Annuities

EXERCISE 01

A five-year ordinary annuity has a present value of €10,000 at 5%.

A.Calculate its annual payment. What happens to its value if the rate falls to 4%?

A.

C=\frac{10{,}000}{[1-1.05^{-5}]/0.05}

Annual payment = €2,309.75.

With the payment fixed, a lower discount rate raises the present value.

PV_{4\%}=2{,}309.747981\frac{1-1.04^{-5}}{0.04}\approx 10{,}282.59

Annuities

EXERCISE 02

A house costs $350,000. Pay $50,000 immediately and finance the remainder with a 30-year mortgage at 7% annually. Equal payments are made at each year-end.

A.Calculate the annual mortgage payment.

A.

300{,}000=C\frac{1-(1.07)^{-30}}{0.07}

C=\frac{300{,}000(0.07)}{1-1.07^{-30}}\approx24{,}175.92

Annual payment: $24,175.92

Only the $300,000 loan is amortised. The $50,000 down payment is not part of the loan balance.

Annuities

EXERCISE 03

An annuity costs $200,000 today and pays $25,000 at each year-end while the investor is alive. The discount rate is 5%.

A.How many payments make its present value exceed its cost?

A.

200{,}000=25{,}000\frac{1-1.05^{-n}}{0.05}

n=-\frac{\ln(0.6)}{\ln(1.05)}=10.47

At least 11 annual payments are needed.

The fractional solution is the algebraic break-even point; payments occur only at year-end.

Annuities

EXERCISE 04

A €60,000 loan accrues at 12% effective annually. Pay €5,000 at each of the first five year-ends, then €10,000 at each year-end. Find the theoretical annuity duration, then settle the remaining balance either as a supplement at the preceding year-end or as a smaller payment at the following year-end.

A.Find the repayment duration and supplementary final payment.

A.

s_{n|i}=\frac{(1+i)^n-1}{i}

D_5=60{,}000(1.12)^5-5{,}000s_{5|0.12}

Balance after five payments: €73,976.26.

D_5=10{,}000a_{n|0.12}

n=-\frac{\ln(1-0.12D_5/10{,}000)}{\ln(1.12)}\approx19.295349

D_{24}=D_5(1.12)^{19}-10{,}000s_{19|0.12}

Make 19 regular €10,000 payments in years 6–24. At year 24, add €2,743.13, making the final payment €12,743.13.

Alternatively, pay only €10,000 at year 24 and settle €3,072.30 at year 25. A fractional annuity duration does not itself specify a contractual payment date.

Annuities

EXERCISE 05

For two years, receive €1,000 at every month-end plus €200 at the end of each even-numbered month. The effective annual discount rate is 5%.

A.Calculate the present value.

A.

a_{n|i}=\frac{1-(1+i)^{-n}}{i}

i_m=1.05^{1/12}-1

i_{2m}=1.05^{1/6}-1

PV=1{,}000a_{24|i_m}+200a_{12|i_{2m}}

PV=1{,}000\frac{1-(1+i_m)^{-24}}{i_m}+200\frac{1-(1+i_{2m})^{-12}}{i_{2m}}

€25,097.10