I have a finance problem that is 99% mathematical.
In finance, the price of a bond could be modelled as the discounted value of its future cash flows, so something like:
Equation #1
MP = C1*D1 + C2*D2 +… + Ci*Di
Where:
MP = Price of bond
Ci = ith cash flow for bond
Di = Discount factor for the ith cash flow, 0 <= Di <=1
Sometimes, the Discount Factor is modeled as a function of time using a polynomial:
Equation #2
Di = B0 + B1*Ti + B2*Ti^2 + … + Bk*Ti^k
Where:
Di = Discount factor at time = i
k = The degree of the polynomial
Ti = Time until the ith cash flow, in years
Bk = The coefficients of the model which describe how the time to cash flow determines the discount factor
Suppose I wanted to instead model the bond prices, MP, with a continously compounded interest rate, such as in the following equation:
Equation #3
MP = C1*e^(-R1*T1) + C2*e^(-R2*T2) +… + Ci*e^(-Ri*Ti)
Where:
MP = Price of bond
Ci = ith cash flow for bond
This interest rate can also be modeled as a function of time with a polynomial:
Equation #4
Ri = B0 + B1*Ti + B2*Ti^2 + … + Bk*Ti^k
Where:
Ri = Interest rate at time = i
k = The degree of the polynomial
Ti = Time until the ith cash flow, in years
Rk = The coefficients of the model which describe how the time to cash flow determines the interest rate.
Problem:
I have the polynomial coefficients for Equation #2. I need the corresponding coefficients for equation #4 (assume both are cubic). I've tried plugging and solving for the beta's for R, but the math becomes pretty messy. Any pointers are appreciated.
My example cubic coefficients: 9.999004e-01, -1.366753e-03, -2.146066e-03, 4.826901e-05
Thank you!