Why Overnight Rates Compound Across Daily Accrual Periods
Summary
The document examines the annualized effective rate for borrowing over a period using backward-looking overnight rates. It presents a formula that compounds each observed annual rate over its applicable day count using simple interest for that accrual interval, then annualizes the total compounded return across the full period. The questioner derives this by equating the accumulated principal to a single simple-interest return over the full period and asks why the daily accruals are not instead compounded as if each rate applied for a whole number of days.
The core distinction is between simple interest within each overnight accrual interval and compounding the resulting amounts from one interval to the next. The document gives the formula and an algebraic framing, but no answer or independent evidence. Its convention and day-count basis are specific to the stated borrowing setup; it does not discuss alternative market conventions, calendar adjustments, or how to handle differing benchmarks and observation rules.
Key ideas
- Each overnight rate applies as simple interest over its associated accrual days.
- The interest-bearing balance compounds from one accrual interval into the next.
- The accumulated return is converted to an annualized effective rate for the full period.
- The document poses the convention question but does not include a response.
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Full text
# How do I derive the effective rate per period for backward looking overnight interbank borrowing?
# How do I derive the effective rate per period for backward looking overnight interbank borrowing?
The formula for the effective rate per period (of say $n$ days) for backward looking overnight interbank *borrowing has been quoted as follows in my textbook:
$$ \left( \prod_{i=1}^{n} \left(1 + \frac{r_i d_i}{360}\right) - 1 \right) \cdot \frac{360}{n} = R $$
Here:
- $r_i$ is the rate of borrowing per annum that applied to $d_i$ days of the borrowing period.
- $n$ is the total number of days in the period, which is the summation of $d_i$ (i.e., $∑_{i=1}^{n} d_i $ )
My approach is that this rate formula has been derived by using the concept of the geometric mean and simple interest
Initially, a principal amount $P_0$ gained simple interest and amounted to
$$ P_0 \left(1 + \frac{r_1 d_1}{360}\right) $$
after $d_1$ days. This amount then gained simple interest and compounded to amount
$$ P_0 \left(1 + \frac{r_1 d_1}{360}\right) \left(1 + \frac{r_2 d_2}{360}\right) $$
at the end of $d_2$ days, and so on...
The initial principal after compounding in such a fashion grew to the same amount which gained simple interest over a period of $n$ days at an $R$ percent per annum:
$$ P_0 \left(\prod_{i=1}^{n} \left(1 + \frac{r_i d_i}{360}\right)\right) = P_0 \left(1 + \frac{R n}{360}\right) $$
I don't understand why the initial principal gained "simple interest" at $r_i$ rate for $d_i$ days and then got compounded again at resp. $r_i$ rate for the rest $d_i$ days, instead of gaining "compound interest" at resp $r_i$ rate for $d_i$ days and then compounding again for the rest of the days, making the previous equation as follows:
$$ P_0 \left(\prod_{i=1}^{n} \left(1 + \frac{r_i}{360}\right)^{d_i}\right) = P_0 \left(1 + \frac{R}{360}\right)^n $$
But I am unable to solve for $R$ using the equation above. Is there a convention in interbank borrowing to use only simple interest since for short periods of borrowing compound interest & simple interest roughly have the same value, and, is my derivation using simple interest and geometric mean correct? Please help.
*Edit: assumption here is that the overall period composed of subperiods 1, 2 etc. lasts less than a year.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.