Perpetuities

EXERCISE 01

An endowment must pay $12,000 annually forever, with the first payment at the end of year 3. The investment return is 6% annually.

0
Invest today2
$200,0003
$12,0004, 5, …
$12,000 yearly

A.How much must be invested today?

A.

PV_2=\frac{12{,}000}{0.06}=200{,}000

PV_0=\frac{200{,}000}{1.06^2}

$177,999.29

The perpetuity formula gives its value one period before the first payment: at time 2, not time 3.

Perpetuities

EXERCISE 02

An annuity pays R every four months in arrears for six years. A second annuity pays €60 annually forever, with the first payment at t = 7. Both are equivalent at t = 6. The effective annual rate is 9%.

A.Find R and the common present value at t = 0.

A.

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

i_{4m}=1.09^{1/3}-1,\qquad n=18

V_6^{perpetuity}=\frac{60}{0.09},\qquad Rs_{18|i_{4m}}=V_6^{perpetuity}

R=\frac{60/0.09}{s_{18|i_{4m}}}

€28.69

V_0=\frac{60}{0.09}(1.09)^{-6}

€397.51

Perpetuities

EXERCISE 03

A perpetuity pays €5,000 every six months, starting immediately. The nominal annual discount rate is 10%, convertible semiannually. Values are measured just before the payment at t = 0.

A.Value the whole perpetuity and its first ten payments.

A.

PV_{due}=R\frac{1+i}{i},\qquad PV_{n,due}=R(1+i)a_{n|i}

i_h=0.10/2=0.05

PV_{\infty}=5{,}000\frac{1.05}{0.05}

€105,000.00

PV_{10}=5{,}000(1.05)a_{10|0.05}

€40,539.11