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Chapter 2

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Chapter 2

Pricing of Bonds

- The price of any security equals the PV of the security’s expected cash flows.
- So, to price a bond we need to know:
- The size and timing of the bond’s expected cash flows.
- The required return (commensurate with the riskiness of the cash flows).

- PV and FV of lump sums and annuities.

- PV of a lump sum:

- PV of an ordinary annuity:

- Future Value
where:

n = number of periods

Pn = future value n periods from now (in dollars)

Po = original principal (in dollars)

r = interest rate per period (in decimal form)

- Future Value of on Ordinary Annuity

- price = PV of all future cash flows
- to find price, you need
- expected CFs
- coupon payments
- par value

- yield

- expected CFs

- We begin with a simple bullet bond:
- Noncallable (maturity is known with certainty)
- Coupons are paid every six months.
- The next coupon is received exactly six months from now.
- The interest rate at which the coupons can be invested is fixed for the life of the bond.
- Principal is paid at maturity (no amortizing).
- Coupon fixed for the life of the bond.

- Notation:
- P = price of the bond (in $)
- n = number of periods (maturity in years 2)
- C = semiannual coupon (in $)
- M = maturity value

Note: All inputs to the bond pricing formula are fixed except for y. As y changes so does P.

- Price a 20-year 10% coupon bond with a face value of $1,000 if the required yield on the bond is 11%.
- Formula inputs:
- The coupon is: 0.10 1,000 = $100.
- The semiannual coupon, C, is: $50.
- n = 40
- y = 0.055

- Projecting cash flows for fixed income securities is relatively straightforward – but sometimes it may be harder, for example:
- if the issuer or the investor has the option to change the contractual due date for the payment of the principal (callable bonds, putable bonds)
- if the coupon payment is reset periodically by a formula based on some value or values of reference rates (floating rate securities)
- if the investor has the choice to convert or exchange the security into common stock (convertible bond)

- Zero-coupon bonds (zeros) are so called because they pay no coupons (i.e., C = 0):
- They have only maturity value:

- Price a zero that expires 15 years from today if it’s maturity value is $1,000 and the required yield is 9.4%
- Formula inputs:
- M = 1,000
- n = 30
- y = 0.047

An investor would pay $252.12 today and receive $1,000 in 15 years.

- A fundamental property of bond pricing is the inverse relationship between bond yield and bond price.

Price

Yield

- For a plain vanilla bond all bond pricing inputs are fixed except yield.
- Therefore, when yields change the bond price must change for the bond to reflect the new required yields.
- Example: Examine the price-yield relationship on a 7% coupon bond.
- For y < 7%, the bond sells at a premium
- For y > 7% the bond sells at a discount
- For y = 7%, the bond sells at par value

- The price-yield relationship can be summarized:
- yield < coupon rate ↔ bond price > par (premium bond)
- yield > coupon rate ↔ bond price < par (discount bond)
- yield = coupon rate ↔ bond price = par (par bond)

- Discount or premium bond prices move toward par value as the bond approaches maturity.
- Market factors – change in yields required by the market.
- Issue specific factors – a change in yield due to changes in the credit quality of the issuer.

- Suppose that you are reviewing a price sheet for bonds and see the following prices (per $100 par value) reported. You observe what seem to be several errors. Without calculating the price of each bond, indicate which bonds seem to be reported incorrectly and explain why.

- We have assumed the following so far:
- Next coupon is due in six months.
- Cash flows are known with certainty
- We can determine the appropriate required yield.
- One discount rate applies to all cash flows.

- What if the next coupon payment is less than six months away?
- Then the accepted method for pricing bonds is:

- For a noncallable bond cash flows are known with certainty (assuming issuer does not default)
- However, lots of bonds are callable.
- Interest rates then determine the cash flow:
- If interest rates drop low enough below the coupon rate, the issuer will call the bond.

- The required yield for a bond is: R = rf + RP
- rfis obtained from an appropriate maturity Treasury security.
- RP should be obtained from RPs of bonds of similar risk.
- This process requires some judgement.

- We have assumed that all bond cash flows should be discounted using one discount rate.
- However, usually we are facing an upward sloping yield curve:
- So each cash flow should be discounted at a rate consistent with the timing of its occurrence.

- Each cash coupon (and principal payment) is a separate zero-coupon bond and should be discounted at a rate appropriate for the “maturity” of that cash flow.

- Coupons for floaters depend on a floating reference interest rate:
- coupon rate = floating reference rate + fixed spread (in bps)
- Since the reference rate is unpredictable so is the coupon.

- Coupon rate = rate on 3-month T-bill + 50bps

Reference Rate

Spread

- Floaters can have restrictions on the coupon rate:
- Cap: A maximum coupon rate.
- Floor: A minimum coupon rate.

- An inverse floater is a bond whose coupon goes up when interest rates go down and vice versa.
- Inverse floaters can be created using a fixed-rate security (called the collateral):
- From the collateral two bonds are created: (1) a floater, and (2) an inverse floater.

- Floater coupon + Inverse floater coupon ≤ Collateral coupon
- Floater par value + Inverse floater par value ≤ Collateral par value

- Consider a 10-yr 15% coupon bond (7.5% every 6 months).
- Suppose $100 million of bond is used to create two bonds:
- $50 million par value floater and $50 million par value inverse floater.

- Floater coupon rate = reference rate + 1%
- Inverse coupon rate = 14% - reference rate

- Problem: if reference rate > 14%, then inverse floater coupon rate < 0.
- Solution: put a floor on the inverse floater coupon of 0%.
- However, this means we must put a cap in the floater coupon of 15%.

- Collateral price = Floater price + Inverse floater price

- We have assumed that the face value of a bond is $1,000 and that is often true, but not always:
- So, when quoting bond prices, traders quote the price as a percentage of par value.
- Example: A quote of 100 means 100% of par value.

- Most bond trades occur between coupon payment dates.
- Thus at settlement, the buyer must compensate the seller for coupon interest earned since the last coupon payment.
- This amount is called accrued interest.
- The buyer pays the seller: Bond price + Accrued Interest (often called the dirty price).
- The bond price without accrued interest is often called the “clean price.”

- Suppose a bond just sold for 84.34 (based on par value of $100) and pays a coupon of $4 every six months.
- The bond paid the last coupon 120 days ago.
- What is the clean price? What is the dirty price?
- Clean price:
- $84.34

- $84.34 + 120/180($4) = $87.01