How can I calculate yearly interest of my investment gain?

Question

I'm trying to figure out the yearly interest of a potential investment. Lets say I invested $10 000, 13 days later i cash out and I now have $10 500. The interest here is 4,76% for 13 days.

But can I convert it to a yearly interest?

Answer 1

The simple interest rate for an investment that costs $10,000 and returns $10,500 in 13 days is 5%.

To calculate the annual effective compound interest rate, the equation is

(1+i)^(365/n) - 1

where i is the simple interest rate and n is the number of days, so a 5% return over 13 days would be 293%.

To calculate the annual effective continuously compounded interest rate, the equation is

e^(i*365/13)

also where i is the simple interest rate and n is the number of days, so a 5% return over 13 days would be 307%.

Continuous compounding is more precise and easier to manipulate but possibly not as intuitive.

Answer 2

Depending on your compounding convention, the annualised rate could be 137%, 140%, 163%, or 293%, or something else. Whenever one talks about an interest rate, it must be tied to two defining components:

1. the compounding method (simple, periodic (annual, quarterly, etc.), continuous)
2. the day count fraction method

The first component determines what formula to use. The second component determines how to compute "T", which appears in all these formulae, from a given start and end date. (In theory, T is just "the time in years" from start to end. In practice, we have a start and end date, and leap years and stuff, and need to compute T somehow from that...)

The balance at the end, B(T), is always the balance at the beginning, B(0), times a growth G. That G can be expressed using a rate r and a time (or "day count fraction") T as follows:

1. Simple compounding: B(T) = B(0) * ( 1 + r*T )
2. Annual compounding: B(T) = B(0) * ( 1 + r )^T
3. Quarterly compounding: B(T) = B(0) * ( 1 + r/4 )^(4*T)
4. Continuous compounding: B(T) = B(0) * exp( r*T )

(Note that the continuous compounding is just the limit of periodic compounding for infinitely small periods, i.e. replacing the 4 above by, essentially, a really large number).

For the day count fraction, there are several methods, commonly denoted by ACT/ACT, ACT/365, 30/360, and other variations. The ISDA Definitions are a good reference here. The simplest one is ACT/365, which is just the number of days from (and including) the start date to (but excluding) the end date, divided by 365.

Now, in your case, we have B(T) = 10,500 = B(0) * G = 10,000 * G, so the growth factor is 1.05.

T = 13/365, using ACT/365.

So, we have, for different compounding methods:

1. Simple: 1 + 13/365 * r = 1.05, thus r = (G-1)/T, so r = 140%
2. Annual: (1 + r)^(13/365) = 1.05, thus r = G^(1/T)-1, so r = 293%
3. Quarterly: (1 + r/4)^(4*13/365) = 1.05, thus r = 4(G^(1/4T)-1), so r = 163%
4. Continuous: exp(r*13/365) = 1.05, thus r = ln(G)/T, so r = 137%

Thus, depending on your compounding, the annualised rate could be 137%, 140%, 163%, or 293% (and this is without even delving into different day count fractions) - a nice illustration that:

• it is somewhat meaningless (or, at any rate, imprecise) to talk about an interest rate without specifying compounding method and day count fraction method
• this problem is exacerbated for high rates. For rates near zero, all these formulae line up pretty well... :-)
• annualised rates for very short periods can be hard to interpret.

Finally, two questions:

1. How do you get a rate of 4.76%? Mysterious.
2. Where can I find an investment like that, please? :-)

Answer 3

Here are a couple of demos to calculate the APR or effective annual rate, depending on your regional terminology (US or EU).