Computation of Bond Price

Bond prices can be expressed as a percentage of par value or as a yield.

Yield to Maturity (YTM)
· A single discount rate that makes the present value of a bond’s cash flows equal to the market price.
· YTM for a semiannual-pay bond is twice the semiannual discount rate and so called as bond equivalent yield.

YTM_annual-pay = (1+YTM _{semianual-pay}/2)²-1
YTM_{semiannual-pay} = [(1+YTM_annual-pay)^1/2-1]x2 = BEY

Bond price of a bond with semi-annual coupon payment:

P = CPN1 / (1-YTM/2) + CPN2 / (1-YTM/2)² +.+ (CPN2N + PAR) / (1-YTM/2)^2N

Where: YTM is BEY

Bond price for a zero-coupon bond
P = Par / (1 + YTM/2)^{2 N}

Where: YTM is BEY


Full price of a bond between coupon payments

P= Σ[Cash flow in period t/(1 + Discount rate)^(t-1+w)]

Where:
w is the no. of days remaining until next coupon (i.e. between settlement date (exclusive) and next copon date(inclusive)/ Total days in coupon period)

Accrued interest = Coupon x (1 - w)

Clean price = Full price - Accrued coupon.

Day count conventions:
· Actual/Actual (US governmentt bond);
· Actual/365 (Bristish government bond),
· 30E/360 (German government bond, Eurobond, most corporate bond, most munis)

Price-Yield Relationship
Inversely related, convex ( bond prices go up faster than they go down)

Pull to par
As time passes, the price of a bond trading at a premium will fall back to par and the price of a bond trading at a discount will rise to par.

Deficiency of traditional approach to valuation:
Each cash flow is unique. Valuing all cash flows of a security using a single discount rate is incorrect unless for a flat term structure. YTM is only a an approximate or weighted average of a set of spot rates.

Arbitrage-free valuation
Each individual cash flow is valued by discounting it at a spot rate corresponding to its maturity.

P = CPN1 / (1-S1) + CPN2 / (1-S2)² +.+ (CPNN + PAR) / (1-SN)^N

Where Si is the corresponding spot rate.
Arbitrage: If the sum of the PVs of cash flows is more than the price of the security, then buy the security and sell the cash flows individually, and vice versa.
Cash flows are valued using the spot rate corresponding to their maturity. It is more accurate than the traditional approach.