Question (BPP Q20, Ch2)

[hokensuri]

Perpetuity A has payment of $5 that are made at the end of each year forever. The first payment occurs at the end of the third year.

Perpetuity B has payment of $Y that are made at the beginning of each year forever. The first payment occurs at the beginning of the sixth year.

The present value at time 0 of both perpetuities is the same assuming an annual effective interest rate of 4%. Determine Y.

Answer: 5.408

The solution indicates that 5(v^2)a_angle_infinity = Y(v^5)a(double dots)_angle_infinity. What I don’t understand is those discount factors. Why is it v^2 (not v^4) for the perpetuity A? Also, why is it v^5 (not v^6) for the perpetuity B? I am thinking we have to discount the perpetuity A for 4 years to get the present value at time 0 because the payment starts at the end of the third year (i.e., there are 4 years between time 0 and the time of the first payment). I used a similar argument for the perpetuity B. But I know I am wrong. What am I missing?

[Gandalf]

You didn't specify whether the formulas were working with perpetuities (or infinite annuities) -due or -immediate. Assuming perpetuity-immediate, one with 0 years deferred has first payment at end of one year, so one with two years deferred (a v^2 factor) has first payment at the end of three years.

Edited: Oops. You did. But the reasoning is correct. First is a perpetuity immediate, as just described. Second is a perpetuity due. 0 years deferred has first payment at start of year 1, so 5 years deferred (a v^5) factor, has first payment at start of year 6.

[hokensuri]

You didn't specify whether the formulas were working with perpetuities (or infinite annuities) -due or -immediate. Assuming perpetuity-immediate, one with 0 years deferred has first payment at end of one year, so one with two years deferred (a v^2 factor) has first payment at the end of three years.

Edited: Oops. You did. But the reasoning is correct. First is a perpetuity immediate, as just described. Second is a perpetuity due. 0 years deferred has first payment at start of year 1, so 5 years deferred (a v^5) factor, has first payment at start of year 6.
Thanks, Gandalf, for your quick response. But I am still confused. When I draw a time line for the perpetuity A, I clearly see 4 time-periods between time 0 and the first payment (the end of the third year). So why we need to discount for just 2 years to get the present value? I know I am asking you a basic question, but I don't get it. Thanks again for your help!

[Gandalf]

For perpetuity A, there are only 3 intervals before the payment: from 0 to 1, from 1 to 2, from 2 to 3. A perpetuity immediate always has payment at the end of an interval, so here we have two extra intervals, so we need v^2.

[atomic]

To clarify, the notation

$$a_{\overline{\infty|}}$$

indicates a perpetuity-immediate whose first payment of 1 is made at the end of the first payment period. The first payment is made at t = 3, so there are two periods, 0 < t < 1, corresponding to the first year, and 1 < t < 2, corresponding to the second year, before the first payment is made. That is why the present value discount factor for the first perpetuity is v^2. Equivalently, we could consider it as a perpetuity-due whose first payment is at the beginning of the year 4, but we would have to use the alternative

$${\ddot a}_{\overline{\infty|}}$$.

Another way to look at it is to think of how much time will elapse between the present (t = 0) and the first payment, and this value determines the first present value discount factor. In this case, the first payment is made at t = 3, and so the first payment must be discounted by a factor of v^3. This sets up the valuation as

$$5v^3 + 5v^4 + 5v^5 + \cdots = 5v^2 (v + v^2 + v^3 + \cdots) = 5v^2 a_{\overline{\infty|}0.04}$$.

Similarly, the second perpetuity has its first payment at the beginning of the 6th year, so t = 5 (the sixth year has not yet passed!). So the valuation of this annuity is

$$Yv^5 + Yv^6 + Yv^7 + \cdots = Yv^5 (1 + v + v^2 + \cdots) = Yv^5 {\ddot a}_{\overline{\infty|}0.04}$$.

Your confusion seems to stem from not knowing how to determine the discount factor of the first payment when the question mixes annuities-immediate and -due. The key is proper valuation is to always observe the timing of the first payment relative to the valuation point, and the rest follows.

[hokensuri]

Thank you both!