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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 l.jpg

Chapter 2

Pricing of Bonds

Time value of money tvm l.jpg

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.

    • 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.

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    Two Important PV Formulas

    • PV of a lump sum:

    • PV of an ordinary annuity:

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    Time Value

    • Future Value


      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

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    Bond Pricing

    • price = PV of all future cash flows

    • to find price, you need

      • expected CFs

        • coupon payments

        • par value

      • yield

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    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.

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    Bond Pricing Formula

    • 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.

    Example l.jpg


    • 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

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    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)

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    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:

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    • 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.

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    Price-Yield Relationship

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



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    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

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    Price-Yield Relationship

    • 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.

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    • 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.

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    Complications to 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.

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

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    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:

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    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).

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    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 l.jpg

    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.

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    Pricing Floaters

    • 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.

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    Pricing Inverse Floaters

    • 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.

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    Inverse Floater Example

    • 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

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    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.

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    Prices Converted into a Dollar Price

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    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.”

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    Clean vs. 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

  • Dirty price:

    • $84.34 + 120/180($4) = $87.01

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