The dirty price Pᵈ of a bond at a given settlement date T equals the invoice amount A of that bond's purchase transaction at T scaled down to a hypothetical notional of 100.
The exact relation between Pᵈ and A is:
Pᵈ = (A/N)100
and inversely the invoice amount A is given by:
A = PᵈN/100
where N is the bond's (perhaps time-dependent) notional at T
The theoretical (fair) value of Pᵈ equals the present value of all cash flows received after T scaled down to a hypothetical notional of 100.
In formula terms:
Pᵈ = (100/N)[P(t₁)c₁ + P(t₂)c₂ + ... + P(tᵥ)cᵥ + P(tᵥ)R]
P(t) is the discount factor for maturity 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.
cᵢ is the iᵗʰ coupon paid to the purchaser of the bond after T
tᵢ is the payment time of the coupon cᵢ
v is the number of coupons scheduled to be paid after T
R is the redemption amount paid at T, which in the non-amortizing case equals the bond's notional.
If the Pᶜ is known, Pᵈ is given by:
Pᵈ = Pᶜ + I
where I is the interest amount (based on a notional of 100) that has been accrued until T during the coupon accrual period containing T