Fractional-Year Compounding Depends on Rate and Compounding Convention
Summary
This explanation examines whether an account compounded annually at rate r can be valued after a fractional number of years using the power formula for compound growth. It first clarifies that compound interest is not linear in time: each period’s interest becomes part of the balance on which later interest accrues.
The response cautions that extending an annual compounding formula to fractional periods depends on how the rate is defined and how often compounding occurs. A stated nominal annual rate compounded semiannually produces a different result from applying that same figure as an effective annual rate. It illustrates the distinction with examples and notes that the simpler fractional-power approach may be adequate for rough comparisons when applied consistently. The discussion is conceptual and does not specify a universal day-count convention or cover market-specific fixed-income accrual rules, so precise calculations require the instrument’s stated conventions.
Key ideas
- With compounding, interest accrues on the accumulated balance, so growth is not linear in time.
- A fractional-year growth formula depends on the rate definition and compounding frequency.
- A nominal annual rate compounded within the year differs from an effective annual rate.
- Simplified formulas may support comparisons when used consistently, but exact valuation requires the applicable conventions.
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# Accrued interest on yearly compounded instrument after less than a year
# 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.
## Answer by D Stanley (score 0, accepted)
https://quant.stackexchange.com/a/53939
> 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).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.