- What is the formula for an annuity due bond given the price and the spot rates?
- what is the formula for an annuity immediate bond given the price and spot rates?
- what is the formula for the above 2 if the “annuity” is increasing by x% each term?

This is a strange question, since you don’t even say what the formula you want is for, but most likely you’re trying to solve for the periodic payment (or first period payment in the third one). This absolutely isn’t a formula you should be trying to memorize, you just want to understand how spot rates work.

The Price (or Present value) is just the sum of the payments discounted at the appropriate spot rates. So (using 3 year annuities as an example, with spot rates one-year spot rate i1; two-year i2; three-year i3):

3 year annuity due with payment P:

PV = P + P/(1+i1) + P/(1+i2)^2; no discounting of the initial payment

3 year annuity immediate with payment P:

PV = P/(1+i1) + P/(1+i2)^2 + P/(1+i3)^3

3 year annuity immediate with first payment P, increasing x% each term

PV = P/(1+i1) + P(1+x%)/(1+i2)^2 + P(1+x^%)^2/(1+i3)^3

Your post refers to “annuity due bond” and “annuity immediate bond”. Neither term sounds like normal US usage to me. If you are thinking of a bond with periodic coupons and principal returned at maturity, again, you just take the present value of the payment amounts, including the principal repayment at maturity, each discounted using the spot rate for when the payment is made.

I **really, really** would be surprised to see you asked for a future value calculation in a spot rate question. But if you were asked one in an SOA/CAS exam question, you need to guess something, and this is what I would try. Someone else may propose a different approach, and I don’t think there are any sample questions that will confirm what you should do.

Step 1: Calculate the present value. (No doubt about how to do that).

Step 2: if you want the future value k years later, multiply the PV by (1 + spot rate)^k, using the k-year spot rate. (So in this approach, step 2 uses a single spot rate, even if there were cash flows at many points in time).

But if you spend time worrying about future values in spot rate questions, your effort would be better spent elsewhere. IMO. FWIW. YMMV.