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Chapter 2 - PowerPoint PPT Presentation

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

• 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).

• You must be comfortable with TVM:

• PV and FV of lump sums and annuities.

• PV of a lump sum:

• PV of an ordinary annuity:

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

• price = PV of all future cash flows

• to find price, you need

• expected CFs

• coupon payments

• par value

• yield

• 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

• The bond price is:

• 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)

• Bond prices change for the following reasons:

• 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.

• These assumptions may not be true and therefore complicate bond pricing.

• 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

• 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.

• Also, CFs on floaters and inverse floaters change over time and are not known (more on this later).

• 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

• 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.

• In other words, we can view a bond as a package of zero-coupon bonds:

• 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.

• Example:

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

• Reference Rate

• 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.

• These bonds are created so that:

• Floater coupon + Inverse floater coupon ≤ Collateral coupon

• Floater par value + Inverse floater par value ≤ Collateral par value

• 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.

• 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.

• Assume a 6-mo coupon reset based on the formula:

• Floater coupon rate = reference rate + 1%

• Inverse coupon rate = 14% - reference rate

• Notice: Floater coupon rate + Inverse coupon rate = 15%

• 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%.

• The price of floaters and inverse floaters:

• 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

• Dirty price:

• \$84.34 + 120/180(\$4) = \$87.01