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Say I invested \$100 every month into the S&P 500 since January 2004. If I sold out on 2019-01-11 (not considering fees or dividends), I will have invested \$18,100 and my investment would be worth \$32,331.15.

How do I calculate my average yearly return (interest rate) on this investment?


What I've tried so far

My starting equation is $$ totalValue = monthlyAmount \cdot interest^{numYears} + monthlyAmount \cdot interest^{numYears - 1/12} + \dots + monthlyAmount \cdot interest^{1/12} $$

I can then extract $monthlyAmount$ to simplify the expression:

$$ totalValue = monthlyAmount \cdot (interest^{numYears} + interest^{numYears - 1/12} + \dots + interest^{1/12}) $$

The part in the parentheses, if I'm not mistaken, seems to be the sum of a geometric sequence - where both the first term and the common ratio is $interest^{1/12}$. If so, then I can use the formula for calculating the sum of the first $n$ terms of a geometric sequence:

$$ totalValue = monthlyAmount \cdot \frac{interest^{1/12} \cdot (1-interest^{1/12})}{1-interest^{numYears}} $$

Alright, given the values in the introduction, I have an equation with a single unknown - the $interest$:

$$ 32331.15 = 100 \cdot \frac{interest^{1/12} \cdot (1-interest^{1/12})}{1-interest^{15}} $$

I gave this to WolframAlpha, and it gave me a solution of 1.074 (i.e. 7.4%). That's nice, but how do I arrive at the solution?

I'm writing a program that calculates this in the general case, so I'm ideally looking for some sort of algorithm that can be written in code.


Source of financial information: https://finance.yahoo.com/quote/%5EGSPC/history?p=%5EGSPC

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  • $\begingroup$ Hint: write it out as a sum involving the rate and note that you have a geometric series. $\endgroup$ – lulu Jan 22 '19 at 23:04
  • $\begingroup$ @lulu No, because the stock price changes. $\endgroup$ – Acccumulation Jan 22 '19 at 23:07
  • $\begingroup$ @Acccumulation Why does that matter? You are looking for a single rate $r$ which would match the return over the period. $\endgroup$ – lulu Jan 22 '19 at 23:09
  • $\begingroup$ I'ver misread the problem a couple of times now, so let me state how I am currently reading it. I believe the OP is looking for the rate, $r$, such that a monthly investment of $100$ over $181$ periods would result in a total value of $32,331.15$ Thus, the first $100$ will have become $100\times (1+\frac r{12})^{181}$, the second will have grown to $100\times (1+\frac r{12})^{180}$ and so on. Is this what was intended? $\endgroup$ – lulu Jan 22 '19 at 23:14
  • $\begingroup$ @lulu yes, that's what I meant. See my update for more information $\endgroup$ – Zoltán Jan 22 '19 at 23:15
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Trusting that the schedule of payments I outlined in the comments are correct, the problem comes down to the following: $$100\times \sum_{i=1}^{181}\left(1+\frac r{12}\right)^i=32,311.15$$

Now, it's possible to evaluate that sum as a rational function, but as that leads to a polynomial of degree $182$ in $r$ you'll be stuck with numerical methods anyway. Instead, I'd leave the function as it stands and work numerically. A simple binary search converges rapidly and we get $$r\approx 7.056\%$$

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Considering the general case, you want to solve for $x$ the equation $$k=\frac TM=\sum_{i=1}^n(1+x)^i=\frac{(x+1) \left((x+1)^n-1\right)}{x}$$ where $T$ is the total value, $M$ the monthly amount, $n$ the number of months and $x$ the monthly interest.

As lulu explained, you need some numerical method (Newton method would be the easiest).

You can have good estimates of the solution considering that $x \ll 1$. So, let us expand the summation as a Taylor series (or use the binomial theorem) and get $$k=n+\frac{n (n+1)}{2} x+\frac{(n-1)n(n+1)}{6} n x^2+O\left(x^3\right)$$

Neglecting the higher order terms, you just face a quadratic equation in $x$ anf then $$x_0=\frac{\sqrt{24(n-1)n(n+1)k-3n^2(5 n^2-6 n-11) }-3n(n+1) } {2(n-1)n(n+1) }$$

For your example, this would give as an estimate $x=0.00626955$ corresponding to $7.52$%. Newton iterates would be

$$\left( \begin{array}{cc} n & x_n \\ 0 & 0.0062695483 \\ 1 & 0.0058852014 \\ 2 & 0.0058753116 \\ 3 & 0.0058753053 \end{array} \right)$$ and then the answer $7.050366$%.

We could have a better estimate (still at the price of a quadratic equation in $x$) using, instead of Taylor series, a simple Padé approximant. This would give $$k=\frac{n+\frac{1}{10} \left(n^2+13 n\right) x+\frac{1}{60} \left(n^3+3 n^2+20 n\right) x^2 } {1-\frac{2}{5} (n-2) x+\frac{1}{20} \left(n^2-3 n+2\right) x^2 }$$ Applied to the example, this would give as an estimate $x=0.00587785$ corresponding to $7.053$%.

Edit

We can even avoid solving quadratic equations building at $x=0$ the $[1,p]$ Padé approximant of $$\frac{(x+1) \left((x+1)^n-1\right)}{x}-k=\frac{(k-n)+a_{(p)} x}{1+\sum_{j=1}^p b_j x^j}\implies x_{(p)}=\frac{n-k}{a_{(p)} }$$

For the simplest , the required coefficient is given by $$a_{(1)}=\frac{1}{6} n (n+5)+\frac{1}{3} k (n-1)$$ $$a_{(2)}=\frac{3 n^2 (n+3)+2 (n-1) n (n+4)k+ (n-2) (n-1)k^2 } {2 n (n+5)+4 (n-1)k }$$ The formula becomes really long for $p \geq 3$ but they will be very easy to code.

Applied to the example, this would give $$x_{(1)}\approx 0.00568499\implies 6.82198\text{ %}$$ $$x_{(2)}\approx 0.00585822\implies 7.02987\text{ %}$$ $$x_{(3)}\approx 0.00587585\implies 7.05102\text{ %}$$ $$x_{(4)}\approx 0.00587561\implies 7.05073\text{ %}$$ $$x_{(5)}\approx 0.00587532\implies 7.05039\text{ %}$$

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