Question

4.    Continuous Streams

        You win a lottery and the lottery commission will pay you $3 million a year for 10 years. They also offer you a choice to take $18 million as one lump sum payment.

            

        a.     What is the long-form expression for this problem if the discount factor is unknown? (Just give the 1st, 2nd, and terminal terms in the equation.) (3)

        b.     Compute the discount factor the lottery commission used to get the $18 million lump-sum payment?   (3)

Answer 1

Solution:

We have two options: either $3 million per year for 10 years, or $18 million as lump sum amount

a) Let the unknown discount factor be denoted by d.

First term: When received $3 million in first year, we have $3 million with us.

Second term: The $3 million received next year, will hold a present value = d*$3 million

Similarly, $3 million received next to next year (or 2 years from now), will hold a present value = d*d*$3 million = d²*$3 mn

Going this way, the 10th (last term) amount of $3 millions, received 9 years from now, will have the value = d⁹*$3 mn

So, option 1 of $3 million for 10 years will give us total amount in today's period = $3 mn + d*$3 mn + d²*$3 mn + ... + d⁹*$3 mn

This is the required long form expression

b) Notice that the above obtained expression is a Geometric progression (GP) series, with first term, a = $3 mn and common ratio, r = d, and total number of terms, n = 10

So, using formula for sum of GP series = a(1 - rⁿ)/(1 - r)

Thus, the above long-form expression can be shortened to

= $3 mn*(1 - d¹⁰)/(1-d)

Now, the lottery commission will not prefer to give out more as the lump-sum amount than what it gives out in installment payments

So, to find value of d required in the question, we have:

Total payment we get under installments = total payment in lump-sum

$3 mn*(1 - d¹⁰)/(1-d) = $18 mn

(1 - d¹⁰) = 6*(1 - d)

1 - d¹⁰ - 6 + 6*d = 0

So, solving this, we get d = 0.675 (approx) (using hit and trials)