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Suppose I borrow a dollar from you and I’ll pay you 100% interest at the end of the year.  How much money will you have then?

$1 * (1 + 1) =$2

What happens if instead the interest is calculated as  50% twice in the year?

$1 * (1.5 * 1.5) =$2.25

After 6 months I owe you $1.50 and then at the end of the year I pay 50% interest on that amount. Or 25% four times per year?$1 * (1.25 * 1.25 * 1.25 * 1.25) = $2.4414 After 3 months I owe you$1.25.  At 6 months I pay you 25% interest on that (which yields \$1.5625).  At 9 months I pay you 25% interest on that, and so on.

## More generally

The formula is:

(1 + 1/n)^n

where n is the number of periods.

We can use R to look at the more general case. Because of R’s vectorization we can do the formula with lots of different n‘s all at once.  And we can easily plot the results.

> eseq <- 1:1000
> plot(eseq, (1+1/eseq)^eseq, type="l", col="blue", lwd=3)

Figure 1: Resulting amount for compounding frequency. From Figure 1 it becomes believable that as the number of periods increases the amount of money converges to a specific value.  That is indeed the case and that number is e.

If we put the x-axis on a logarithmic scale, then we can see better what happens in the plot.  We’ll also add a horizontal line at the value e.

> plot(eseq, (1+1/eseq)^eseq, type="l", col="blue", lwd=3, log="x")
> abline(h=exp(1), lwd=3, col="gold")

Figure 2: Resulting amount for compounding frequency with logarithmic x-axis. From Figure 2 we see that daily compounding is virtually the same as continuous compounding.

Natural logarithms are those that use e as their base.  Note that log returns use natural logarithms.

We’ve just seen why log returns are also called continuously compounded returns.

## Epilogue

Money is the seed of money, and the first franc is sometimes more difficult to acquire than the second million.

Jean-Jacques Rousseau 