Chapter 6: Valuing Bonds

Notation

  • CPN = Coupon Payment on a Bond
  • rn = Interest Rate or Discount Rate for a Cash Flow that Arrives in Period n
  • n = # of Periods
  • PV = Present Value
  • YTM = Yield to Maturity
  • NPER = Annuity Spreadsheet Notation for # of Periods or Dates of the Last Cash Flow
  • P = Initial Price of a Bond
  • RATE = Annuity Spreadsheet Notation for Interest Rate
  • FV = Face Value of a Bond
  • PMT = Annuity Spreadsheet Notation for Cash Flow
  • YTMn = Yield to Maturity on a Zero-Coupon Bond with n Periods to Maturity
  • APR = Annual Percentage Rate

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.

Coupon Payment (CPN)=Coupon Rate×Face ValueNumber of Coupon Payments per Year\text{Coupon Payment }(CPN)=\frac{\text{Coupon Rate}\times\text{Face Value}}{\text{Number of Coupon Payments per 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.

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:

P=FV(1+YTMn)nP=\frac{FV}{(1+YTM_n)^n}

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

YTMn=(FVP)1/n1YTM_n=\left(\frac{FV}{P}\right)^{1/n}-1

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

rn = YTMn

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.

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…

P(30 years to maturity)=$1001.0530=$23.14P(\text{30 years to maturity})=\frac{\$100}{1.05^{30}}=\$23.14

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

P(25 years to maturity)=$1001.0525=$29.53P(\text{25 years to maturity})=\frac{\$100}{1.05^{25}}=\$29.53

When the YTM stayed constant, as time went on the discount rate had to decrease because there’s now less time until the face value will be received. If you purchased it at $23.14 and sold it at $29.53, the IRR of your investment would be:

(29.5323.14)1/5=5.0%\left(\frac{29.53}{23.14}\right)^{1/5}=5.0\%

Your return is the same at the YTM. This is a general property for bonds: If a bond’s yield to maturity does not change, then the IRR of an investment in the bond equals its yield to maturity even if you sell the bond early.

This effect of time on price also hold for coupon bonds, but it’s a bit more complicated.

Problem: The Effect of Time on the Price of a Coupon Bond

Consider a 30-year risk-free $100 bond with a 10% coupon rate (annual payments) and an unchanging 5% (EAR) YTM. What is the effect of time on the price of this bond?

Solution: The Effect of Time on the Price of a Coupon

Initial Price=P=$10×10.05(111.0530)+$1001.0530=$176.86\text{Initial Price}=P=\$10\times\frac{1}{0.05}\left(1-\frac{1}{1.05^{30}}\right)+\frac{\$100}{1.05^{30}}=\$176.86

Now consider the cash flows of this bond in one year, immediately before the first coupon is paid.

Price is computed by discounting the cash flow by the YTM. Let’s treat the first cash flow separately.

Price before 1st Coupon=P=$10+$10×10.05(111.0529)+$1001.0529=$185.71\text{Price before 1st Coupon}=P=\$10+\$10\times\frac{1}{0.05}\left(1-\frac{1}{1.05^{29}}\right)+\frac{\$100}{1.05^{29}}=\$185.71

The price has gone up because we are no longer discounting the first cash flow and the discount rate has fallen. We could also compute the price by noting that because the YTM remains at 5% for the bond, investors in the bond should earn a return of 5% over the year: $176.86 × 1.05 = $185.71.

What happens to price right after the first coupon is paid?

Price after 1st Coupon=P=$10×10.05(111.0529)+$1001.0529=$175.71\text{Price after 1st Coupon}=P=\$10\times\frac{1}{0.05}\left(1-\frac{1}{1.05^{29}}\right)+\frac{\$100}{1.05^{29}}=\$175.71

The price falls by $10, and we see that the price is now below the initial price (which was above par). This means that the premium is declining as time goes on. An investor who buys the bond initially, receives the first coupon, and then sells it, still earns a 5% return if the YTM did not change:

10+175.71176.85=1.05\frac{10+175.71}{176.85}=1.05

Clean and Dirty Prices for Coupon Bonds

Since bond prices fluctuate around the time of each coupon payment in a jagged pattern, bond traders often do not quote the price of a bond in terms of its actual cash price (dirty price or invoice price). Instead, bonds are often quoted in terms of a clean price which is the bond’s cash price less an adjustment for accrued interest, the amount of the next coupon payment has already accrued.

Clean Price=Cash (Dirty) PriceAccrued Interest\text{Clean Price}=\text{Cash (Dirty) Price}-\text{Accrued Interest}

Accrued Interest=Coupon Amount×(Days Since Last Coupon PaymentDays in Current Coupon Period)\text{Accrued Interest}=\text{Coupon Amount}\times\left(\frac{\text{Days Since Last Coupon Payment}}{\text{Days in Current Coupon Period}}\right)

Before a coupon payment, the accrued interest will equal the full amount of the coupon. Immediately after a coupon is paid out, the accrued interest will be 0.

Interest Rate Changes and Bond Prices

As interest rates in the economy fluctuate, the yields that investors demand to invest in bonds will also change. Consider a 30-year zero-coupon $100 bond with a YTM of 5%.

P=$1001.0530=$23.14P=\frac{\$100}{1.05^{30}}=\$23.14

Suppose the interest rates suddenly rise so that investors now demand a 6% YTM.

P=$1001.0630=$17.41P=\frac{\$100}{1.06^{30}}=\$17.41

There is a substantial price drop. This illustrated a general phenomenon; higher YTM implies a higher discount rate for a bond’s remaining cash flows, reducing their present value and hence the bond’s price. Therefore, as interest rates & bond yields rise, bond prices will fall, and vice versa.

The sensitivity of a bond’s price to changes in interest rates depends on the timing of its cash flows. Cash flows to be received in the near future are less dramatically affected. Thus, shorter-maturity, zero-coupon bonds are less sensitive to changes in interest rates. Similarly, bonds with higher coupon rates (because they pay higher cash flows upfront) are less sensitive to interest rate changes than otherwise identical bonds with lower rates. The sensitivity of a bond’s price to changes in interest rates is measured by the bond’s duration. Bonds with high durations are highly sensitive to interest rate changes.

6-3: The Yield Curve and Bond Arbitrage

Replicating a Coupon Bond

It is possible to replicate the cash flows of a coupon bond using zero-coupon bonds by using the Law of One Price. Let’s replicate a three-year, $1000 bond that pays 10% annual coupons.

Because the coupon bond cash flows are identical to the cash flows of the portfolio of zero-coupon bonds, the Law of One Price states that the price of the portfolio of zero-coupon bonds must be the same as the price of the coupon bond. Assume the following prices and bond yields for zero-coupon bonds, shown in the following table.

The three-year coupon must trade for a price of $1153. If the price of the coupon bond was higher, you could earn an arbitrage profit by selling the coupon bond and buying the zero-coupon bond portfolio. If the price of the coupon bond were lower, you could earn an arbitrage profit by buying the coupon bond and short selling the zero-coupon bonds.

Valuing Coupon Bonds: Zero-Coupon Yields or Spot Rates of Interest

Recall that the YTM of a zero-coupon bond is the competitive market interest rate for a risk-free investment a term equal to the term of the zero-coupon bond. It is known as the spot rate of interest. We can use this to get the price of a coupon bond, as well. The price of a coupon bond must equal the present value of its coupon payments and face value discounted at the appropriate spot rates of interest.

Price of a Coupon Bond Using Spot Rates

P=PV(Bond Cash Flows)=CPN1+r1+CPN(1+r2)2+...+CPN+FV(1+rn)nP=PV(\text{Bond Cash Flows})=\frac{CPN}{1+r_1}+\frac{CPN}{(1+r_2)^2}+...+\frac{CPN+FV}{(1+r_n)^n}

CPN is the bond coupon payment, r­n is the spot rate calculated from the YTMn (the YTM of a zero-coupon bond that matures at the same time as the nth coupon payment), and FV is the face value. For the bond considered earlier, we can calculate the price as such:

P=$1001.035+$1001.042+$100+$10001.0453=$1153P=\frac{\$100}{1.035}+\frac{\$100}{1.04^2}+\frac{\$100+\$1000}{1.045^3}=\$1153

Basically, the information in the zero-coupon yield curve is enough to price all other risk-free bonds.

Coupon Bond Yield

Using the formula above, we can price a coupon bond. Earlier, we saw how to compute the YTM of a coupon bound from its price. Combining these, we can determine the relationship between the yields of a zero-coupon bonds and coupon bonds.

P=$1153=$100(1+YTM3)+$100(1+YTM3)2+$100+$1000(1+YTM3)3P=\$1153=\frac{\$100}{(1+YTM_3)}+\frac{\$100}{(1+YTM_3)^2}+\frac{\$100+\$1000}{(1+YTM_3)^3}

Using an annuity spreadsheet, we can solve for the YTM; it’s 4.44%. Because the coupon bond provides cash flows at different points in time, the YTM of a coupon bond is a weighted average of the yields of zero-coupon bonds of equal & shorter maturities. The weights depend on the magnitude of the cash flows each period. In this example, the zero-coupon yields were 3.5%, 4.0%, and 4.5%. For this coupon bond, most of the value in the PV calculation comes from the third cash flow because it includes the principal, so the yield is closest to the three-year, zero-coupon yield of 4.5%.

This is significant because bonds with the same maturity can have different yields depending on their coupon rates. As the coupon increases, earlier cash flows become relatively more important than later cash flows in the calculation of the PV. If the yield curve is upward sloping the resulting YTM decreases with the coupon rate of the bond. Alternatively, when the zero-coupon yield curve is downward sloping, the YTM will increase with the coupon rate. When the yield curve is flat, all zero-coupon and coupon-paying bonds will have the same yield, independent of their maturities & coupon rates.

Coupon-Paying Yield Curve

The plot of the yields of coupon bonds of different maturities is called the coupon-paying yield curve. The Bank of Canada reports the yields on 2-, 3-, 5-, 7-, 10-, and 30-year bonds.

Two coupon bonds with the same maturity may have different yields; therefore, it is important to know the specific bond issues that are used by the Bank of Canada and practitioners––these are called the benchmark bonds. Using similar methods to those employed in this section, we can apply the Law of One Price to determine the zero-coupon bond yields using the coupon-paying yield curve.

6-4: Corporate Bonds

Corporate bonds (issued by corporations), have a default risk. The issuer may default and might not pay back the full amount promised in the bond indenture. This is known as the credit risk of the bond.

Corporate Bond Yields

The most you can receive from a bond are the cash flows promised, but what you’d expect to receive may be less. As a result, investors pay less for bonds with credit risk. Since the YTM is calculated using promised cash flows, the yield of bonds with credit risk will be higher than default-free bonds.

No Default

Suppose a one-year, zero-coupon Treasury Bill has a YTM of 4%. What is the price & yield of a one-year, zero-coupon $1000 bond issued by Loblaws where all investors agree there is no possibility that Loblaw’s bond will default within the next year. The yield will be the same as the Treasury Bill!

P=$10001+YTM1=$10001.04=$961.54P=\frac{\$1000}{1+YTM_1}=\frac{\$1000}{1.04}=\$961.54

Certain Default

Now suppose investors believe Loblaw will default with certainty at the end of the year and will only be able to pay 90% of its outstanding obligations. Thus, bondholders know they will receive only $900. The $900 is risk-free, since bondholders know that they will receive it. The price falls.

P=$9001+YTM1=$9001.04=$865.38P=\frac{\$900}{1+YTM_1}=\frac{\$900}{1.04}=\$865.38

The bond’s YTM is computed using the promised rather than actual cash flows. So, we get this:

YMT=FVP1=$1000865.381=15.56%YMT=\frac{FV}{P}-1=\frac{\$1000}{865.38}-1=15.56\%

Loblaw’s YTM is higher than the Treasury Bill’s, but it does not mean that investors will earn a return a 15.56%. Since Loblaw will default, the expected return of the bond equals its 4% cost of capital:

$900$865.38=1.04\frac{\$900}{\$865.38}=1.04

Note: The YTM of a defaultable bond is not equal to the expected return of investing in the bond! It will be higher, since we use the promised cash flow instead of the expected cash flow.

Risk of Default

Following the example from above, but let’s assume that the bond payoffs are uncertain. There’s a 50% chance that the bond will repay its face value in full and a 50% chance it will default and pay $900. Thus, on average, you will receive $950.

To determine the price of this bond, we must discount this expected cash flow using a cost of capital equal to the expected return of other securities with equivalent risk. Thus Loblaw’s debt cost of capital, which is the expected return Loblaw’s debt holders will require to compensate them for the risk of the bond’s cash flows, will be higher than the 4% risk-free interest rate.

Suppose investors demand a risk premium of 1.1% for this bond, so that the appropriate cost of capital is 5.1%.

P=$9501.051=$903.90P=\frac{\$950}{1.051}=\$903.90

YTM=FVP1=1000903.901=0.1063YTM=\frac{FV}{P}-1=\frac{1000}{903.90}-1=0.1063

Of course, the 10.63% promised yield is the most investor will receive. If Loblaw default, YTM is:

Default YTM=900903.901=0.43%\text{Default }YTM=\frac{900}{903.90}-1=-0.43\%

The average return is 0.50(10.63%) + 0.50(-0.43%) = 5.1%, the bond’s cost of capital.

Summary

The bond’s price decreases & its YTM increases, with a greater likelihood of default. Conversely, the bond’s expected return, which is equal to the firm’s debt cost of capital, is less than the YTM if there is a default risk. Moreover, a higher YTM doesn’t necessarily imply a bond’s expected return is higher.

Bond Ratings

Several companies rate the creditworthiness of bonds & make this information available to investors. The 3 best-known bond-rating companies in Canada are the Dominion Bond Rating Service (DBRS), Standard & Poor’s, and Moody’s Investors Service. Bonds with the highest rating are judged to be least likely to default.

Bonds in the top four categories are often referred to as investment-grade bonds. Bonds in the bottom five categories are often called speculative bonds, junk bonds, or high-yield bonds. The rating depends on the risk of bankruptcy as well as the bondholders’ ability to lay claim to the firm’s assets in the event of such a bankruptcy.

Corporate Yield Curves

We can plot a yield curve for corporate bonds.

We refer to the difference between the yields of the various bonds and the Government of Canada yields as the default spread or credit spread. Credit spreads fluctuate as perceptions regarding the probability of default change. The higher it is, the more default risk there is.

6-5: Sovereign Bonds

Sovereign bonds are bonds issued by national governments. While Government of Canada bonds are generally considered to be default free, the same cannot be said for bonds issued by many other countries. In 2012, Greece defaulted and wrote off over $100 billion, or about 50%, of its outstanding debt, in the largest sovereign debt restructuring in world history.

Because most sovereign debt is risky, the prices and yields of sovereign debt behave much like corporate debt: The bonds issued by countries with high probabilities of default have high yields & low prices.

However, there’s a difference. Unlike a corporation, a country facing difficulty meeting its financial obligations typically has the option to print additional currency to pay its debts. Doing so is likely to lead to high inflation & a sharp devaluation of the currency, however. So, debt holders carefully consider inflation expectations when determining the yield they are willing to accept because they understand that they may be repaid in money that is worth less than it was when the bonds were issued.


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