## The formulae

If the daily, weekly or monthly compounded return is r then annualise it as follows:

Daily to annual: [(1+r)^365 – 1] * 100

Weekly to annual: [(1+r)^52 – 1] * 100

Monthly to annual [(1+r)^12 – 1] * 100

## Examples

A daily return of 0.02% to annual: [(1+0.0002) ^ 365 – 1] * 100 = 7.36%

A weekly return of 0.5% to annual: [(1 + 0.005) ^ 52 – 1] * 100 = 29.6%

A monthly return of 1% to annual: [(1 + 0.01) ^ 12 – 1] * 100 = 12.7%

### On a calculator

Taking the first example, remember that 0.02% is 0.0002. Then, on a calculator, you would type:

1 “+” 0.0002 “=”

This gives 1.0002. Then you need to find your “to the power of” key. Annoyingly this is labelled differently on different calculators. On my phone I use RealCalc where the button is labelled “y^x”, on other calculators such as Casio, it’s labelled “x^y”. This is what it looks like on RealCalc:

Once you’ve found the key, and assuming your screen shows 1.0002 you then type

“y^x” 365 “=”

And you should see 1.07572268516. We subtract one

“-” 1 =

And you should now see 0.07572268516. Then we convert to a percentage by multiplying by a hundred

“x” 100 =

The answer is 7.57%.

### In Spreadsheets

If cell A1 contains your daily percentage (say 0.02%) then we can annualise with:

=(1+A1)^365-1

If the rate in cell A1 is weekly then we raise to the power 52 (number of weeks in a year)

=(1+A1)^52-1

And if the rate in cell A1 is monthly then we raise to the power 12 (number of months in a year)

=(1+A1)^12-1

## Explanation

As an investor, you care about growth. When you compare investments you want to compare them in a standard way, which is why we need to annualise returns. There are two things we have to think about when we work out compound interest: the *time* that our money is invested and the *compounding period*. If our interest is compounded daily for a year then we have 365 compounding periods (days) per year. If our interest is compounded weekly for a year then we have 52 compounding periods (weeks) per year. The more frequently we compound the more interest we get. But to compare different compounding periods we use a standard compounding period of one year.

So let’s say we bought a share for £100 yesterday and it’s worth £100.10 today. Its value has risen by 10 pence or 0.1% and the daily return is 0.1%. To turn this into an annual return we increase our initial amount by 0.1% repeatedly by the number of days in a year.

After one day our £100 is worth (1 + 0.1%) * £100 = (1 + 0.001) * £100 = £100.10

After two days our £100 is worth (1 +0.1%) * (1 +0.1%) * 100 = 1.001 * 1.001 * £100 = 1.002001 * £100 = £100.2001

After three days our £100 is worth (1 +0.1%) * (1 +0.1%) * (1 + 0.1%) * 100 = 1.001 * 1.001 * 1.001 * £100 = 1.003003 * £100 = £100.3003

…

A more compact way of writing this is to use power notation.

After n days our £100 is worth 1.001^n * £100

In a year there are 365 days, so to annualise 1% we would raise it to the power 365.

After 365 days our £100 is worth 1.001^365 * £100 = £144.03, and this is a 44.03% increase. This is the annual compound interest rate.

## Mathematical Detail

Ignore this if you don’t like maths!

The formula for geometric compounding relates present value and future value. If we compound at a frequency $latex f &s=1$ at rate $latex r &s=1$ for time $latex t &s=1$ (note that $latex t &s=1$ is always in years and the frequency $latex f &s=1$ is always compounding periods per year) then

$latex \mbox{FV} = \mbox{PV} \left(1 + \frac{r}{f} \right)^{f t} &s=2$

If we want to convert from *any* compounding rate to *any* other compounding rate then we make use of the law of one price: the future value must be the same regardless of our compounding frequency. Let’s say our two compounding frequencies are $latex f_1 &s=1$ and $latex f_2 &s=1$ and the two rates with those compounding frequencies are $latex r_1 &s=1$ and $latex r_2 &s=1$ then

$latex \mbox{FV} = \mbox{PV} \left(1 + \frac{r_1}{f_1} \right)^{f_1 t} = \mbox{PV} \left(1 + \frac{r_2}{f_2} \right)^{f_2 t} &s=2$

If we re-arrange that we get the general equation for converting rates with different compounding frequencies:

$latex r_2 = \left[ \left( 1 + \frac{r_1}{f_1} \right)^\frac{f_1}{f_2} – 1 \right] f_2 &s=2$

As the convention is to *always* use annual compounding $latex f_2 &s=1$ is always in years so $latex f_2 = 1 &s=1$ and $latex f_1 &s=1$ is always measured “per year”. That’s how we get the simple equation at the top of the page.

If that’s not complicated enough for you then we can also convert between different types of interest using the same trick. For example if we use continuous compounding $latex \mbox{FV} = \mbox{PV} e^{rt} &s=1$. And if we use simple interest then $latex \mbox{FV} = \mbox{PV} \left( 1 + rt \right) &s=1$. You can convert between simple, compound and continuous interest by equating the future values and re-arranging.