The present value, at interest rate i, of an n-payment annuity immediate of 1 per period is commonly expressed as which of the following in terms of v and d?

Study for the SOA Fundamentals of Actuarial Mathematics (FAM) Exam. Prepare with flashcards and multiple choice questions with detailed explanations. Get ready for your future as an actuary!

Multiple Choice

The present value, at interest rate i, of an n-payment annuity immediate of 1 per period is commonly expressed as which of the following in terms of v and d?

The main idea is that the present value of a level annuity-immediate with n payments of 1 per period is the sum of the discounted payments: v^1 + v^2 + ... + v^n, where v = 1/(1+i). Using the geometric-series result, this sum equals v(1 − v^n)/(1 − v). Since 1 − v = d, the present value in terms of v and d is v(1 − v^n)/d. Note that (1 − v^n)/d would be the present value of an annuity-due with n payments (payments at the beginning of each period), not an immediate annuity. So for the annuity-immediate case, the correct expression in terms of v and d is v(1 − v^n)/d. If a choice matches (1 − v^n)/d, it corresponds to the due form, not the immediate form; this suggests a mismatch or misprint in the provided options.

The main idea is that the present value of a level annuity-immediate with n payments of 1 per period is the sum of the discounted payments: v^1 + v^2 + ... + v^n, where v = 1/(1+i).

Using the geometric-series result, this sum equals v(1 − v^n)/(1 − v). Since 1 − v = d, the present value in terms of v and d is v(1 − v^n)/d.

Note that (1 − v^n)/d would be the present value of an annuity-due with n payments (payments at the beginning of each period), not an immediate annuity. So for the annuity-immediate case, the correct expression in terms of v and d is v(1 − v^n)/d. If a choice matches (1 − v^n)/d, it corresponds to the due form, not the immediate form; this suggests a mismatch or misprint in the provided options.

More practice questions