FM FM Annuities and Loan Amortization 3

Question 1

A deferred annuity-immediate pays $1,000 per year for 10 years, with the first payment occurring at the end of year 5. At 7% annual effective interest, the present value today is approximately:

Accepted Answer

$5,358

Answer

PV = 1000 × [a(14,7%) - a(4,7%)] = 1000 × [8.7455 - 3.3872] = 1000 × 5.3583 ≈ $5,358.

Question 2

For an annuity-due of n payments of 1 per period at interest rate i, the relationship between ä(n,i) and a(n,i) is:

Accepted Answer

ä(n,i) = a(n,i) × (1+i)

Answer

Since an annuity-due shifts each payment one period earlier, its PV is (1+i) times that of the corresponding ordinary annuity.

Question 3

An arithmetic increasing annuity-immediate has payments of $100, $200, $300, $400, $500 at the end of years 1 through 5. Which expression gives its present value at 5% annual interest?

Accepted Answer

100 × (Ia)(5, 5%)

Answer

An arithmetic increasing annuity with first payment P and common difference P uses (Ia)(n,i), so PV = P × (Ia)(5,5%) = 100 × (Ia)(5, 5%).

Question 4

For a continuously payable annuity at a constant force of interest δ, the present value of payments made continuously at rate 1 per year for T years is:

Accepted Answer

ā(T) = (1 - e^(-δT)) / δ

Answer

The continuous annuity present value ā(T) = integral of e^(-δt) dt from 0 to T = (1 - e^(-δT))/δ.

Question 5

An investor wants to accumulate $100,000 in 15 years by making level annual deposits at the end of each year into an account earning 6% annually. How much should each annual deposit be?

Accepted Answer

$4,296.28

Answer

Deposit = 100,000 / s(15, 6%) = 100,000 / 23.2760 ≈ $4,296.28, where s(15,6%) = (1.06^15 - 1)/0.06.

FM - Certifications in Financial Mathematics Practice Test

FM - Certifications in Financial Mathematics Practice Test

FM FM Annuities and Loan Amortization 3