Suppose you receive for graduation a gift of $1200 from your favorite relative. You are required to invest at least $800 of the gift in a no-withdrawal savings program for at least two years.

Question

Plan B: Employee's credit union (ECU) also has some plans in which interest is compounded continuously. You are still comparing with an investment of $1000 at 6% annual interest at First Savings Bank (FSB).

Investment at FSB annual interest: $1123.60

1. Supposed your savings will earn 5% interest compounded continuously at ECU. How much would your initial deposit there need to be to have the amount you could have with FSB in 2 years?
2. If your initial deposit at ECU is $1000, what continuous compound interest rate would ECU need to pay for you to have the amount you could have with FSB in 2 years?
3. You speculate about making no additional deposits and no withdrawals from the savings account for 10 years. What continuous compound interest rate would ECU have to pay so that your initial $1000 doubles in 10 years? How would that compare with the amount in your account at FSB after 10 years?

Answer 1

Hi Chloe! You've got a good start here with having computed the account balance from FSB after two years. Let's add to that the $200 that weren't invested at all, to see that after 2 years with $1000 of the $1200 in FSB's 6% annual interest rate account, we have $1323.60 total

We'd now like to compare to investing at ECU, which it seems to suggest is using a continuous compounding setup in contrast to the annual rate that FSB advertises (this would be easier to determine with certainty, with Part A of the problem also included for context/comparison). Let's set up a continuous compounding formula:

P*e^(rt)

... I'm going to modify this slightly, since again we'd like to add the remainder of the cash we did not invest, back into our total at the end - if P out of $1200 is invested, that leaves 1200-P uninvested, for a total of:

P*e^(r*t)+1200-P

1. In this question are given r=.05 and t=2 years, and are looking to determine what P will leave us with the same final total as the FSB situation. We set up an equation:

P*e^(.05*2)+1200-P = $1323.60

and use algebra to solve. We're trying to isolate P, so let's begin by bringing everything without a P over to the right:

P*e^(.1) - P = 123.6

now factor P out of the lefthand side

P(e^.1 - 1) = 123.6

and divide to solve for P

P = 123.6/(e^.1 - 1) ≈ $1175.23, nearly all of the $1200.

We can double-check by working forward with this found value: if we invest $1175.23 in a continuously compounding account with a 5% interest rate for 2 years, we'll have

$1175.23*e^(.05*2) ≈ 1298.83

adding the final 1200-1175.23 = $24.77 on to this account balance, we indeed get back to 1298.83+24.77 = $1323.6... the same total if we invest $1000 with the FSB account! So our answer of investing $1175.23 with ECU in this scenario checks out.

2. Ok, what if instead we invest the same $1000 at ECU, in an account of currently-unknown rate? We want to see what rate, under continuous compounding, works out to the same after 2 years as the 6% annual rate with FSB. This time, since we're leaving the same $200 out of the account in either scenario, we don't have to add the extra step of adding it back in... functionally, if we did add it back in to both, the first step of our algebra when we go to solve the relationship will be subtracting it off both sides anyways.

So we again have a continuous compounding setup

P*e^(r*t)

but this time instead of P being the unknown, we're given P=$1000 and are looking for r when t=2:

1000*e^(r*2) = 1123.6

Begin by dividing through by 1000

e^(2r) = 1.1236

then take the natural log of both sides to get rid of the base of e^

2r = ln(1.1236)

and finally, divide by 2 to isolate our rate:

r = ln(1.126)/2 ≈ 0.0593357649

I'm keeping many decimals here because even very slight discrepancies in interest rates can compound to big differences in total interest over time. The rate is around 5.93357649% - pretty close to the 6% at FSU! but notice that continuous compounding does allow a lower advertised rate than annual does.

This is generally not how a bank would advertise its savings rates - they'll go with the 6%, to make it sound like you're getting more, even when the point is that we just figured out how to get the exact same amount =P Where a bank will instead want to make it sound like there's lower rates is instead on loans - if you've ever seen/heard ads that mention terms like APR vs. APY, that's what's going on: shifting the word problem slightly to make it sound like they're gonna give you lots of interest in your investment accounts, and not ask as much in interest when you're the one who owes it after taking out a loan from them.

3. Final problem! We set up another continuous compounding scenario, beginning with

P*e^(r*t)

and this time we're given P=$1000 but t=10. We're looking for r to double our $1000 after 10 years

1000*e^(10r) = 2000

We're nearly there, let's do some algebra! This will feel pretty similar to part (2). We can start by dividing through by 1000

e^(10r) = 2

take ln of both sides

10r = ln(2)

and divide by the coefficient of 10 to isolate our rate:

r = ln(2)/10 ≈ 0.0693147181

We find we'd need an interest rate of about 6.93147181% to double our $1000 in 10 years with ECU's continuous compounding.

Well ok, that's less than a point's difference from FSB's rate, how much will we make there in 10 years?

1000*1.06^10 ≈ $1790.85... more than $200 less. That's a greater than 10% loss in potential savings over that time period! See, I told you little disparities in interest rate add up to big differences in savings over time

I hope this was helpful! Please feel free to comment with follow-up questions as needed.