FM Annuities and Loan Amortization Flashcards

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:

Answer: $5,358

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

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:

Answer: ä(n,i) = a(n,i) × (1+i)

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

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?

Answer: 100 × (Ia)(5, 5%)

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%).

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:

Answer: ā(T) = (1 - e^(-δT)) / δ

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

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?

Answer: $4,296.28

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

Which of the following statements about perpetuities is CORRECT?

Answer: The future value of a perpetuity evaluated at time n grows without bound as n → ∞

Since a perpetuity pays forever, its future value (accumulated to any point in the future) grows without bound — it has no finite future value.

The yield rate (internal rate of return) of an investment is defined as the interest rate i at which:

Answer: The present value of all cash outflows equals the present value of all cash inflows

The yield rate satisfies NPV = 0, meaning the PV of inflows equals the PV of outflows — it is the break-even discount rate.