Download Financial Math: Bond Pricing Formulas & Concepts (UConn, Math 3615, Fall 2008, Mod 4) - Pr and more Study notes Mathematics in PDF only on Docsity!
University of Connecticut
Math 3615: Financial Mathematics Problems
Fall 2008
Summary ā Module 4
- BONDS -
Definitions: F = face value or par value r = coupon rate (per coupon payment period) F r = coupon amount n = number of coupon payment periods remaining until redemption date i = effective interest rate (āyieldā) per coupon payment period (based on the bondās price) v = 1 / (1+ i ) RV = redemption value (ACTEX manual uses C for redemption value)
g = coupon rate based on Redemption Value:
F
RV
r g
CPN = coupon
Formulas for the price of a bond on a coupon date:
in all cases if RV = F
Concept: PV(RV) + PV(CPNs)
Basic formula: RV ā
vn + F ā
r a ā
(^) n (^) | = RV( v n + g a ā
(^) n |) F ā
v n + Fā
r a ā
n |
Premium/Discount Formula:
|
F
RV RV
RV n
r i a
F + F( r ā i a ) (^) n |
= RV + (CPN ā RV ā
i a ) (^) n | = F + (CPN ā F ā
i a ) (^) n |
Makeham Formula^1 :
F
K F K
RV
r i
K (F K)
r i
where K = F vn ( =PV[RV])
For a bond purchased between coupon payment dates:
Total price = Po (1+ i ) t = Price on prior coupon date, accumulated to settlement date with compound interest
Po = price on prior coupon date i = effective interest rate per coupon period t = fraction of coupon period between prior coupon date and settlement date = (days between prior coupon date and settlement date)/( days in coupon period) (Note that this calculation may be based on actual days or on a 360-day year.)
(^1) Makehamās formula is not normally used where F ā RV.
Bond prices are typically quoted excluding the accrued coupon. Thus the quoted price is equal to the above-calculated Total price , less the amount of the accrued coupon. The amount of the accrued coupon equals the coupon payable at the next coupon date times the fraction of the current coupon period that has elapsed prior to the transaction date (i.e., the accrued coupon is calculated using simple interest methods).
Price excluding accrued coupon = Po (1+ i ) t^ ā Couponā
t
Terminology for bond prices:
Term for the Price Corresponding Term for Price including accrued coupon excluding accrued coupon Total sale price Price Flat price Market price Premium-plus-accrued True price Dirty price Clean price
Amortization of Premium or Discount^2 in a Bondās Price:
Amount of premium (or discount) amortized in k th^ period = F( r-i ) vn - k +
Note that the amount by which the bondās price changes during a period (if the interest rate remains constant) is equal to the change during the prior period times (1+ i ).^3
If the bondās market price exceeds its par value (because the bondās coupon rate exceeds the market interest rate), then it will be called at the earliest possible call date, unless market interest rates rise. Exception: If there is a call premium that exceeds the bondās current market premium (i.e., if the issuer would have to pay more than the market price for the bond), then it will not be called.
If the bondās market price is less than its par value (because the bondās coupon rate is less than the market interest rate), then it will not be called before maturity, unless market interest rates fall.
(^2) Technically, the premium in a bondās price is said to be āamortizedā over the life of the bond, but the
discount in a bondās price is said to be āaccruedā over the bondās life. However, the ACTEX manual uses the term āamortizedā for both premium and discount. In either case (premium or discount), the difference between the bondās current value and its par value decreases over time, reaching 0 at maturity, and the amount by which the premium or discount changes in a given period equals (1+ i ) times the change in the prior period.
(^3) The practical use of this concept (of amortizing a bondās premium or discount) lies in determining the
bondās book value for accounting purposes. The reported value must be consistent with the purchase price on the purchase date, and it must be consistent with the maturity value on the maturity date. Book values calculated in this way (i.e., at an unchanging interest rate) do not represent market values, and the bondās value on the companyās balance sheet (the bondās ābook valueā) will generally not match the market value of the bond except on the date of purchase and the maturity date, unless the market interest rate for the bond happens to match the interest rate that the bondowner is using to determine the book value.
