Question

Problem 3. Some bond calculus. (Hint: for this exercise, the formula for a geometric series might be useful: if qメ1, then 1-q I will assume in the future, that this formula is well known.

Answer 1

Total lottery money = $ 1 mn

To be paid out in 20 equal installments over t=1 to t=20

N = Nos. of years = 20

Value of each installments, V = $ 1 mn / 20 = $ 0.05 mn

Yield = Discount rate = Y = 5% = 0.05

Current price for this lottery winning = P = PV of all the future payment on the lottery

$=V \times \sum_{i=1}^{N = 20}(1+Y)^{-i} = \frac{V}{Y}\times [1-(1+Y)^{-N}] = \frac{0.05}{0.05}\times [1-(1+0.05)^{-20}]$

= $ 0.62311 mn

if it's paid out over 50 years then,

V = $ 1 mn / 50 = $ 0.02 mn

N = 50

Hence, current price of lottery today

$=V \times \sum_{i=1}^{N = 50}(1+Y)^{-i} = \frac{V}{Y}\times [1-(1+Y)^{-N}] = \frac{0.02}{0.05}\times [1-(1+0.05)^{-50}]$

=$ 0.36512 mn