Finance question on discount rate

Question

What discount rate would make you indifferent between receiving $3,320.00 per year forever and $5,907.00 per year for 26.00 years? Assume the first payment of both cash flow streams occurs in one year. Please show in depth how to solve it and how to solve it on a financial calculator step by step. Thanks!

Answer 1

You are indifferent between both cash flow streams when the present value the cash flow streams are equal.

First, let's consider how to calculate the PV of the first stream of cash flows:

$3,320 per year forever is known as a perpetuity, and the equation for the present value of a perpetuity is:

PV= C/r

where: C=cash flow per period r=discount rate per period

Now, let's look at the PV of the second stream:

This stream is an annuity. The equation for the PV of an annuity is:

PV=C*[(1+r)ⁿ-1]/r

where:C=cash flow per period r=discount rate per period n=number of periods​

We need to set the equations equal to each other and solve for r.

C*[(1+r)ⁿ-1]/r=C/r

Plugging in known values:

5907[(1+r)²⁶-1]/r=3320/r

Cancel 1/r from both sides and divide both sides by 5907:

(1+r)²⁶-1= 0.56

Add 1 to both sides, take the natural log of both sides, divide both sides by 26:

26*ln(1+r)=ln(1.56)

ln(1+r)=0.017

Raise both sides by e:

1+ r = e^0.017

Solve for r:

r=0.017