Combining Annual Stock Returns into a Multi-Year Annualized Return
Summary
The document distinguishes a stock’s return over an individual year from its annualized return across multiple years. It describes an example with an initial investment, a change in year-end price, and no purchases or sales during the holding period. The question is whether the overall annualized return can be derived from the two yearly returns.
The accepted response says to combine the annual growth factors multiplicatively and take their geometric average over the number of full years, then subtract one. It also points out that the result depends slightly on the exact ending date used in the XIRR calculation: a year-end date and the following January 1 do not represent precisely the same elapsed time. The example uses a two-year period and shows the calculation with the stated yearly returns. This approach assumes reinvestment through compounding and a consistent definition of the period; irregular cash flows or different annualization conventions require corresponding adjustments.
Key ideas
- Multi-year annualized returns compound yearly growth factors rather than add yearly percentage returns.
- For equal-length periods, the annualized return is the geometric average of the period growth factors minus one.
- The elapsed dates supplied to XIRR can affect the calculated annualized rate.
- A comparison of yearly and overall returns assumes a consistent holding period and compounding convention.
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# 12974 # Is there any relationship between investment return of a stock, for individual year (Yearly return) and multiple years (Overall annualized return) currently, I'm calculating the return of a stock, for individual year and multiple years. I tend to answer the following question. - Return of a stock for end of year 2010 (Individual year) - Return of a stock for end of year 2011 (Individual year) - Return of a stock for beginning of year 2010 till end of year 2011 (Multiple years) Assuming, - I purchase 1 unit of stock at 1st January 2010, at price $1 - No further buy transaction & sell transaction in between, in beginning of year 2010 till end of year 2011 Here's my calculation ### Return for end of year 2010 (Stock price reach $2.5 at end of year) ``` Date Price 01/01/2010 -$1.0 (Invest) 31/12/2010 $2.5 XIRR([01/01/2010, 31/12/2010], [-1.0, 2.5]) = 1.506 ``` ### Return for end of year 2011 (Stock price reach $1.8 at end of year) We assume stock opening price at the beginning of year is $2.5. ``` Date Price 01/01/2011 -$2.5 (Invest) 31/12/2011 $1.8 XIRR([01/01/2011, 31/12/2011], [-2.5, 1.8]) = -0.2806 ``` ### Return for beginning of year 2010 till end of year 2011 (Multiple years) ``` Date Price 01/01/2010 -$1.0 (Invest) 31/12/2011 $1.8 XIRR([01/01/2010, 31/12/2011], [-1.0, 1.8]) = 0.3422 ``` I was wondering, is there any relationship between 0.3422 (Return for multiple years) and 1.506 (Return for individual year), -0.2806 (Return for individual year). Is there a way for me to derive 0.3422, based on value 1.506 & -0.2806 ? ## Answer by RJT (score 3, accepted) https://quant.stackexchange.com/a/12975 Are you sure the return for two years is 0.7214? It should be 0.3422 per year if you are using 31/12/2011, and 0.3416 if you are using 01/01/2012 as the end date. Assuming the last number (because it makes for two full years, therefore easier to calculate), yes, there is a formula to derive it from the return of the individual years. It's the geometric average: ``` =SQRT((1+1.506) * (1-0.2806)) - 1 ```
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