# Interest Anually, Compounded Continuosly

I decide that a year from now I`m going to invest in $100 in an investment account that pays 8% interest anually, compounded continuosly. What will the investment be worth in 12 years?

I tried LN(1,08) = 0,076961041

i = 7,696104114

PV = 100 CHS

N = 12

0 PMT

FV = 243,4444

Why is it wrong???

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When you use the TVM buttons on your calculator, what sort of interest rate do you need to use? Continuous? Effective? Nominal?

Simplify the complicated side; don't complify the simplicated side.

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I believe it should be 100e

^{(}^{0.08x12) }= 261.17Look at the example in the TVM chapter, for continuous compounding.

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In addition to Analyst0718, you can work backwards to 8% annual interest rate by:

-> ln(261.17/100) = 0.96

-> 0.96 divided by 12 = 0.08 or 8%

The answer is 241.xx through….

But thats `s because you need to use N = 11 I guess

Continuous, But I don`t know a way of doing it using the TVM buttons, maybe only applying the formula…

that part is easy to miss if you try and rush the problem. you are investing 100 a year from now (that is, at t = 1) an not right-away. 12 years from now, 11 years will have passed since you invested 100 dollars, which means N should be 11 as you pointed out.

You’re right, because it says “A year from now”.

Not continuous.

Next guess?

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How it`s not?

pays 8% interest anually, compounded continuosly.

If it`s not continuous it`s probably nominal than, but that doesn`t make sense…

Guess again.

You got that part right.

Simplify the complicated side; don't complify the simplicated side.

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the magician is asking what value would you have to enter in I/Y so that, after entering N and PV you can find the future value using FV. it is worth noting that there is no dedicated button for continuous compound rate, though you could use iconv, set 8% as the nominal rate and an arbitrarily high value for the C/Y (99999 should be good enough), then asking the calculator for the effective rate. the EFF rate given by iconv is slightly below the true effective rate, but should be very close, and certainly close enough to the true effective rate for solving the problem.

having said that, i see no real reason to use the TVM functionality for this problem when it is cumbersome (you have to use iconv to even get the interest rate you need) and prone to error (if your P/Y was set to a number other than 1 as a result of solving another problem and forgot to change it back to 1, you would get a wrong answer as a result, even if your other inputs were correct)

So, Effective?

I decide that a year from now I`m going to invest $100 in an investment account that pays 8% interest anually, compounded continuosly. What will the investment be worth in 12 years?

I figured out a way to do this in HP12c thorugh is a little tricky:

8 enter 1 million divide - i

11*1 million - N

PV = CHS 100

0 PMT

FV I get the right answer: 241,0899622

Too bad I`m not a BA II Plus user to do that :(

Try 525,600.

“Mmmmmm, something…” - H. Simpson

Yes.

I’m getting the impression that you may not have a solid understanding of what an effective interest rate is. Or a nominal interest rate. Or a continuously compounded interest rate.

You would benefit from such understandings, I believe.

In any case, you’re given an 8% (nominal) rate, compounded continuously. You need an effective rate. How do you get from a (nominal) continuously compounded rate to an effective rate?

Simplify the complicated side; don't complify the simplicated side.

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Better still in knowing how to do it properly. The formula’s in the curriculum, and the approved calculators each have a button for it.

Simplify the complicated side; don't complify the simplicated side.

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I did 525.600 in a exercise before, but it wasn`t as consistent as doing 1M….

“Mmmmmm, something…” - H. Simpson

The thing is that I always use EAR and If I do I get a rate that is above 8…

maybe I´m using a formula that I shouldn`t…

I found another one in the book

FV

_{n}= PVe^{rsN}I don`t think this formual is easy to remember through, I always gets confused with this e in my calculator (I read the manual of the hp12c that says that)

from the manual:

Natural antilogarithm.

Raises and (approximately 2,718281828)

to the power of the number

shown in record X

But I need to get that number e raised over 11*0,08 not the 11*0,08 raised to e….

I get so confused when I have calculations involving e

for the BA2 plus, and maybe the HP calculator, you need to type the exponent first,

and then hit 2nd -> e

^{x}this is different from the TI-30XS, and most non-financial calculators, where you first type e

^{x}and then the exponent.knowing how to find the effective rate of interest given a nominal rate with arbitrary number of compounding is fundamental. incidentally, continuously compounded is just like regular compounding, but the number of times the interest rate is compounded is infinity. this is why you can approximate the effective rate the way i suggested in a previous post, by making the number of compounding large enough.

the e

^{x}formula you may have in your notes comes from taking the limit of the compounding formula.there is a really neat mathematical proof here: http://www.havensmath.com/math-extras/definitionofe

This makes more sense, I guess I will need to know the formula + the e

^{x}that is inversed, typing the exponent first.I really hate this continuous compounding thing but I`m getting better with it over the time (I don`t see how continuous compounding is usefull in real life, I never see a bond that changes value (due to the rate changing) in the same day, maybe there`s something like that in Fixed Income but I still don`t got to this content.

As you should.

Simplify the complicated side; don't complify the simplicated side.

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EAR = (1+0,08)

^{1/1000000}-1?I clearly understand what they`re, the exercise gives me a Effective Annual Rate and I need to transform it on a Stated Annual Rate (so if I do it the stated needs to be lower the value of effective), my problem is the continuous compounding thing.

I`m really confused tbh

The best way for me would find what would be the nominal rate compounded annual, but the exercise gives me compounded continuous, how can I convert the continuous to annual?

Nope.

We’ll fix that.

A nominal rate compounded annually is an effective annual rate. You’re correct: that’s what you want.

The continuously compounded rate is 8%. This is a nominal rate; i.e., at the end of one year the value of your investment

have increased by exactly 8%.will notTo convert from a continuously compounded rate (

r) of 8% to an effective annual rate (EAR), use:_{cc}EAR =

e− 1=^{rcc}e^{0.08}– 1 = 0.083287068 =8.3287068%You can use this rate with the TVM buttons to get the correct answer.

Simplify the complicated side; don't complify the simplicated side.

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There’s the problem: they

give you an effective annual rate. They give you adon’tnominal(orstated),continuously compoundedrate.An effective

rate compoundsannual, notannuallysemiannually, notmonthly, notdaily, notcontinuously.No, you don’t.

When using the TVM buttons on your calculator,

iis always –– analways!effectiverate, not a stated rate. And it is always –– the effective rate for one (payment) period.always!Here, you’re given a stated (continuously compounded) rate, so you need to convert it to an effective rate. And, because you want to measure time in years (you’re using

n= 11 (years)), you want to convert it to an effectiverate.annualSimplify the complicated side; don't complify the simplicated side.

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Thanks a lot magician that`s very usefull!

I found this topic really hard because in some situations as this the exercise gives the stated anual rate, compounded continuosly.

Some other questions gives like: A bank compounds interest continuosly on its deposit and offers an effective annual interest rate of 14%, than it asks what is the annual interest rate.

I found this really confusing…

^ So the bank says it offers an EAR of 14% and you have to solve for the

annual rate:nominal1 +0.14 = e

^{r}where r is the nominal annual rate, compounded continuouslyln (1.14) = ln [ e

^{r}]r = ln (1.14)

r = 0.131028262

There is an ICONV worksheet on the BAII which lets you enter EAR, nominal rate and compounding frequency. If I use frequency = 525,600, EAR =14, I get a NOM of 13.10282787, which is pretty darned close to the actual answer.

“Mmmmmm, something…” - H. Simpson