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Chapter 2. Pricing of Bonds. Time Value of Money (TVM). 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.

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### Chapter 2

Pricing of Bonds

Time Value of Money (TVM) You must be comfortable with TVM:

- 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.

Time Value

- 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

Bond Pricing

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

- yield

- expected CFs

Pricing A Bond

- 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.

Bond Pricing Formula The bond price is:

- 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.

Example

- 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

A few good points…

- 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)

Pricing Zero-Coupon Bonds

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

Example

- 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.

Price-Yield Relationship

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

Price

Yield

Price-Yield Relationship

- 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

Price-Yield Relationship Bond prices change for the following reasons:

- 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.

Example

- 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.

Complications to Bond Pricing These assumptions may not be true and therefore complicate bond pricing.

- 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.

Complications to Bond Pricing:Next Coupon Due < 6 Months

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

Complications to Bond Pricing:CFs May Not Be Known Also, CFs on floaters and inverse floaters change over time and are not known (more on this later).

- 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.

Complications to Bond Pricing:Determining Required Yield

- 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.

Complications to Bond Pricing:Cash Flow Discount Rates In other words, we can view a bond as a package of zero-coupon bonds:

- 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.

Pricing Floaters Example:

- 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.

Pricing Inverse Floaters These bonds are created so that: Equivalently, the bonds are structured so that the cash flows from the collateral bond is sufficient to cover the cash flows for the floater and inverse floater.

- 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

Inverse Floater Example Assume a 6-mo coupon reset based on the formula: Notice: Floater coupon rate + Inverse coupon rate = 15% The price of floaters and inverse floaters:

- 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

Price Quotes on Bonds

- 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.

Clean vs. Dirty Price

- 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.”

Clean vs. Dirty Price Dirty 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

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