BOND PRICES AND YIELDS

FIN 366: INVESTMENTS © JOSEPH FARIZO

DEBT AND EQUITY

Firms finance operations and growth through a mixture of borrowing (liabilities) and through offering ownership interest (equity).We’ve talked at length about equities, including:

• their markets (IPOs, primary, secondary)

• trading strategies (short sales, margin)

• securities that derive their value from equities (derivatives- calls and puts)

• valuation (dividend discount models, FCFE, comps)

Now we consider firms’ longer-term borrowing, the other component of the firm’s capital structure or mix of debt and equity.

BOND TERMS AND CHARACTERISTICS

CORPORATE BOND BASICS

A corporate bond is a security that obliges the issuer (the borrower) to pay specified interest

payments and principal to the bondholder (the lender) over a period.

• Par or face value or principal: the amount borrowed by the firm, to be repaid at maturity

• Coupon: periodic (most frequently semiannual or twice yearly) interest payments, expressed as a percentage of par, paid over the life of the bond

Features of the bonds are disclosed in the bond’s indenture, the contract between the issuer and lender.

EXAMPLE: A bond with a $1,000 par value, a coupon rate of 8%, and a 30-year term is issued at

par for $1,000.

The issuer pays the bondholder 8% of $1,000 = $80 each year in two $40 semiannual installments.

The borrower or issuer pays the lender or bondholder $40 for every 6 months for 30 years, then

pays back the par, face, or principal of $1,000 at maturity.

Notably, bonds are tradable. Coupon payments and principal at maturity are made to the holder of

the security. The firm itself does not realize any profits on transactions on the secondary market to

which it is not a party, similar to common stock transactions.

Given bonds are tradable, there is an active secondary market. The price of the bond can be

determined based on its characteristics (its cash flow, principal, and maturity.)

Figure 1: Bond Example

THE CORPORATE BOND MARKET

Why would a bond investor be willing to pay $116 for a bond that pays only $100 at

maturity?

Price is a percentage of

par: $116.28 indicates the

buyer paid 116.28% of the

bond’s par to another bond

trader for this bond.

Figure 2: Bond Markets and Trading

Supply and demand forces

result in price changes

through time.

The Financial Industry Regulatory Authority’s (FINRA’s) Bond Center presents bond

market and trading data.1

Most bond trading takes place over the counter (OTC) electronically among dealers. Transactions

are often privately negotiated between parties, with little centralized reporting of trading or

transactions. A firm can issue several bonds with varying maturities and characteristics.

US corporate bond issuance in 2020 raised $2.28 trillion in new borrowing, nearly 6 times as much

as equity issuance raised through IPOs and secondary offerings ($390 billion). The market value

of US corporate bond debt was $10.5 trillion relative to the equity market value of $65.2 trillion.

The average daily trading volume of corporate bonds is $38.9 billion while equities is $479 billion.

The Securities Industry and Financial Markets Association’s (SIFMA’s) Capital Markets

Factbook presents bond market and trading data.2

Individual or retail investors may invest in bonds through a broker, but given the higher costs,

illiquidity, and inability to easily diversify relative to equities, the easiest way to invest is through

a bond mutual fund or asset allocation mutual fund. Vanguard3, Fidelity4, and BlackRock5 each

have options for investors.

BOND PROVISIONS

Convertible bonds allow the bondholder (lender) to exchange the bond for a specified number of

shares of common stock in the firm.

If you are a bond holder, would you want this opportunity? What do we expect the coupon

on this debt would be relative to bonds that are not convertible, all else equal?

Callable bonds may be repaid early by the issuer (borrower) at their discretion.

If you are a bond holder, would you want this call provision on the bond you hold? What

do we expect the coupon on this dept would be relative to bonds that are not callable, all

else equal?

Example: A firm issued 30-year callable bonds 10 years ago, and it agreed to pay a coupon of 3%

annually (in semiannual installments) on the principal. Since then, interest rates in the market

overall have fallen significantly. The firm calls those 3% coupon bonds by paying back the existing

bondholders 20 years early. It then issues new bonds with a coupon of 1.0%, lowering the firm’s

cost of their debt.

Puttable bonds allow the lender/bondholder, at their discretion, to demand early repayment of the principal or extend the bond past its maturity.

If you are a bond holder, would you want this opportunity? What do we expect the coupon

on this debt would be relative to bonds that are not puttable, all else equal?

Example: 10 years ago, you bought a 30-year corporate bond with a 5% coupon (paid in

semiannual installments). You notice that since that time, interest rates have risen, and other

similar debt is being issued by firms that are paying 7% coupons. You require the firm pay you

early on your 5% bonds so that you can invest in bonds with the 7% coupon.

BOND PRICING

The value of a bond is the present value of coupon payments plus the present value of the par value

paid at maturity.

𝐵𝑜𝑛𝑑 𝑉𝑎𝑙𝑢𝑒 = ∑𝐶𝑜𝑢𝑝𝑜𝑛 𝑃𝑎𝑦𝑚𝑒𝑛𝑡𝑠

(1 + 𝑟)𝑡

𝑇

𝑡=1

+𝑃𝑎𝑟 𝑉𝑎𝑙𝑢𝑒

(1 + 𝑟)𝑇

Notice that the coupon rate is used to determine the coupon payments. The discount rate

or yield is used to determine the present value or price. We will discuss the discount rate

and yields in detail later.

PROBLEM: What is the value of an 8% coupon bond, making semiannual interest payments, that

matures in 30 years with a par or face value of $1000? Assume we discount our cash flows at a

10% annual rate.

SOLUTION: By the formula

𝐵𝑜𝑛𝑑 𝑉𝑎𝑙𝑢𝑒 = ∑𝐶𝑜𝑢𝑝𝑜𝑛 𝑃𝑎𝑦𝑚𝑒𝑛𝑡𝑠

(1 + 𝑟)𝑡

𝑇

𝑡=1

+𝑃𝑎𝑟 𝑉𝑎𝑙𝑢𝑒

(1 + 𝑟)𝑇

And given our

• Coupon Payments are ($1,000 × 8%) ÷ 2 = $40

• Discount rate r is 10% ÷ 2 = 5%

• Number of periods T = 30 years × 2 periods per year = 60

We have:

𝐵𝑜𝑛𝑑 𝑉𝑎𝑙𝑢𝑒 = ∑$40

(1 + .05)𝑡

60

𝑡=1

+$1000

(1 + .05)60= $810.71

Where

∑$40

(1 + .05)𝑡

60

𝑡=1

=$40

(1 + .05)1+

$40

(1 + .05)2+ ⋯ +

$40

(1 + .05)60

Given corporate bonds often have long maturities, doing this calculation in a financial calculator

or in Excel is practically a necessity.

We will assume in all cases, unless otherwise stated, that coupon payments are received at the end

of the period. This is most common in practice. (If you calculator has BGN above the display, it

assumes payments at the beginning. Toggle this option with the keystrokes 2ND BGN 2ND SET.)

BOND YIELDS

DISCOUNT RATE

The discount rate or yield used in bond pricing often depends on the prevailing market rate and

bond characteristics. Consider:

• Rates on bonds of similar risk and similar characteristics (opportunity cost)

• Unique features of the bond (callability, puttable)

• Default risk

For example, you purchased a 30-year 6% coupon bond 5 years ago from AirRide, a large

commercial aircraft manufacturer. Now, 5 years later, interest rates in the economy have risen.

AirRide’s primary competitor, Jet Stream Aviation, issues 30-year bonds paying 10% coupons at

par. Assuming AirRide and Jet Stream Aviation are similarly risky and the bonds have similar

characteristics, you might use that 10% as your 6% coupon bond’s discount rate.

If you are the holder of the AirRide 6% coupon bond, what do you think happens to the

value of your bond if other similar bonds are now paying 10% coupon?

2ND

Figure 3: Bond Price Calculator Inputs

CLR TVM

N

I/Y

FV

PMT

CPT

Number of periods = 60

PV

Discount rate = 5%

Face or par = $1000

Coupon payments = $40

Price = Present value = -$810.71 (negative, an outflow)

YIELD TO MATURITY

A bond’s yield to maturity (YTM) is the annual discount rate that make the present value of the

bond equal to its price. It is an average annual compound rate of return over the life of the bond,

assuming you purchase the bond at the price and reinvest all coupon payments at that same yield.

PROBLEM: What is the YTM of an 8% coupon bond, making semiannual interest payments, that

matures in 30 years with a par or face value of $1,000 that currently trades at $810.71?

SOLUTION: In this case, we solve for r in the formula:

𝐵𝑜𝑛𝑑 𝑉𝑎𝑙𝑢𝑒 = ∑$40

(1 + 𝑟)𝑡

60

𝑡=1

+$1000

(1 + 𝑟)60= $810.71

Note that the YTM must be expressed as the annual rate by multiplying the calculator

output by 2 (if semiannual payments), by 4 (if quarterly payments), etc.

2ND

Figure 4: Bond YTM Calculator Inputs

CLR TVM

N

PV

FV

PMT

CPT

Number of periods = 60

I/Y

Price = -$810.71

Face or par = $1000

Coupon payments = $40

=5 × 2 = YTM = 10.0%

The Excel file Bond Price Calculator available at josephfarizo.com/fin366.html provides a

useful time value of money calculator for you to work additional practice problems.

CURRENT YIELD

The current yield is the total of the bond’s coupons paid over a year divided by its current price.

It provides a useful point of comparison across bonds: how does the bond’s total annual payment

relate to the current price?

Example: The current yield for a $1,000 bond paying $40 coupons semiannually that trades for

$1,276.76 would be 80 ÷ 1276.76 = 6.276%.

YIELD TO CALL

Firms pay a call price rather than the principal on callable bonds when they are retired prior to

maturity. On callable bonds, investors may be interested in determining the bond’s yield to call

(YTC) if the likelihood of the bond being called is high.

In what interest rate environment is a firm more likely to call its callable bond?

The calculation is similar to computing the YTM, except the time to maturity and the par are

updated to reflect the terms of the callable bond.

Example: An investor purchases a $1,000 par value 9% coupon bond (paid semiannually),

maturing in 20 years for $1,098.66. The bond is callable in 5 years at a call price of

$1050.

YTM: N = 40 , FV = 1000 , PMT = 45, PV = -1098.66

<CPT> I/Y × 2 = 8%

YTC: N = 10 , FV = 1050 , PMT = 45, PV = -1098.66

<CPT> I/Y × 2 = 7.44%

Similar to the YTC is the yield to worst (YTW) which is computed for the most unfavorable terms

permitted in the bond’s indenture (called at earliest possible date). The YTC and YTW may be the

same.

HOLDING PERIOD RETURN

Investors may also with to determine the holding period return associated with buying and selling

bonds prior to maturity. As before, the HPR will account for the income and capital gains

associated with the investment. If yields or prevailing interest rates change, the bond price changes

as well.

𝐻𝑃𝑅 =𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝐼𝑛𝑐𝑜𝑚𝑒 + 𝐵𝑜𝑛𝑑 𝑃𝑟𝑖𝑐𝑒𝑁𝑒𝑤 − 𝐵𝑜𝑛𝑑 𝑃𝑟𝑖𝑐𝑒𝑂𝑙𝑑

𝐵𝑜𝑛𝑑 𝑃𝑟𝑖𝑐𝑒𝑂𝑙𝑑

Note that we make significant simplifying assumptions here (i.e., coupons are the end of the

holding period with no reinvestment or taxes).

PROBLEM: An investor purchases an 8% coupon, 30-year, $1,000 bond making semiannual

payments at a yield of 6%. They hold this security for 3 years and sell when yields are 5%.

SOLUTION: To find the HPR, we compute the bond’s beginning and ending price as well as the

sum of the coupons received:

Purchase Sale

N 30 years × 2 = 60 (30 years - 3 years held) × 2 = 54

I/Y 6% ÷ 2 = 3% 5% ÷ 2 = 2.5%

FV $1000 $1000

PMT ($1000 × 8%) ÷ 2 = $40 ($1000 × 8%) ÷ 2 = $40

PV <CPT> = $1,276.76 <CPT> = $1,441.85

𝐻𝑃𝑅 =( ) + ( − )

( )= 31.72%

INTERPRETATION: This investor’s return was approximately 31.72% given they transacted in this

bond prior to its maturity. Notice this differs from the coupon rate and other yields.

The Excel file Yields and Returns available at josephfarizo.com/fin366.html helps to

illustrate the computation of different yields and HPRs for a bond with inputs that you

provide.

THE RELATIONSHIP BETWEEN BOND PRICES AND YIELDS

Firms often issue bonds at prevailing market rates. That is, they are issued such that their coupon

rate equals their yield, or coupon rate = discount rate.

What is the price of a bond that is issued with a coupon rate equal to its discount rate? Hint:

you can prove this for yourself in your calculator.

Bonds are issued at par when the coupon rate equals the discount rate. As prevailing market rates

change, the price of already existing bonds changes:

• Prevailing interest rates higher than a previously issued bond: previously issued

bonds become less attractive. Investors sell and prices fall.

• Prevailing interest rates lower than an already issued bond: previously issued bonds

become more attractive. Investors buy and prices rise.

This leads to an important point regarding bond prices and yields:

BOND PRICES AND YIELDS ARE INVERSELY RELATED.

Consider a simple example: Which bond offers a greater yield to an investor? One purchased for

$1 that pays $1,000 tomorrow, or one purchased for $999.99 that pays $1,000 tomorrow?

AS INVESTORS BID UP BOND PRICES, THE YIELD TO HOLDING THEM FALLS. AS BOND PRICES FALL,

THE YIELD TO HOLDING THEM RISES.

PREMIUM AND DISCOUNT BONDS

Premium bonds have a price greater than par. Additionally:

Premium Bonds : Coupon % > Current Yield > YTM

Discount bonds have a price less than par. Additionally:

Discount Bonds : Coupon % < Current Yield < YTM

Additionally, bonds that pay no coupons, or zero-coupon bonds, qualify as discount bonds.

Given they pay no coupon, they often are issued at and sell at discounts.

Example: Assume there is a $1,000 par value 10-year zero-coupon bond issued for $900. In the

calculator, we input N = 10, PV = -900, FV = 1000 <CPT> I/Y = 1.0592%. The YTM

is 1.0592%, the average rate of return earned if bought at $900 and held until maturity

when you pay $1,000. Notice: 900 × (1+0.010592)10 = 1000. That is, if you buy at $900

and earn $1,000 in ten years at maturity, you’ve made 1.0592% per year with

compounding.

Yields, Discount Rates, YTM Recap

Given the numerous yields, returns, rates, and relationships associated with bonds, it’s helpful to

summarize each in a table where they can be easily compared.

Figure 5: Bond Returns and Rates Summary Table

Term Definition

Coupon Rate Annual rate (often paid semiannually) as % of par

Yield Discount rate of the coupon payments

Yield to Maturity Annual yield equating price and present value of bond payments

Yield to Call Yield assuming bond is called early at the call price

Current Yield Annual coupon payments divided by current price

HPR Bond return over a period given change in price and coupons received

Premium Price > Par or CR>CY>YTM

Discount Price < Par or CR<CY<YTM

At Par Price = Par or CR = CY = YTM

NOTES

END NOTES

1 FINRA’s Bond Center: http://finra-markets.morningstar.com/BondCenter 2 SIFMA’s Capital Markets Fact Book: https://www.sifma.org/resources/research/fact-book/ 3 Vanguard VLTCX Bond Fund: https://investor.vanguard.com/mutual-funds/profile/portfolio/VLTCX/portfolio-

holdings 4 Fidelity FCBFX Bond Fund: https://fundresearch.fidelity.com/mutual-funds/summary/316146596 5 BlackRock BFMCX Bond Fund: https://www.blackrock.com/us/individual/products/227526/blackrock-core-

bondinstitutional-class-fund