BA453 Ch2 Pricing of Bonds
The price of a floater depends on
(i) the spread over the reference rate and (ii) any restrictions that may be imposed on the resetting of the coupon rate.
Present Value When Payments Occur More Than Once Per Year
If the future value to be received occurs more than once per year, then the present value formula is modified so that (i) the annual interest rate is divided by the frequency per year, and (ii) the number of periods when the future value will be received is adjusted by multiplying the number of years by the frequency per year.
A bond's price changes in the opposite direction from the change in the required yield.
The reason is that as the required yield increases (decreases), the present value of the cash flow decreases (increases).
Price of an Inverse Floater: In general, an inverse floater is created from a fixed-rate security. The security from which the inverse floater is created is
called the collateral. From the collateral two bonds are created: a floater and an inverse floater.
One Discount Rate Applicable to All Cash Flows: A bond can be viewed as a package of zero-coupon bonds, in which
case a unique discount rate should be used to determine the present value of each cash flow.
The price of a floating-rate bond will trade close to par value
if the spread required by the market does not change and there are no restrictions on the coupon rate.
The price of an inverse floater depends on
the price of the collateral from which it is created and the price of the floater.
Determining the price of any financial instrument requires an estimate of
(i) the expected cash flows, and (ii) the appropriate required yield. The required yield reflects the yield for financial instruments with comparable risk, or alternative investments.
Relationship Between Bond Price and Time if Interest Rates Are Unchanged: For a bond selling at par value, the coupon rate is equal to the required yield.
As the bond moves closer to maturity, the bond will continue to sell at par value. Its price will remain constant as the bond moves toward the maturity date.
The price of a bond will not remain constant for a bond selling at a premium or a discount.
The discount bond increases in price as it approaches maturity, assuming that the required yield does not change. For a premium bond, the opposite occurs. For both bonds, the price will equal par value at the maturity date.
A bond will be priced below, at par, or above par depending the bond's coupon rate and the required yield required by investors.
When the coupon rate is equal to the required yield, the bond will sell at its par value. When the coupon rate is less (greater) than the required yield, the bond will sell for less (more) than its par value.
The framework for pricing a bond assumes the following:
(i) the next coupon payment is exactly six months away; (ii) the cash flows are known; (iii) the appropriate required yield can be determined; and, (iv) one rate is used to discount all cash flows.
The price of a bond can change for three reasons:
(i) there is a change in the required yield owing to changes in the credit quality of the issuer; (ii) there is a change in the price of the bond selling at a premium or a discount, without any change in the required yield, simply because the bond is moving toward maturity; or, (iii) there is a change in the required yield owing to a change in the yield on comparable bonds (i.e., a change in the yield required by the market).
Price Quotes
A bond selling at par is quoted as 100, meaning 100% of its par value. A bond selling at a discount will be selling for less than 100; a bond selling at a premium will be selling for more than 100.
Determining the Appropriate Required Yield: All required yields are benchmarked off yields offered by Treasury securities.
From there, we must still decompose the required yield for a bond into its component parts.
The price of a bond without accrued interest is called the clean price.
The exceptions are bonds that are in default. Such bonds are said to be quoted flat, that is, without accrued interest.
For an option-free bond, the cash flows are the coupon payments and the par value or maturity value.
The higher (lower) the required yield, the lower (higher) the price of a bond.
For example, a floater may have a maximum coupon rate called a cap or a minimum coupon rate called a floor.
The price of a floater will trade close to its par value as long as the spread above the reference rate that the market requires is unchanged, and neither the cap nor the floor is reached.
Accrued Interest: When an investor purchases a bond between coupon payments, the investor must compensate the seller of the bond for the coupon interest earned from the time of the last coupon payment to the settlement date of the bond.
This amount is called accrued interest. For corporate and municipal bonds, accrued interest is based on a 360-day year, with each month having 30 days.
Cash Flows May Not Be Known: For most bonds, the cash flows are not known with certainty.
This is because an issuer may call a bond before the stated maturity date.
The amount that the buyer pays the seller is the agreed-upon price plus accrued interest.
This is often referred to as the full price or dirty price.
Present Value of a Series of Future Values
To determine the present value of a series of future values, the present value of each future value must first be computed. Then these present values are added together to obtain the present value of the entire series of future values.
Present Value of an Ordinary Annuity
When the same dollar amount of money is received each period or paid each year, the series is referred to as an annuity. When the first payment is received one period from now, the annuity is called an ordinary annuity. When the first payment is immediate, the annuity is called an annuity due.
The cash flow is not known for
either a floating-rate or an inverse-floating-rate security; it will depend on the reference rate in the future.
Price of a Floater: The coupon rate of a floating-rate security (or floater) is
equal to a reference rate plus some spread or margin.
Price-Yield Relationship: A fundamental property of a bond is that
its price changes in the opposite direction from the change in the required yield. The reason is that the price of the bond is the present value of the cash flows.
The cash flows for a bond that the issuer cannot retire prior to
its stated maturity date consist of periodic coupon interest payments to the maturity date, and the par (or maturity) value at maturity.
Accrued interest is
the amount that a bond buyer who purchases a bond between coupon payments must pay the bond seller. The amount represents the coupon interest earned from the time of the last coupon payment to the settlement date of the bond.
The price of an inverse floater equals
the collateral's price minus the floater's price.
The price of a bond is the present value of the bond's expected cash flows,
the discount rate being equal to the yield offered on comparable bonds.
Over time, the price of a premium or discount bond will change even if the required yield does not change. Assuming that the credit quality of the issuer is unchanged,
the price change on any bond can be decomposed into a portion attributable to a change in the required yield and a portion attributable to the time path of the bond.
Relationship Between Coupon Rate, Required Yield, and Price: When yields in the marketplace rise above the coupon rate at a given point in time,
the price of the bond falls so that an investor buying the bond can realizes capital appreciation. The appreciation represents a form of interest to a new investor to compensate for a coupon rate that is lower than the required yield. When a bond sells below its par value, it is said to be selling at a discount. A bond whose price is above its par value is said to be selling at a premium.
For a zero-coupon bond, there are no coupon payments. The price of a zero-coupon bond is equal
to the present value of the maturity value, where the number of periods used to compute the present value is double the number of years and the discount rate is a semiannual yield.
