MF807 Corporate Finance

Prof. Thomas Chemmanur

Topic Note 2

1. Bond Valuation

A bond represents borrowing by firms from investors.

F º Face Value of the bond (sometimes known as "par value"); this is the amount the company will pay back at the maturity of the bond.

Ct º Coupon or interest payment at date t. Usually coupon amounts are the same at each date, in which case we will simply use C for coupon. In this latter case, coupon is often expressed as a percentage c of the face value. This percentage is called "coupon rate". Then, C = c.F, where c is expressed in decimals. Notice that the coupon rate is specified as part of the bond (it is not something that changes according to economic conditions).

n º The number of periods to maturity.

r º The rate of return per period at which the bond cash flows are to be discounted, determined by financial market conditions. For the present, we will assume that cashflows at all dates are to be discounted at the same rate. r changes as the general level of interest rates in the economy changes.

The price that you are willing to pay for the bond is simply the present value of all the cash flows that the bond entitles you to; Thus,

Usually coupon payments are made semi-annually; this means that the length of a period is a half-year, and r should be in terms of return per half-year; n should also be in half-years.[1]

If the price of the bond is less than the face value, we say that the bond is selling at a "discount"; if they are the same, the bond is selling at par; if the price is higher than the face value, it is selling at a "premium".

The following facts are of interest:

i) When c = r, the bond sells at par.

ii) When c > r, the bond sells at a premium (provided n > 0). When c < r, the bond sells at a discount (provided n > 0).

iii) When r goes up, the price of bonds fall; when r goes down, the price of bonds go up. This is because the present value of any cash flow stream goes down (up) as the discounting rate increases (decreases).

iv) Longer the time to maturity of a bond, the more sensitive the bond price to interest rate changes.

v) The market price of a bond approaches its face value as its maturity date approaches (i.e., as n goes to 0). At n = 0, P = F.

2. Yield to Maturity (YTM)

The yield to maturity of a bond is the rate of return we would earn if we bought the bond at a price P and held it till it matures. i.e., it is that rate of return r that solves the equation (1) for a known P.[2] Clearly YTM will be different for different values of P. (Usually, yield to maturity is computed and quoted in the financial press assuming semi-annual compounding).

3. The "Term-Structure" of Interest Rates

Often, the discounting rate to be applied to a one year loan is different from that for a two year loan, etc. The relationship between the length of a loan (or, to talk in terms of a bond, the time to maturity) and the rate of return you can earn on a loan (the yield to maturity in the case of a bond) is called the term-structure of interest rates or the yield curve. To make this relationship clear, we will think in terms of 'spot rates.'

A spot rate is the rate of return on a loan or a bond which has only one cash flow to the investor: the investor makes a loan of an amount P and gets back the amount F at the maturity of the loan. Remember that this is exactly the same as investing in a bond with a zero coupon rate, which has a current price P and face value F. Thus the one-year spot rate is the yield to maturity on zero coupon bonds (sometimes referred to as 'pure discount' bonds) of one year maturity. Let us denote the one year spot rate by r1. Then, the price of a one-year maturity zero-coupon bond with face value F is given by

Similarly, the two-year spot rate r2 is the rate of return on a two-year loan, or equivalently, the yield to maturity on a two year zero-coupon bond. Then, the price of a two-year zero coupon bond is,

We can write down similar relationships between the prices of a three year pure discount bond, a four year zero coupon bond etc., and the corresponding spot rates r3, r4, r5 etc.

Now, if r1 < r2 < r3 ...etc., the term structure is uniformly upward sloping, which is very often the case. On the other hand, if r1 > r2 > r3... the term structure is uniformly downward sloping. The term structure is 'flat' if r1 = r2 = r3 ...etc. The term-structure can also take other shapes as well.

Once we know all the spot rates, we can price any riskless bond with cash flows of any magnitude and at arbitrary points in time. This is because we can show that all cash flows of similar riskiness occurring at the same point in time should be discounted at the same rate. Thus, consider a bond with coupons C1, C2, C3, ...Cn occurring at time periods t=1, 2, 3, ...n. In addition, let the face value of the bond be F. The price of the bond is then given by,

4. The law of one price

What happens if the bond price is different from the above computed price? We can show that in that case, 'the law of one price' is violated, and there are opportunities for making arbitrage profits which can be taken advantage of by investors, and the price of the bond will be driven back to the above price. This is a consequence of the law of one price which states that securities (or portfolios of securities) which have the same riskiness, and which entitle the holder to the same stream of cash flows should have the same current price. Whenever this law is violated, there are arbitrage opportunities which can be exploited by investors, and which drive prices back to conformity with this law. Remember that whenever there are arbitrage opportunities, the security markets cannot be in equilibrium: thus the equilibrium price of the above bond should be given by the above formula.

Problem Consider the following bonds:

Bond Cash flow at date

1 2 Price

A 80 1080 982

B 1100 --- 880

C 120 1120 1010

Is there is an arbitrage opportunity? If so, what price for bond C will eliminate it?

5. Value of a Perpetuity

If a bond pays a coupon C each period for ever (no face value), its market value (at a discounting rate r) is, P = C/r, since the coupon stream forms a perpetual annuity.

6. Value of Preferred Stock

"Preferred stock" usually promise a fixed payment for ever (however, unlike corporate bonds, the company can miss payments without going bankrupt). Thus they can be treated as a perpetuity: P = C/r. Here C denotes the "preferred dividend".

[1] For consistency, bond traders apply semi-annual compounding to the par value also.

[2] The assumption here is that you are able to re-invest all coupons you receive also at the rate of return r.