## dirty price

The

*dirty price***of a bond at a given settlement date**

*Pᵈ***equals the invoice amount**

*T***of that bond's purchase transaction at**

*A***scaled down to a hypothetical notional of 100.**

*T*The exact relation between

**and**

*Pᵈ***is:**

*A*

*Pᵈ = (A/N)100*and inversely the invoice amount

**is given by:**

*A*

*A = PᵈN/100*where

**is the bond's (perhaps time-dependent) notional at**

*N*

*T*The theoretical (fair) value of

**equals the present value of all cash flows received after**

*Pᵈ***scaled down to a hypothetical notional of 100.**

*T*In formula terms:

*Pᵈ = (100/N)[P(t₁)c₁ + P(t₂)c₂ + ... + P(tᵥ)cᵥ + P(tᵥ)R]*where

**is the discount factor for maturity**

*P(t)***as implied by a yield curve bootstrapped out of market prices of similar (same issuer) bonds so that it reflects the yield of the referenced bond.**

*t***is the iᵗʰ coupon paid to the purchaser of the bond after**

*cᵢ*

*T***is the payment time of the coupon**

*tᵢ*

*cᵢ***is the number of coupons scheduled to be paid after**

*v*

*T***is the redemption amount paid at**

*R***, which in the non-amortizing case equals the bond's notional.**

*T*If the clean price

**is known,**

*Pᶜ***is given by:**

*Pᵈ*

*Pᵈ = Pᶜ + I*where

**is the interest amount (based on a notional of 100) that has been accrued until**

*I***during the coupon accrual period containing**

*T*

*T*