# Accrued interest on yearly compounded instrument after less than a year

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).

• 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. May 7 '20 at 14:56
• @Jemlin95 It's not - it's a rough approximation in some cases but it's not mathematically correct. May 7 '20 at 15:02
• 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? May 7 '20 at 15:17
• It would be closer, yes, but day count conventions can also make slight differences. But conceptually, yes, that would be correct. May 7 '20 at 15:29