MONEY & BANKING (Eco 340)

Prof. Ranjit Dighe

Lecture notes to accompany Cecchetti’s Chapter 6

(Bonds, Bond Prices, and the Determination of Interest Rates)

Last revised 12-March-2008.

In these notes:

I. Bonds: An introduction

II. Bond prices and PDV

III. How changes in market interest rates affect bond prices

IV. Yield to maturity, or Internal rate of return

V. Understanding changes in interest rates: Supply & demand in the bond market

— A. Introduction: Bond prices are inversely related to interest rates

— B. Asset demand in general: The Theory of Asset Demand

— C. Shifts in the demand for bonds

— D. Shifts in the supply of bonds

— E. Summary and more examples

VI. Supply & demand in the bond market: Applications

— A. Effect of changes in the expected rate of inflation on interest rates

— B. Effects of business-cycle expansions (and recessions) on interest rates

I. BONDS: AN INTRODUCTION

Recall that a bond is a formal IOU, with a promise to repay an initial amount borrowed plus a fixed amount of interest, according to a particular schedule.

— The market price of a bond is the present-day value of its future payments (ignoring for now such considerations as default risk and tax treatment).

There are two main types of bonds — coupon bonds and discount bonds (or zero-coupon bonds ).

(1) COUPON BONDS

— These sell at (or near) their face value ( at par) . So a $1000 coupon bond would sell for about $1000. The reason someone would want to own it is for the coupons (regular interest payments).

— A coupon bond offers a particular interest rate, i _coupon. also known as the coupon rate. or CURRENT YIELD.

Coupon rate = dollar value of the yearly coupon payment, divided by the face value of the bond.

Annual coupon payment = i _coupon * (face value of bond)

Ex. A $1000-face-value coupon bond that pays a 5% coupon rate will make annual coupon (interest) payments of

05 * $1000 = $50

Equivalently, a $1000-face-value coupon bond that pays a $50 annual coupon has a coupon rate of

Some coupon bonds (such as U.S. Treasury coupon bonds) are sold at auction and, as auction prices are not set in advance, a those coupon bonds will not necessarily sell for exactly their face value. Unless the coupon rate and the market interest rate are exactly the same, a $1000 coupon bond would sell for a bit more or a bit less than $1000. If so, the effective interest rate on the bond will be a bit different from its coupon rate.

Most Treasury bonds and corporate bonds are coupon bonds.

(2) DISCOUNT BONDS — These sell at less than their face value (FV) and pay no interest/coupons until the bond matures. Ex. a U.S. savings bond.

The price (P_d ) and interest rate (i_d ) of a new discount bond are jointly determined. Whatever price the bond sells for, we can calculate its interest rate by applying the internal-rate-of-return formula we learned earlier:

Interest rate on an n -year discount bond = i _d = < [(FV/P_bond ) (1/n) ] — 1 > (*100% )

The simplest case is a 1-year discount bond, since the 1/n term becomes 1 and therefore drops out:

Interest rate on a one-year discount bond = i _d = [(FV/P_bond ) — 1 ] (*100% )

Ex. a one-year, $1000 discount bond that sells for $900 would have an interest rate of

i _d = ($1000)/($900)^(1/1) — 1 = 1.111 — 1 = .111 = 11.1%

Ex. a two -year, $1000 discount bond that sells for $900 would have an interest rate of

i _d = ($1000)/($900)^(1/2) — 1 = (1.111)^(1/2) — 1 = 1.054 — 1 = .054 = 5.4%

(The bond’s face value is a constant, by the way. For a discount bond, the variables are the price and the interest rate.)

Just as the formula for i_d is the same as the internal-rate-of-return formula, a discount bond’s price P_d is equal to its present-day value (PDV. And recall that the PDV and internal-rate-of-return formulas are themselves just rearrangements of the future value (FV) formula).

P_d = (Face Value) / [(1+ i_d ) n ]

—> A discount bond’s price and its interest rate have an exact inverse relation. The higher the interest rate, the lower the bond’s initial price (the greater the discount ) relative to its face value.

II. BOND PRICES AND PDV

Again, the appropriate price of a bond is its PDV. (PDV, after all, is defined as what it’s worth now. )

For a discount bond, calculating the PDV is simple, because there is just one payment. We can just apply the basic PDV formula. It’s identical to the previous formula, except it uses the market interest rate i instead of the bond’s interest rate i_d (these will be the same anyway, because in a competitive bond market all bonds with the same attributes will offer the same interest rate — this is the law of one price from Econ 101)

PDV_{discount bond} = (Face Value)/[(1+i) n ]

PDV of 1-year discount bond with face value (FV) of $1000 = $1000/[1.05 1 ] = $1000/1.05 = $952.38

For a coupon bond, calculating the PDV is more complicated, because there are multiple payments, and they are made at different times. Recall that the PDV of any future payment is the dollar amount of the payment divided by (1+i) n (which is the compounded gross interest rate), which implies that you have to compute the PDV’s of each of those future payments separately and then add them up:

[Refer to the in-class handout, Everything You Always Wanted to Know About PDV but Were Afraid to Ask. The appropriate formula for a coupon bond is number (2).]

Exs.:

PDV of a 1-year coupon bond with FV of $1000 and coupon payment of $50 (made 1 year from now), if the market interest rate i is 5% =

PDV of a 2-year coupon bond with FV of $1000 and coupon payments of $50 (made at the end of each year) if the market interest rate i is 5% =

— Again, PDV = FV = $1000.

As long as the coupon rate ( i _coupon ) is the same as the market interest rate i. the PDV of the coupon bond will be the same as the bond’s face value, and the coupon bond will sell at its face value.

— This is true no matter how long or how short the bond’s term length is. A longer-term bond like a 30-year bond pays out more money, but you get most of it, especially the final payment of principal, much later than you would with a short-term bond like a 1-year bond. The larger amount of payments with a long-term bond is exactly canceled out by the fact that the later payments have lower PDV’s.

—- Remembering this fact could save you a lot of computations in the future.

—- If the market interest rate were higher than the coupon rate, then the bond’s PDV or resale value would be less than its face value and you would need to go through the full battery of computations. Likewise, if the market interest rate were lower than the coupon rate, then the bond’s PDV would be higher than its face value.

Special case of coupon bonds: Consols

Time for an auction. On the block: I and my descendants will pay you and your descendants $50 every year, forever and ever. What would you pay for that? (Assume that you care just as much about your descendants as you do about yourself. Also assume the market interest rate is 5%.)

— This is actually a special type of bond called a CONSOL (a bond that pays a coupon forever but never makes that final lump-sum payment of the principal). Like any other bond, its appropriate price is its PDV. But calculating its PDV, at first blush, would seem to involve adding infinitely many terms:

PDV of getting $50 per year forever, when i = 5%:

= $50/(1+.05) + $50/(1+.05) 2 + $50/(1+.05) 3 +.

There is a shortcut, however, because those terms get smaller and smaller as n gets larger (e.g. the PDV of a $50 payment to be received 100 years from now is only 38 cents), and their sum converges into a finite number. A series like this one is a geometric sum. which you may have learned about in high school. Here is the general formula [see number (3) on the Everything You Always Wanted to Know. handout]:

PDV of a fixed payment (FP) in perpetuity (i.e. forever):

PDV = FP/(1+i) + FP/(1+i) 2 +. + FP/(1+i) n +.

= FP * 1/i

= FP/i

So that $50 perpetual yearly payment would be worth, if the market interest rate is 5%,

PDV = $50/.05 = $1000,

the same as a 1-, 2-, or n- year coupon bond making a $50 annual coupon payment when i = 5%.

It’s been a couple centuries since anyone actually did auction off consol bonds in any significant quantity, but this formula does have some very practical uses, such as for pricing stocks.

[Not covered yet, but possibly of interest: For something that pays a fixed amount over a finite time span — like, say, a 10-year, $10-million-a-year NBA contract — finding the PDV the standard way would involve adding up 10 separate terms and would be very time-consuming. There is, however, a shortcut, which makes use of that consol formula. Let us denote those fixed yearly payments as FP, for fixed payment. The time-consuming way to compute the combined PDV of all those payments over n years is PDV = FP/(1+i) + FP/(1+i) 2 +. + FP/(1+i) n

(formula (2) on PDV handout). The shortcut involves applying the consol formula; we can calculate the PDV of an n -year series of fixed payments (FP) as the difference between the PDV of a consol that pays $FP per year forever and a consol that pays $FP per year starting n+1 years from now. See formula (4) on the PDV handout.]

[Ex. A college hoops star signs a 10-year, $10-million-a-year NBA contract, starting one year from now. Total payments add up to $100 million, but PDV will be considerably less, since all of those payments are in the future. What is the PDV (if i =.05)?

PDV = ($FP)/i * [ 1 — 1/(1+i ) n ]

= $10/.05 * [ 1 — 1/(1+.05) 10 ] (in millions)

= $200 * [ 1 — 1/1.6289] (in millions)

= $200 * (1 — .6139) (in millions)

= $200 * .3861 (in millions)

= $77.2 million]

III. HOW CHANGES IN MARKET INTEREST RATES AFFECT BOND PRICES

Ex. Consider a 1-year, $1000-face-value discount bond. Recall: P_d = (Face Value) / [(1+ i_d ) n ]. Let’s consider a range of possible interest rates and what the bond’s price will be at each interest rate: