Question

6. Explain the various inputs to the present value of an annuity (i.e. C, r, t, and PVA). How is the value of the annuity impacted by each input?

Explain what happens to the value of a perpetuity over time, assuming the perpetuity has begun.

Explain what happens to the value of a growing perpetuity over time, assuming the growing perpetuity has begun.

Explain why the value today of a perpetuity that doesn’t start until sometime in the future is not simply C/r.

thank you.

Answer 1

Explain the various inputs to the present value of an annuity (i.e. C, r, t, and PVA). How is the value of the annuity impacted by each input?

PVA=C/r*(1-1/(1+r)^t)
C is the cash flow per period
r is the discount rate
t is the time period for which payments will occur
PVA is the value of annuity or money you will have to spend today to buy the annuity which provides cash flow of C per period, starting next period, for t periods
As evident from the formula:
C increases, value increases
r increases, value decreases
t increases, value increases

Explain what happens to the value of a perpetuity over time, assuming the perpetuity has begun.

Value of perpetuity at time t=C/r
So, as seen from the formula it is independent of t and hence value of perpetuity does not change over time and it is always the same

Explain what happens to the value of a growing perpetuity over time, assuming the growing perpetuity has begun.

Value of perpetuity at time t=C0*(1+g)^t/(r-g)
For growing perpetuity, g>0 hence (1+g)^t will increase with t
As seen from the formula, growing perpetuity value will increase with time

Explain why the value today of a perpetuity that doesn’t start until sometime in the future is not simply C/r.
The value is C/r for perpetuity which start at the end of the period. As one would not earn interest from the end of the period until sometime in the future, the value of perpetuity is not simply C/r. Rather if the cash flow starts at time t, the value will be C/r*1/(1+r)^(t-1). If the cash flow starts at the end of the period t=1 and the value becomes simply C/r but otherwise it is not simply C/r.