Question

You want to set aside some money for your friend Pip.

(a) You have great expectations for Pip and want to give him $250,000 a year for 20 years. If you’re able to lock in an effective annual nominal interest rate of 8%, how much do you need to put down today if you want the payments to begin next year?

(b) Before signing the documents to set up Pip’s trust, you decide that you’re feeling even more generous. How much do you need to put down today if you want the payments to go on forever, and you want the payment to begin this year instead of next?

(c) Before signing the more generous contract, your banker points out that $250,000 in 20 years won’t have the same purchasing power as $250,000 today. Assuming the inflation rate is 2%, what is the value of $250,000 in 20 years expressed in today’s dollars?

(d) Terrified by inflation, you ask your banker to set up an annuity that will pay Pip the sum of $250,000 today, $255,000 (= $250, 000(1.02)) next year, $260,100 (= $250, 000(1.02)2 ) in two years, ..., and $371,486.85 (= $250, 000(1.02)20) in twenty years. How much will this annuity set you back?

(e) How much more will the annuity in part (d) cost if the payments go on forever (with the payment in year n equal to $250, 000(1.02)n )?

Answer 1

(a) This is a question to determine the Present Value (PV) of a stream of future cash flows starting next year. Assuming current year (today) is Y₀and first cash flow, C₁ is coming after a year on Y₁ , then the value of C₁ in today's terms (i.e in Y_o) is C₁/(1+i)¹.

Similarly, the value of second cash flow C₂ in today's terms (i.e in Y₀) is C₂/(1+i)². We therefore get the present value of all future cash flows as:

PV = C₁/(1+i)¹ + C₂/(1+i)² +...+ C₂₀/(I+I)²⁰ , here C₁ = C₂ = C₃ =...= C₂₀ = C

OR

PV = C*(1 - (1+i)^-n)/i = $250,000*(1-1/(1.08)²⁰)/0.08 = $2,454,537

(b) This question talks about valuing a constant stream of cash flows till perpetuity. The formula is:

PV = C + [C/(1+i)¹ + C/(1+i)²+ C/(1+i)³ +...] = C+ C/i

OR

PV = $250,000 + $250,000/0.08 = $3,375,000

(c) To determine the future value of a cash flow on the basis of a rate of inflation is given by the formula:

Future Value = PV*(1+i)²⁰ = $250,000 * (1+2%)²⁰ = $371,486.8.

(d) To determine the Present Value of a growing yet finite number of annuity payments, growing at growth rate g and yielding interest i, the formula is

PV = First payment * [1-[(1+g)/(1+i)]ⁿ] /(i-g)

OR

PV = 250,000*(1-(1.02/1.08)²⁰/(0.06) = $2,838,303

(e) This question is essentially asking us to calculate the present value of perpetual cash flows assuming the cash flows grow at rate g and earn interest i. The formula for these kind of cash flows is

PV = C/(1+r) + C*(1+g)/(1+r)² + C*(1+g)²/(1+r)³ = C/(r-g)

In our case, the cash flows are as follows:

C + [C*(1+g)/(1+r) + C*(1+g)² / (1+r)² +...]= C+ [C*(1+g)/(r-g)] = $250,000 + [$250,000*(1+2%)/(8%-2%)] = $250,000 + $4,250,000 = $4,500,000.

Note: whenever there is an immediate cash flow (today, now, etc), no discounting is done for this cash flow. Only future cash flows are discounted to arrive at present value