5: Bond Valuation∗

Dr. Bhanu Pratap Singh†

IIM Sirmaur, 2024

Learning Objectives
• Bonds & Bond Valuation.
• Inflation & Interest Rates.
• Determinants of Bond Yields.

Introduction
• Corporations and governments frequently borrow money by issuing or selling debt securities
called bonds.

• These bonds can be valued using the cashflows associated with these bonds.
– By discounting these cashflows.
– Hence, bond values depend, in large part, on interest rates.

Bonds & Bond Valuation

Bond Features & Prices

• A bond is normally an interest-only loan.

– Borrower pay interest every period.
– The principal is repaid only at the end of the loan.

• Example: Beck Corporation wants to borrow $1,000 for 30 years.

• The interest rate on similar debt issued by similar corporations is 12%.

• Beck will thus pay .12 × $1,000 = $120 in interest every year for 30 years.
– $120 regular interest payments are called the bond’s coupons.
– Coupon is constant and paid every year: Level coupon bond.

• At the end of 30 years, Beck will repay the $1,000.

– The amount repaid at the end of the loan is called the bond’s face value, or par value.
– However, the price of a bond can be equal, less or greater than the Face/Par value.
– Annual coupon divided by the face value is called the coupon rate on the bond (12%).

• The number of years until the face value is paid is called the bond’s time to maturity.
– Once the bond has been issued, the number of years to maturity declines as time passes.
∗ Reference: Corporate Finance, Ross, Westerfield, Jaffe & Jordan, Chapter 8
† bhanupratap.singh@iimsirmaur.ac.in

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Bonds & Bond Valuation
Bond Values & Yields

• With time, interest rates change in the market, however the cash flows from a bond stay
same.
– When interest rates rise, PV of the bond’s cash flows declines, and the bond is worth
less.
– When interest rates fall, the bond is worth more.

• To determine the value of a bond at a particular point in time, we need to know

– The number of periods remaining until maturity.
– The face value, the coupon, and the market interest rate for bonds with similar features.
– The interest rate required in the market on a bond is called the bond’s yield to maturity
(YTM).

Bonds & Bond Valuation

Bond Values & Yields: Example

• Xanth Co. were to issue a bond with 10 years to maturity.

• Annual coupon is $80: Bond will pay $80 per year for the next 10 years in coupon interest.

• In addition, Xanth will pay $1,000 to the bondholder in 10 years.

• Assuming similar bonds have a yield of 8%, what will this bond sell for?

Bonds & Bond Valuation

Bond Values & Yields: Example

• Total bond value = PV of Annuity + PV of Face value.

• 1000 = 536.81 + 463.19.

– pv.annuity(r = .08, n = 10, pmt = -80) + 1000/1.08ˆ10 (using FinCal
Library)

• This bond sells for exactly its face value. This is not a coincidence.

• The going interest rate in the market is 8%.

• With an $80 coupon, this bond pays exactly 8% interest only when it sells for $1,000.

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Bonds & Bond Valuation

Bond Values & Yields: Example

• Suppose that a year has gone by. 9 years to maturity.

• Interest rate in the market has risen to 10%. Bond Price?

• PV of Annuity + PV of Face value = 460.72 + 424.10 = 884.82.

– Therefore, the bond should sell for about $885.
– pv.annuity(r = .1, n = 9, pmt = -80) + 1000/1.1ˆ9
– In other words, this bond, with its 8% coupon, is priced to yield 10% at $885.

• Why the bond is selling for less than its face value?
– Because market interest rate is 10% & coupon is only 8%.
– To compensate, the price is less than the face value: Discount Bond.

Bonds & Bond Valuation

Bond Values & Yields: Example

• Why the bond is discounted by $115?

– Current coupon is $80.
– The bond would be worth $1,000 only if it had a coupon of $100 per year.
– In a sense, an investor who buys and keeps the bond gives up $20 per year for nine
years.
– At 10%, this annuity stream is worth: $115.18 (the discount).
– pv.annuity(r = .1, n = 9, pmt = -20)

• What would be the bond price if interest rates had dropped by 2%?
– The bond would sell for more than $1,000.
– Such a bond is said to sell at a premium and is called a premium bond.

Bonds & Bond Valuation

Bond Values & Yields: Semiannual Coupons

• In practice, bonds usually make coupon payments twice a year.

– So, if coupon rate of 14%, the owner will receive $140 per year, but in two payments of
$70 each.

• Bond yields are quoted like annual percentage rates (APRs).

– With a 16% quoted yield and semiannual payments, the true yield is 8% per six months.

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Bonds & Bond Valuation

Bond Values & Yields: Semiannual Coupons

• If this bond matures in 7 years, what is the bond’s price?

– Total value = PV of annuity + PV of face value
– 917.56 = 577.10 + 340.46
– pv.annuity(r = .08, n = 14, pmt = -70) + 1000/1.08ˆ14

• What is the effective annual yield on this bond?

– EAR = (1 + .08)2 − 1 = 16.64%.
– ear(r = .16, m = 2)

Bonds & Bond Valuation

Bond Values & Yields

• Overall, the bond prices and interest rates always move in opposite directions.

• When interest rates rise, a bond’s value, like any other present value, declines.

• Similarly, when interest rates fall, bond values rise.

Bonds & Bond Valuation

Interest Rate Risk

• The risk that arises for bond owners from fluctuating interest rates is called interest rate
risk.

• How much interest rate risk a bond has depends on how sensitive its price is to interest rate
changes.

• This sensitivity directly depends on two things: the time to maturity and the coupon rate.

• All other things equal:

– The longer the time to maturity, the greater the interest rate risk.
– The lower the coupon rate, the greater the interest rate risk.

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Bonds & Bond Valuation

Interest Rate Risk

Bonds & Bond Valuation

Interest Rate Risk

• Intuitively, short-term bonds are less sensitivity because cashflows are received quickly.

• The reason that bonds with lower coupons have greater interest rate risk is essentially the
same.

• Bond with the higher coupon has a larger cash flow early in its life, so its less sensitive to
changes in the discount rate.

Bonds & Bond Valuation

Finding the Yield to Maturity (YTM)

• Suppose we are interested in a 6-year, 8% coupon bond.

– A broker quotes a price of $955.14.
– What is the yield on this bond?
– By trial and error or by other tools.
– irr(cf = c(-955.14, 80,80,80,80,80,1080)) OR
– a <- rep(80, 5)
– irr(cf = c(-955.14, a, 1080))
– A bond’s YTM should not be confused with its current yield.
– Current yield is simply a bond’s annual coupon divided by its price.

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Bonds & Bond Valuation

Finding the Yield to Maturity (YTM)

• Example 8.2
– A bond has a price of $1,080.42, face value of $1,000, a semiannual coupon of $30, and
a maturity of 5 years.
– What is its current yield? What is its yield to maturity?
– Current yield: 60/1080.42 = 5.55%.
– YTM: 4.2%.
– irr(cf = c(-1080.42, 30,30,30,30,30,30,30,30,30,1030)) * 2 OR
– b <- rep(30, 9)
– irr(cf = c(-1080.42, b ,1030)) * 2

Bonds & Bond Valuation

Zero Coupon Bonds

• A bond that pays no coupons at all must be offered at a price much lower than its face
value.

• Such bonds are called zero coupon bonds, or just zeroes.

• Example 8.4:
– Suppose that the Geneva Electronics Co. issues a $1,000 face value, 8-year zero coupon
bond.
– What is the yield to maturity on the bond if the bond is offered at $627? Assume
annual compounding.
– 627 = 1000/(1 + r)8
– YTM: 6%.
– discount.rate(n = 8, pv = -627, fv = 1000, pmt = 0)

Inflation & Interest Rates

Real vs Nominal Rates

• Suppose the one-year interest rate is 15.5%.

– Anyone depositing $100 in a bank today will end up with $115.50 next year.

• Further imagine a pizza costs $5 today, implying that $100 can buy 20 pizzas.

• Finally, assume that the inflation rate is 5%, leading to the price of pizza being $5.25 next
year.

• How many pizzas can you buy next year if you deposit $100 today?

• You can buy $115.50/$5.25 = 22 pizzas.

– This is up from 20 pizzas, implying a 10% increase in purchasing power.

• Overall, the nominal rate of interest is 15.5%, the real rate of interest is only 10%.

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 • Nominal Rate: Percentage change in the number of dollars.

• Real Rate: Percentage change in how much you can buy with your dollars.
– Percentage change in buying power.

Inflation & Interest Rates

Relation between nominal rates, real rates, and inflation.

• 1 + R = (1 + r) × (1 + h).

• R is the nominal rate, r is the real rate, h is the inflation rate.

• Approximate Relationship: R = r + h

• Also known as Fisher equation.

• Fisher Effect: A rise in the rate of inflation causes the nominal rate to rise just enough so
that the real rate of interest is unaffected.

• In other words, the real rate is invariant to the rate of inflation

Determinants of Bond Yields

The Term Structure of Interest Rates

• At any point in time, short-term and long-term interest rates generally differ.

• Sometimes short-term rates are higher, sometimes lower.

• The relationship between short- and long-term interest rates is known as the term structure
of interest rates.

• The term structure tells us the nominal interest rates on default-free, pure discount bonds
of all maturities.

• These rates are, in essence, “pure” interest rates because they contain no risk of default and
involve just a single, lump-sum future payment.

• In other words, the term structure tells us the pure time value of money for different lengths
of time.

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Determinants of Bond Yields

Shape of Term Structure

• When long-term rates are higher than short-term rates, we say that the term structure is
upward sloping.

• When short-term rates are higher, we say it is downward sloping.

• The term structure can also be “humped.” When this occurs, it is usually because rates
increase at first, but then decline at longer-term maturities.

• The most common shape of the term structure, particularly in modern times, is upward
sloping, but the degree of steepness has varied quite a bit.

Determinants of Bond Yields

What determines the shape of the term structure?

• Real interest rate.

• Prospect of future inflation: Inflation premium.

• Interest rate risk: Interest rate risk premium.

Determinants of Bond Yields

Yield Curve: A plot of Treasury yields relative to maturity

• This plot is called the Treasury yield curve (or just the yield curve).

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Determinants of Bond Yields

Yield Curve

• The shape of the yield curve is a reflection of the term structure of interest rates.

• In fact, the Treasury yield curve and the term structure of interest rates are almost the
same thing.

• The only difference is that the term structure is based on pure discount bonds, whereas the
yield curve is based on coupon bond yields.

• As a result, Treasury yields depend on the three components that underlie the term struc-
ture.