Converting Annual Interbank Rates to Monthly Returns
Summary
The document explains how to convert an annual effective interest rate into a rate for a shorter period, in the context of using an interbank rate as a proxy for the monthly risk-free rate. It gives compounding-based conversions for monthly and quarterly periods, and distinguishes those conversions from simply dividing the annual rate by twelve or four.
For an annual rate, the monthly effective rate is obtained by taking the twelfth root of one plus the annual rate, then subtracting one. The answer notes that dividing by twelve is a close linear approximation when rates are small. The example says an annual rate of 3% produces approximately 0.25% per month, so the proposed excess-return subtraction is nearly equivalent in that case. The document does not discuss day-count conventions, whether the quoted interbank rate is effective or annualized, or timing alignment between the rate observation and monthly stock return.
Key ideas
- Convert an annual effective rate to a monthly rate using compounding over twelve periods.
- A quarterly rate can likewise be converted using the appropriate fractional-year exponent.
- Dividing a small annual rate by twelve is an approximation to the compounded monthly conversion.
- The example’s 3% annual rate corresponds to approximately 0.25% monthly.
- The document does not address rate quoting conventions or day-count and observation timing details.
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Full text
# convert three months interbank rate into monthly rate # convert three months interbank rate into monthly rate I have a time series of the three month interbank rate each month and I suppose that the rate is has yearly frequency. I need these interbank rates to be on a monthly basis because I want to us these rate as a proxy for the risk- free rate, so I can subtract the rates from my monthly return to get the excess rate. Let's consider an example: by assuming the monthly return of stock x in April is 6%. The interbank rate at the end of April is 3% annually. Can I simply subtract o.25%(3/12) from 6% resulting in 5.75% of excess return. Any help would be much appreciated. ## Answer by mt_christo (score 1) https://quant.stackexchange.com/a/19202 To get one-month rate X from three-month rate Y, you use this formula: `1 + X = (1 + Y)^(1/3)` To get one-month rate X from annual (12-month) rate Y, you use this formula: `1 + X = (1 + Y)^(1/12)` To get three-month rate X from annual (12 month) rate Y, you use: `1 + X = (1 + Y)^(3/12)` 0.25% is annual 3% converted to a monthly rate, i.e. (1+0.03)^(1/12) - 1 By the way, multiplying by 1/12 or 3/12 etc. is just a linear approximation of raising numbers like 1.03 (that are close to 1) to the power of 1/12 and 3/12 respectively. ## Answer by Rime (score 0) https://quant.stackexchange.com/a/18744 I think you are on the right track... To double check I tried to get the Monthly return by using the following formula: `Monthly Return = [(1 + Annual Rate)^(1/12)]-1` by using this formula with the stated 3% Annual Return, the Investor receives approximately 0.25% per month....
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