This post is about an Android application I wrote to calculate the Annual Equivalent Rate (AER) of a portfolio.
The AER of a portfolio is the annualised interest rate that, when applied to the portfolio contributions, results in the current value of the portfolio.
Simple example
If I invested £100 exactly one year ago and the value now is £105, the AER would be 5%.
This is because £100 x (1 + 5%) = £105.
Not so simple Example
However, things get significantly more complicated when there are multiple contributions on different dates.
What is the AER of a portfolio with contributions of £100 one year ago, £50 ten months ago that is worth £160 today?
It turns out to be approximately 7.04%. This is a typical example that is simply stated but has a not-so-obvious solution.
The Algorithm
To compute the AER, I used a numerical method called the Newton–Raphson method.
The method starts with an initial guess to the root of a function. From here, we follow the derivative of the function in order to converge on the true root of the function.
In this case, the function we are interested in is:
Where 
The derivative is:
![f'(r) = \sum_i C_i \left[ \frac{D_i - D_t}{365} \right] (r + 1)^{\left[ \frac{D_i - D_t}{365} - 1\right] }](https://s0.wp.com/latex.php?latex=f%27%28r%29+%3D+%5Csum_i+C_i+%5Cleft%5B+%5Cfrac%7BD_i+-+D_t%7D%7B365%7D+%5Cright%5D+%28r+%2B+1%29%5E%7B%5Cleft%5B+%5Cfrac%7BD_i+-+D_t%7D%7B365%7D+-+1%5Cright%5D+%7D&bg=ffffff&fg=000000&s=0&c=20201002 "f'(r) = \sum_i C_i \left[ \frac{D_i - D_t}{365} \right] (r + 1)^{\left[ \frac{D_i - D_t}{365} - 1\right] }")
The algorithm starts with an initial estimate 
This process is repeated until the difference between successive estimates is less than some threshold value or the number of iterations hits some predefined limit.
AER Calculator
This led me to write the Android application AER Calculator, which can be found on the Google Play Store.
The (open source!) project can be found on GitHub.
