I am reading a book on fixed income instruments and don't quite understand one of the examples on compounded rates. Let's say the investement is compounded yearly at rate $r$. Then after $T$ years, where $T$ is an integer, the account should contain $(1+r)^T$ times it's initial balance. The book claims that even for non-integer values of $T$ the formula of $(1+r)^T$ is correct.

I don't really understand why this is the case. I would have thought that the accrued interest is linear in time and proportional to the balance at the latest compounding date.


I would have thought that the accrued interest is linear in time and proportional to the balance at the latest compounding date.

No it's not linear - in fact it's pretty easy to demonstrate. Say you start with \$100 compounded at 10% annually. After 1 year You'll have $\$100 * (1+0.10) = \$110$. Since the interest is compounded (meaning that the interest is added to the balance), After 2 year You'll have $\$110 * (1+0.10) = \$121$. After 3 years, $\$133.1$. So the interest earned is not linear.

That said, the claim that $(1+r)^T$ is also correct for non-integer values of $T$ is not quite true. It depends on how often the interest compounds and what the interpretation of $r$ is. It's a decent approximation for smaller values of $r$, but look at a $1,000 investment at 20% interest that compounds semiannually (meaning 10% every 6 months).

After 1 year, you'll have $(\$1,000 * (1+0.1)) * (1+0.1) = \$1,210 $ versus $\$1,000 * (1+0.2) = \$1,200 $. As the $r$ goes up, the error between the two methods increases.

Alternately if you interpret 20% as an annualized rate, the semi-annual calculation would be $\$1,000 * (1+0.2)^{1/2} = \$1,095 $ versus $\$ 1,100$ for the equivalent semi-annual rate.

But, from an investment standpoint, these differences are usually negligible and the easier formula is "good enough" when making comparative analyses (so long as the usage is consistent).

  • $\begingroup$ I understand that a 20% rate compounded annually means after each year they add 20% to the balance and then give you 20% on that the following year and so on. So for integer time T the balance will have increased by a factor of 1.2^T. However I still don't quite see why after half a year the balance should have increased by a factor of 1.2^0.5. $\endgroup$
    – Jemlin95
    May 7 '20 at 14:56
  • $\begingroup$ @Jemlin95 It's not - it's a rough approximation in some cases but it's not mathematically correct. $\endgroup$
    – D Stanley
    May 7 '20 at 15:02
  • $\begingroup$ so for the example of 20% compounded annually; Ff we were to graph the function 1.2^x and then connect the functional values at integer values of x with straight lines the resulting graph would be the true accrued interest correct? $\endgroup$
    – Jemlin95
    May 7 '20 at 15:17
  • $\begingroup$ It would be closer, yes, but day count conventions can also make slight differences. But conceptually, yes, that would be correct. $\endgroup$
    – D Stanley
    May 7 '20 at 15:29

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