital only once (hence the $105 result). What if interest accrues

every 6 months? What if interest accrues every 12 months?

• FACT: For a given interest rate figure (e.g. 5%), more frequent accrual of interest

yields a higher final payoff

• FACT: for a given final payoff, more frequent accrual of interest implies a lower interest

rate figure

4.2.1 Discount factors, interest rates, and compounding frequencies

• Discount factors and interest rates are intimately related, once we make explicit the

compounding frequency

• Two compounding frequencies are particularly important:

– Semi-annual compounding, because it matches the frequency of coupon payments

of US Treasury notes and bonds

– Continuous compounding is also important, mainly for analytical convenience

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• Example: Let t1 = September 19, 2011 and let t2 = September 19, 2012 (one year from

now). Consider an investment of $100 today at the semi-annually compounded rate

of interest r = 5%, for one year. This means the investment grows by 5%/2 = 2.5%

every six months, so that at t2 the payoff is

Payoff at t2 = $100 × (1 + r/2) × (1 + r/2) = $100 × (1 + r/2)2 = $105.0625

The relation between money at t1 and money at t2 establishes a discount factor between

the two dates, given by

Z(t1, t2) =$100

payoff at t2=

1

(1 + r/2)2

• More generally, let n be the number of times per year that interest accrues, and let

rn(t1, t2) be the (annualized) n-times compounded interest rate. Given the discount

factor Z(t1, t2), rn(t1, t2) is defined by the equation

Z(t1, t2) =1(

1 + rn(t1,t2)n

)n×(t2−t1)

Solving for rn(t1, t2) gives

rn(t1, t2) = n×

(1

Z(t1, t2)1

n×(t2−t1)

− 1

)

• The continuously compounded interest rate is obtained by increasing the com-

pounding frequency n to infinity. It is given by the formula

Z(t1, t2) = exp−rcc(t1,t2)(t2−t1)

Solving for rcc(t1, t2), we obtain

rcc(t1, t2) = − ln(Z(t1, t2)

t2 − t1

• Given a discount factor, we can define interest rates of any compounding requency by

using the above equations. In order to compare apples to apples, we often use the

effective annual rate, defined as the reciprocal of the discount factor minus one

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4.3. Coupon bonds

• US government treasury bills involve only one payment from the Treasury to the in-

vestor at maturity–their coupon rate is zero

• Knowing the price of zero coupon bonds allows us to easily determine the discount

factor for the maturity of the bond:

Pz(t1, t2) = 100 × Z(t1, t2)

• The Treasury issues zero coupon bonds for maturities up to only 52 weeks. For longer

maturities, the Treasury issues securities that carry a coupon. That is, they pay a

series of cash flows (the coupons) between the issue date an maturity, in addition to

the final principal.

4.3.1 From zero coupon bonds to coupon bonds

• Note that a coupon bond can be represented by the sequence of its cash payments

• For example, consider a 2-year 4.375% Treasury note issued in Sepctember of 2011.

What is the sequence of cash flows?

• Given the sequence of cash flows, we can compute the value of the bond if we know

discount factors for each of the four payment dates

Bt(t, T, c, n) =n∑

i=1

c× 100 × Z(t, ti) + 100 × Z(t, T ) =n∑

i=1

cPz(t, ti) + Pz(t, T )

• In the example above, suppose te 6-month, 1-year, 1.5-year, and 2-year discount factors

are 0.97862, 0.95718, 0.936826, and 0.91707, respectively. What should the price of

the bond be?

• A bond which trades at a price of $100 is said to be at par

• No arbitrage argument for bond pricing based on zero coupon rates: suppose the bond

in the above example were trading at $98. How could you take advantage of this

situation?

4.3.2 From coupon bonds to zero coupon bonds

• We can also go the other way around: If we have enough coupon bonds, we can compute

the implicit value of zero coupon bonds from the prices of the coupon bonds

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• Example: in June 2005 (t), the 6-month T-bill, expiring in December of 2005 (T1),

was trading at $98.3607. On the same date, the 1-year, 2.75% note was trading at

$99.2343, which matured at T2 = June 2006. What is the 6-month zero coupon price?

What is the 1-year zero coupon price?

• Suppose that on the same date, a 3% note maturing in December 2006 trades at

$99.1093. What is the 1.5-year zero coupon price?

• This procedure is called bootstrapping, and can be used as long as we have as many

distinct bonds (i.e. different coupons) as there are maturities for which we need to find

discount factors.

4.3.3 Expected return and yield-to-maturity

• How can we measure the expected return on an investment in Treasury securities?

• Assuming the investor will hold the bond until maturity, computing the expected return

on an investment in a zero coupon bond is straightforward (it is the reciprocal of the

discount factor, minus 1).

• For coupon bonds it is more complicated. A popular measure is yield-to-maturity

ym, which is the internal rate of return on the bond. It is defined by

Bt(t, T, c, n) =n∑

i=1

c

(1 + ym)i+

1

(1 + ym)n

• Note that the yield-to-maturity is the return the investor would earn if he holds the

bond to maturity. If he buys a 10-year note and sells it after one year, however, his

realized return may be higher or lower than what the YTM was when he bought the

note, and will depend on how interest rates change over the course of the year.

• Thus, YTM is not really the expected return from buying and holding the bond, but

is really just a convenient way of quoting a bond price.

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