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A rich relative has bequeathed you a growing perpetuity. The first payment will occur in a year and will be $2 000. Each year after​ that, you will receive a payment on the anniversary of the last payment that isnbsp 5% larger than the last payment. This pattern of payments will go on forever. If the interest rate is 16% per​ year, a.  What is​ today's value of the​ bequest? b.  What is the value of the bequest immediately after the first payment is​ made?

Question

A rich relative has bequeathed you a growing perpetuity. The first payment will occur in a year and will be $2 000. Each year after​ that, you will receive a payment on the anniversary of the last payment that isnbsp 5% larger than the last payment. This pattern of payments will go on forever. If the interest rate is 16% per​ year, a.  What is​ today's value of the​ bequest? b.  What is the value of the bequest immediately after the first payment is​ made?

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Solution

a. To calculate the present value of a growing perpetuity, we can use the formula:

PV = D / (r - g)

where:

  • PV is the present value
  • D is the cash inflow (dividend) in the first period
  • r is the discount rate
  • g is the growth rate

In this case, D = $2,000, r = 16% or 0.16, and g = 5% or 0.05. Plugging these values into the formula, we get:

PV = 2,000/(0.160.05)=2,000 / (0.16 - 0.05) = 20,000

So, the present value of the bequest is $20,000.

b. The value of the bequest immediately after the first payment is made can be calculated by subtracting the first payment from the present value and then dividing by the difference between the discount rate and the growth rate. This is because the first payment has already been received, so it is no longer part of the future cash flows that need to be discounted. The formula is:

V = (PV - D) / (r - g)

Substituting the given values, we get:

V = (20,00020,000 - 2,000) / (0.16 - 0.05) = $18,000

So, the value of the bequest immediately after the first payment is made is $18,000.

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