calculus question

Question

Your rich uncle bequests to you a continuous, constant income stream of $6000 per year for the next 10 years. The terms of the bequest require that this income stream be paid continuously into a specific savings account that will not be available to you for 10 years. This account earns 5.2% interest, compounded continuously.

1) What is the present value of the bequest? 46786.09061

2) How much money would the bequest be worth (including all interest accrued) after 10 years? 78695.49804

3) You discover that a bank is offering 5.7% interest compounded continuously on a certificate of deposit (CD) that matures in 10 years. What is the cost of a CD at the above interest rate that would provide the same amount of money as the bequest after 10 years? 

4) Because the CD earns more interest than the savings account specified in the will, you feel that you are losing out on interest. So you ask the executor of the estate to use funds from the estate to buy a CD that will be worth the same as the bequeathed income stream in 15 years. You ask her to pay you today the difference between the present value of the original bequest and the amount invested in the CD.

How much should she pay you today? 

I got the first two answers right. I just need help with the last two #3 and #4.

Answer 1

Your question does not need just Math and Calculus, but also knowledge

of financial terminology, which is not my area.

But it appears that you have been trying to get some help for several days,

so here is my two cents worth.

I will be careful to state the definitions of financial terminology,

so you can see whether I am proceeding on the bases of correct assumptions.

I am assuming the following two formulas:

(A)

If an initial investment earns at a continuous interest rate r per year, then

z(t) = z(0)e^rt

where

z(0) is the initial investment,

z(t) is the investment after t years.

(B)

If a continuous income stream of m per year earns at a continuous interest rate of r per year, then

z(t) = (m/r)(e^rt - 1)

where

z(t) is the investment after t years

Problem 2)

I assume that you solved problem 2) by using formula (B) with

m = 6000,

r = 0.052

t = 10

Problem 1)

I assume that you solved problem 1) by applying formula (A) with

r = 0.052

t = 10

z(10) = 78695.49804

Problem 3)

I would use formula (A) with

r = 0.57

t = 10

z(10) = 78695.49804

From formula (A)

z(0) = z(t)e^-rt

Substituting actual numbers

z(0) = 78695.49804×e^-0.057×10 = 44504.30605

Problem 4)

Let

m = 6000 per year be the constant income stream,

p = 0.052 be the interest rate in the savings account,

q = 0.057 be the interest rate in the CD,

t = 15 be the time period.

Use formula (B) with r = p = 0.052

to see how much the bequeathed income stream would earn in 15 years

z(t) = (m/p)(e^pt - 1)

Substituting actual numbers

z(15) = (6000/0.052)(e^0.052×15 - 1) = 136323.7229

Now use formula (A) to see how much the executor would have to pay for a CD

earning 5.7% to get the same amount in 15 years

z(0) = z(t)e^-qt = 136323.7229×e^-0.057×15 = 57976.18791

So I am getting that the original bequest applied for 15 years is better than the CD for 15 years.

In other words, if you wanted the CD you would have to pay the executor

the difference between the $57976.18791 and your answer to question 1).

In contrast, if the period were 10 years then the CD would be better.