Question

If an amount of $200 is compounded at a rate of 5% annually, what is the interest paid after 2.5 years.

I know the formula is \(P *(1 + r/n)^{nT} - P\).

Here P is 200
r = 0.05
n = 1
and T = 2.5 ??

Is it permitted to have fractional T? Can you explain or give some sources where it is explained why T can/ can't be fractional.

Answer 1

T is time in years,
if you are compounding annually then T should be a whole number

Answer 2

Hmm, then what about in this particular question? Will T be 2.5?

Btw, the original question is from a previous year CFA question paper.

Answer 3

@misty1212

Answer 4

@satellite73 @dan815 @shrutipande9

Answer 5

how many months is half a year?

Answer 6

ugh, how many years is .5 years is what i meant to ask lol

Answer 7

B (1+r) (1+r) (1+r/2)
yr1 yr2 yr.5

youve only aquired half the interest for the partial year

Answer 8

@amistre64 , so the calculation would be

\[P (1 + r)^{2}(1+\frac{r}{2})^{0.5}\]

Am I right?

Answer 9

not the .5 exponent that only occurs once, not 1/2 a time

Answer 10

I see what you're saying.

So it would just be \[P(1+r)^{2}(1+\frac{r}{2}) \]

So basically we treat the half year as being compounded half-yearly

Answer 11

\(n \times T = 1 \times 2.5\)

Answer 12

now what im reading is telling me that\[B(1+r)^{k}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)\]

"in the mathematical theory of interest, if we say that an account earns compound interest
at a rate i, we are implicitly stating that we use formula (1) for partial periods as
well"

https://www.math.purdue.edu/~rcp/MA170/InterestTheory.pdf

in practice, institutions tend to just do simple interest for the partial year it says as well

so the thing is, if this is for a course, what does your course work give?

Answer 13

B(1+r)^2 is the begnining balance, and if we close out our account, we receive the extra simple interest for the partial period.

B(1+r)^2 * (1 + r/2)

Answer 14

The book doesn't explicitly state anything for partial periods. Thanks for the link though.

Answer 15

page 4 starts talking of your issue here :)

Answer 16

Yep found it :)
Thanks for the help.

Answer 17

youre welcome