Department

ManagementCourse Code

MGT338H5Professor

Gabor ViragChapter

6This

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Chapter 6 – Valuing Bonds

Chapter 6 – Valuing Bonds

Bond Cash Flows, Prices, and Yields; Dynamic Behaviour of Bond Prices;

Yield Curve and Bond Arbitrage; Corporate Bonds; and Sovereign Bonds

→ NOTATION

→ 6-1: BOND CASH FLOWS, PRICES, AND YIELDS

Bond Terminology

A bond is a security sold by governments & corporations to raise money from investors in exchange for

promised future payment. The terms of the bond are described as part of the bond indenture, which

indicates the amounts & dates of all payments to be made. These payments are made until a final

repayment date, the maturity date. The time left till repayment is the term of a bond.

Bonds typically make 2 types of payments to their holders. The promised interest payments of a bond

are called coupons. The principal or face value of a bond is the notional amount we use to compute

the coupon payments. Usually, the face value is repaid at maturity. The amount of each coupon payment

is determined by the coupon rate, which is the percentage of the face value that is paid as coupons

each year.

Example: A “$1000 bond with a 10% coupon rate and semiannual payments” will pay coupon

payments of (10% × $1000)/2 = $50 every six months.

Zero-Coupon Bonds

The simplest type of bond is a zero-coupon bond, a bond that does not make coupon payments and

only pays the face value at maturity. Treasury Bills, which are Government of Canada bonds with a

maturity of up to one year, are zero-coupon bonds. Zero-coupon bonds always trade at a discount (a

price lower than the face value), so they are also called pure discount bonds.

Yield to Maturity

The IRR of an investment opportunity is the discount rate at which the NPV of the investment

opportunity is equal to zero.

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Chapter 6 – Valuing Bonds

The IRR of an investment in a zero-coupon bond is the rate of return that investors will earn on their

money if they buy the bond at its current price and hold it to maturity. The IRR of an investment in a

bond is given a special name, the yield to maturity (YTM) or just the yield:

The yield to maturity of a bond is the discount rate that sets the present value of the

promised bond payments equal to the current market price of the bond.

Suppose you pay $96,618.36 for a $100,000 one-year, risk-free, zero-coupon bond.

Thus, by the Law of One Price, the competitive market risk-free interest rate is 3.5%. That means all

one-year, risk-free investments must earn 3.5%.

Yield to Maturity for Multiple Periods

The YTM for a zero-coupon bond with n periods to maturity, current price P, and face value FV is:

Rearranging, we get the Yield to Maturity of an n-Year Zero-Coupon Bond:

This is the effective rate of return per period for holding the bond from today until maturity on date n.

Spot Rates of Interest

The spot rate of interest is the risk-free interest rate. Because a default-free zero-coupon bond that

matures on date n provides a risk-free return over the same period, the Law of One Price guarantees

that the spot rate of interest equals the yield to maturity on such a bond.

Risk-Free Interest Rate (Spot Rate of Interest) with Maturity n

The yield curve plots the risk-free interest rate for different maturities, which corresponds to the yields

of risk-free zero-coupon bonds. Thus, the yield curve is also called the zero-coupon yield curve.

Coupon Bonds

Coupon bonds make regular coupon interest payments on top of the face value at maturity. The yield

to maturity for a bond is the IRR of investing in it and holding it to maturity; it is the single discount

rate that equates the present value of the bond’s remaining cash flows to its current price.

The coupon payments represent an annuity. The YTM is the interest rate y that solves the following

equation for the Yield to Maturity of a Coupon Bond.

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Chapter 6 – Valuing Bonds

Unfortunately, there is no simple formula to solve for the YTM directly. We need to use a financial

calculator, trial and error, or the annuity spreadsheet. When the equation is solved, the yield will be an

effective rate per coupon interval. This yield is typically multiplied by the number of coupons per year

and stated as an annual rate as an APR.

If we know the YTMn, we can calculate the price of the bond using the formula above.

Conventions

We can convert any price into a yield, so yield & prices are interchangeable. Bonds are often quoted

with their YTM, since it easier to use for comparisons. Prices can be quoted as well, such as $944.98

per $1000 face value. To make it easier to compare, prices are often quoted as a percentage of their

face value; 94.498.

→ 6-2: DYNAMIC BEHAVIOUR OF BOND PRICES

Coupon bonds may trade at a discount (a price lower than their face value), at a premium (a price

greater than their face value), or at par (a price equal to their face value).

Discounts and Premiums

Bonds trading at a discount mean an investor will earn a return from both the coupons and the face

value. It’s YTM exceeds its coupon rate. The reverse is also true; If a coupon bond’s YTM exceeds its

coupon rate, the PV of its cash flows at the YTM will be less than its face value, and the bond will trade

at a discount.

Bonds trading at a premium mean an investor’s return from coupons is diminished by receiving a face

value less than the price paid for the bond. It’s YTM is less than its coupon rate.

Bonds trading at par means the price equals the face value, and the YTM is equal to the coupon rate.

Summary: Bond Prices Immediately After a Coupon Payment

Most issuers of coupon bonds choose a coupon rate so that the bonds will initially trade at, or very

close to, par or face value.

Time and Bond Prices

Suppose you purchase a 30-year, $100 zero-coupon bond with a yield to maturity of 5%. Initially…

Assume that the YTM remains at 5%. 5 years later, the bond price will be…

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