Present Value of a Growing Perpetuity What is the equation for the present value of a growing perpetuity with a payment of C

73. Present Value of a Growing Perpetuity What is the equation for the present value of a growing
perpetuity with a payment of C one period from today if the payments grow by C each period?
74. Rule of 72 A useful rule of thumb for the time it takes an investment to double with discrete
compounding is the “Rule of 72.” To use the Rule of 72, you simply divide 72 by the interest rate to
determine the number of periods it takes for a value today to double. For example, if the interest rate is 6 percent, the Rule of 72 says it will take 72/6 = 12 years to double. This is approximately equal to the actual answer of 11.90 years. The Rule of 72 can also be applied to determine what interest rate is
needed to double money in a specified period. This is a useful approximation for many interest rates
and periods. At what rate is the Rule of 72 exact?
75. Rule of 69.3 A corollary to the Rule of 72 is the Rule of 69.3. The Rule of 69.3 is exactly correct
except for rounding when interest rates are compounded continuously. Prove the Rule of 69.3 for
continuously compounded interest.
Excel Master It! Problem
Excel is a great tool for solving problems, but with many time value of money problems, you may still need to draw a time line. For example, consider a classic retirement problem. A friend is celebrating her birthday and wants to start saving for her anticipated retirement. She has the following years to retirement and retirement spending goals:
Years until retirement 30
Amount to withdraw each year $90,000
Years to withdraw in retirement 20
Interest rate 8%
Because your friend is planning ahead, the first withdrawal will not take place until one year after she retires.
She wants to make equal annual deposits into her account for her retirement fund.
1. If she starts making these deposits in one year and makes her last deposit on the day she retires, what amount must she deposit annually to be able to make the desired withdrawals in retirement?
2. Suppose your friend has just inherited a large sum of money. Rather than making equal annual
payments, she has decided to make one lump sum deposit today to cover her retirement needs. What
amount does she have to deposit today?
3. Suppose your friend’s employer will contribute to the account each year as part of the company’s
profit sharing plan. In addition, your friend expects a distribution from a family trust several years
from now. What amount must she deposit annually now to be able to make the desired withdrawals in
retirement? The details are:
Employer’s annual contribution $ 1,500
Years until trust fund distribution 20
Amount of trust fund distribution$25,000

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