Compounding Frequency and Annualized Returns in Valuation
Summary
The discussion explains why annualized return figures cannot be carried across different holding periods without specifying the compounding convention. It contrasts an investment growing by an annual rate compounded once with alternatives compounded monthly or daily; each convention produces a different future value from the same quoted annual rate. In the example, the stated one-year value corresponds to annual compounding, so discounting over a partial year uses that annual rate raised over the remaining fraction of a year.
Dividing an annual rate into monthly pieces and multiplying by the number of months is not automatically equivalent to the partial-period discounting calculation. The response suggests that period scaling in applications such as value-at-risk depends on the assumed compounding convention. It offers simple illustrations but does not develop expected-return valuation under uncertainty or distinguish arithmetic from geometric averages in depth, so those topics remain beyond its explanation.
Key ideas
- A quoted annual return needs a compounding convention to determine values over different horizons.
- Annual, monthly, and daily compounding produce different future values from the same stated annual rate.
- Partial-year discounting must follow the convention used to establish the annual value.
- Scaling an annual rate by months assumes a convention and may not match annual compounding.
Tags
Full text
# Monthly and annual arithmetic mean in valuations? # Monthly and annual arithmetic mean in valuations? I know this is back to basics but I am perplexed by it!!! Assume that the future value (FV) of an investment at the end of year 1 is 112, the annual arithmetic expected return is 12%, hence the present value (PV) today is 100 since 112/(1.12)=100. It is also evident that the monthly arithmetic average expected return is 1% since 12%/12months= 1%. Assume that a month has passed, what is the value of the investment? It is known that this is 120/(1.12)^(11/12)= 100.948879293. However, we can also say that the 11 month expected (average) arithmetic return is 1% x 11 =11%. However, if we use this as the discount rate the result is different since 112/1.11= 100.900900901. Why is this approach wrong? At time 0 we use the 12% which is 1% x 12 but when we use this approach at the end of the first month/beginning of the second month, the result is inaccurate. The same holds for other months and the difference can be much larger than my example. I have often seen (e.g., in value at risk calculations) that the annual expected return (in my example 12%) is divided by 12 and then multiplied by 3 to find the 3 month value at risk for instance. What is wrong with my approach in the valuation example? I am confused about the fact that assuming that in ex ante valuations we use the arithmetic average return as the discount rate, if the arithmetic average return is 11% in my example, why is this not the correct discount rate? Or is it because the rate has to always be the 12% and what changes is the time (i.e., 1, 11/12, 11/11, etc) by definition? What is then the significance of the 1% monthly expected return and why is it used in some instances? ## Answer by Tarun Bhasker L (score 1) https://quant.stackexchange.com/a/74203 Returns are usually quoted on annual basis but to arrive at right PV(or FV) you need to know the compounding term. Below are some examples. - 12% compounded annually: \$100 now is \$112 in 1 year. [100*(1+0.12)] - 12% compounded monthly: \$100 now is \$112.682 in 1 year. [100*(1 + 0.12/12)^12] - 12% compounded daily(365 days): \$100 now is \$112.747 in 1 year. [100*(1 + 0.12/365)^365] Coming to numbers in your question. - "Assume that the future value (FV) of an investment at the end of year 1 is 112, the annual arithmetic expected return is 12%, hence the present value (PV) today is 100 since 112/(1.12)=100" -> this is 12% return compounded annually. - "I have often seen (e.g., in value at risk calculations) that the annual expected return (in my example 12%) is divided by 12 and then multiplied by 3 to find the 3 month value at risk for instance." - this probably would have been 12% compounded quarterly. Finally I want to note that these differences in compounding may not be significant depending on the context you are using.
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.