Question

I KNOW OPTION 2 IS BETTER AND i HAVE THE WORK TO SHOW, JUST TRYING TO FIGURE OUT WHY OPTION 2 IS BETTER
You decide that you should invest the money in a long-term CD (certificate of
deposit). The financial section of your local paper has the following advertisements.
Option 1: A 10-year CD with an APR of 5.44% compounded monthly.
Option 2: A 10-year CD with an APR of 5.43% compounded continuously.
Which is the better option?

Answer 1

because the coupon is a tad lower but it is compounded continuously

Answer 2

for option 2 i got F= 1.7212P and then option 3 i got F= 1.7208P

Answer 3

How is option 2 lower?

Answer 4

5.43% is lower than 5.44% but it is compounded continuously so it is a better investment

Answer 5

What does compunded continusously specifically means?

Answer 6

you will understand compounding yearly, monthly, weekly, daily

take that to the extreme, compound hourly, by the minute, by the second, by the nano second -- you keep going and you end up with a mathematical construct, the details of which i do not sadly carry around in my head.

does that make sense or is it jibberish. you can be honest, btw :-)

Answer 7

Use the formulas for compound interest and continuously compounded interest and compare the results.

Answer 8

@IrishBoy123 it sounds a little jibberish lol

Answer 9

lol!!!! nice one

OK :) take your standard compounding formula and reduce the compounding period as i describe, all the way down to a second


you will have to take the interest rate down too.

you will find a convergence, i promise :))

i prolly should look this up before opening my big mouth !

Answer 10

you might find Salman Khan better at explaining stuff than me

https://www.khanacademy.org/economics-finance-domain/core-finance/interest-tutorial/cont-comp-int-and-e/v/continuously-compounding-interest-formula-e

that would not surprise me :)

Answer 11

okay perfect thanks

Answer 12

wait so anytime it is compounded continously its the better option?

Answer 13

i haven't run the numbers but i suspect that for a difference of 1 basis point in return, then yes.

i'd need to crank up a spreadhseet or go find the formulae on the WWW to be definitive. and i can't see me doing that right now :(

Answer 14

it's over a 10 year period, remember

Answer 15

Example:
If you deposit $100 at 8% compounded quarterly.
That means each quarter of a year, each 3 months, you add the interest earned in those 3 months.
Start with a deposit of $100.
After 3 months pass, you add the interest you earned in the quarter year.
In a quarter year, you earned 8%/4 = 2%.
2% of $100 is $2.
Now you have $102.
When the next quarter year passes, you are earning 2% on the $102.
etc.

If the interest were compounded monthly, then every month you add that month's worth of interest, and you earn interest on it.

With continuously compounded interest, every infinitesimal interval of time that passes has interest earned and added to the principal.
Given the same interest rate, continuously compounded interest earns more interest than interest compounded daily.

Answer 16

In this case, you have slightly different rates.
You need to calculate the interest earned in each case to see which one will earn more.

Answer 17

Interest compounded periodically:

\(FV = P(1 + \dfrac{r}{n})^{nt}\)

Interest compounded continuously:

\(FV = Pe^{rt} \)

Answer 18

aha, another math connection! Bernoulli's compounding formulae, & the invention of e!!

\[\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n} = e\]


if you've run some numbers and concluded option 2 is better, i'd easily believe that but i think we all agree you gotta run the numbers :(

tu @mathstudent55

Answer 19

Option 1: 5.44% compounded monthly for 10 years.

\(\Large FV = P \left( 1 + \dfrac{r}{n} \right) ^{nt}\)

\(\Large FV = P \left (1 + \dfrac{0.0544}{12} \right) ^{12 \times 10}\)

\(\Large FV = 1.72077P\)

Option 2: 5.43% continuously compounded for 10 years

\(\Large FV = P e ^{rt}\)

\(\Large FV = P e ^{0.0543 \times 10}\)

\(\Large FV = 1.72116P\)

As you can see, Option 2 yields more interest.

Answer 20

@mathstudent55

great work!!