It is January 1st, 2021, and you are setting up a perpetuity that will allow you to withdraw R dollars at the end
of April and October every year, forever. Suppose that the stated APR of the perpetuity is r > 0.
Question 1 Determine the present value of the perpetuity if interest is compounded monthly. Leave your answer in
terms of R and r.
Question 2 Determine the present value of the perpetuity if interest is compounded continuously instead. Leave
your answer in terms of R and r.
Question 3 Which of your answers (between Question 1 and Question 2) do you expect to be larger? This question
will be hard to answer mathematically, but you should be able to argue it using financial reasoning. In 5 sentences
or less, justify your answer.
Answer 1: When the interest is compunded monthly, the present value of the April perpetuity will be R/(r/12) (since it is forever). Similalry, the presenet value of the October perpetuity will be R/(r/12). The total present value of the April and October perpetuities = R/(r/12)+R/(r/12)=24R/r.
Answer 2: When the interest is compunded continuouslly (say, daily), the present value of the April perpetuity will be R/(r/365) (since it is forever). Similalry, the presenet value of the October perpetuity will be R/(r/365). The total present value of the April and October perpetuities = R/(r/365)+R/(r/365)=730R/r.
Answer 3: Based on Answers 1 and 2, the seond perpetuity (the one compunded continuously) will be larger. The reason for this is that greater the compounding, higher the future value of cash flows and lower the discounting rate in the denomintaor (used to calculate the present value).
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