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The rate of return on a portfolio is the rate of return on a portfolio measured over a period of time.

Calculation

The rate of return on a portfolio can be calculated either directly or indirectly, depending upon the particular type of data available.

Direct measurement

Direct measurement of the rate of return on a portfolio applies one of several alternative methods, such as for example the time-weighted method, or the modified Dietz method.^[1]^[2] It requires knowledge of the value of the portfolio at the start and end of the period of time under measurement, together with the external flows of value into and out of the portfolio at various times within the time period. For the time-weighted method, it is also necessary to know the value of the portfolio when these flows occur (i.e. either immediately after, or immediately before).

Indirect calculation

The rate of return on a portfolio can be calculated indirectly as the weighted average rate of return on the various assets within the portfolio.^[3] The weights are proportional to the value of the assets within the portfolio, to take into account what portion of the portfolio each individual return represents in calculating the contribution of that asset to the return on the portfolio.

The indirect calculation of the rate of return on a portfolio can be expressed by the formula:

r = A₁r₁ + A₂r₂ + ... + A_nr_n

where:

r equals the rate of return on the portfolio,

A_i equals the weight of the ith asset in the portfolio, and

r_i equals the rate of return on the ith asset in the portfolio.

Example

rate of return on a mining stock also called r_m equals 10%
rate of return on a child care centre also called r_c equals 8%
rate of return on a fishing company also called r_f equals 12%

Now suppose that 40% of the portfolio is in the mining stock (weighting for this stock also called A_m), 40% is in the child care centre (weighting for this stock also called A_c) and the remaining 20% is in the fishing company (weighting for this stock also called A_f). To determine the rate of return on this portfolio, first calculate the contibution of each asset to the return on the portfolio, by multiplying the weighting of each asset by its rate of return, and then add these contributions together:

for the mining stock, its weighting is 40% and its rate of return is 10% so its contribution equals 40% x 10% = .04 = 4%
for the child care centre, its weighting is 40% and its rate of return is 8% so its contribution equals 40% x 8% = .032 = 3.2%
for the fishing company, its weighting is 20% and its rate of return is 12% so its contribution equals 20% x 12% = .024 = 2.4%

Adding together these percentage contributions gives 4% + 3.2% + 2.4% = 9.6%, resulting in a rate of return on this portfolio of 9.6%.

Discrepencies

If there are any external flows or other transactions on the assets in the portfolio during the period of measurement, and also depending on the methodology used for calculating the returns and weights, discrepencies may arise between the direct measurement of the rate of return on a portfolio, and indirect measurement (described above).

References

^ *Carl Bacon. Practical Portfolio Performance Measurement and Attribution. West Sussex: Wiley, 2003. ISBN 0-470-85679-3
^ *Bruce J. Feibel. Investment Performance Measurement. New York: Wiley, 2003. ISBN 0-471-26849-6
^ Levy,A 2009, ECON331 'Uncertainty, risky assets (activities) and portfolio choice', lecture notes accessed 22 May 2009 elearning.uow.edu.au

[1] *Carl Bacon. Practical Portfolio Performance Measurement and Attribution. West Sussex: Wiley, 2003. ISBN 0-470-85679-3

[2] *Bruce J. Feibel. Investment Performance Measurement. New York: Wiley, 2003. ISBN 0-471-26849-6

[Levy_2009-3] Levy,A 2009, ECON331 'Uncertainty, risky assets (activities) and portfolio choice', lecture notes accessed 22 May 2009 elearning.uow.edu.au

[1]

[2]

[3]

@@ Line 1: / Line 1: @@
+The '''rate of return on a portfolio''' is the [[rate of return]] on a [[portfolio (finance)|portfolio]] measured over a period of time.
-The '''rate of return on a portfolio''' is "a [[Weighted mean|weighted average]] of the [[Rate of return|rates of return]] on the various assets with the weights being equal to the fractions of the individual's wealth held in these assets" <ref name="Levy 2009">Levy,A 2009, ECON331 'Uncertainty, risky assets (activities) and portfolio choice', lecture notes accessed 22 May 2009 elearning.uow.edu.au</ref> This equates to an average of the average returns on a [[portfolio (finance)|portfolio]] taking into account what portion of the portfolio each individual return represents.
-==Example==
+==Calculation==
+The rate of return on a portfolio can be calculated either directly or indirectly, depending upon the particular type of data available.
+===Direct measurement===
+Direct measurement of the rate of return on a portfolio applies one of several alternative methods, such as for example the [[True time-weighted rate of return|time-weighted method]], or the [[modified Dietz method]].<ref>*Carl Bacon. Practical Portfolio Performance Measurement and Attribution. West Sussex: Wiley, 2003. ISBN 0-470-85679-3</ref><ref>*Bruce J. Feibel. ''Investment Performance Measurement''.  New York: Wiley, 2003. ISBN 0-471-26849-6</ref> It requires knowledge of the value of the portfolio at the start and end of the period of time under measurement, together with the [[True time-weighted rate of return|external flows]] of value into and out of the portfolio at various times within the time period. For the time-weighted method, it is also necessary to know the value of the portfolio when  these flows occur (i.e. either immediately after, or immediately before).
+===Indirect calculation===
+The rate of return on a portfolio can be calculated indirectly as the '''[[Weighted mean|weighted average]] [[rate of return]]''' on the various assets within the portfolio.<ref name="Levy 2009">Levy,A 2009, ECON331 'Uncertainty, risky assets (activities) and portfolio choice', lecture notes accessed 22 May 2009 elearning.uow.edu.au</ref> The weights are proportional to the value of the assets within the portfolio, to take into account what portion of the portfolio each individual return represents in calculating the '''contribution''' of that asset to the return on the portfolio.
+The indirect calculation of the rate of return on a portfolio can be expressed by the formula:
+:''r'' = ''A<sub>1</sub>r<sub>1</sub> + A<sub>2</sub>r<sub>2</sub> + ... + A<sub>n</sub>r<sub>n</sub>''
+where:
+:''r'' equals the rate of return on the portfolio,
+:''A<sub>i</sub>'' equals the weight of the ith asset in the portfolio, and
+:''r<sub>i</sub>'' equals the rate of return on the ith asset in the portfolio.
+====Example====
 *rate of return on a [[mining]] stock also called r<sub>m</sub> equals 10%
@@ Line 7: / Line 27: @@
 *rate of return on a [[fishing]] company also called r<sub>f</sub> equals 12%
-Now assume that 40% of the portfolio is in the mining stock (weighting for this stock also called ''A<sub>m</sub>''), 40% is in the child care centre (weighting for this stock also called ''A<sub>c</sub>'') and the remaining 20% is in the fishing company (weighting for this stock also called ''A<sub>f</sub>''). To determine the rate of return on this portfolio, multiply the weighting of each asset by its rate of return and add these figures together:
+Now suppose that 40% of the portfolio is in the mining stock (weighting for this stock also called ''A<sub>m</sub>''), 40% is in the child care centre (weighting for this stock also called ''A<sub>c</sub>'') and the remaining 20% is in the fishing company (weighting for this stock also called ''A<sub>f</sub>''). To determine the rate of return on this portfolio, first calculate the contibution of each asset to the return on the portfolio, by multiplying the weighting of each asset by its rate of return, and then add these contributions together:
+*for the mining stock, its weighting is 40% and its rate of return is 10% so its contribution equals 40% x 10% = .04 = 4%
+*for the child care centre, its weighting is 40% and its rate of return is 8% so its contribution equals 40% x 8% = .032 = 3.2%
+*for the fishing company, its weighting is 20% and its rate of return is 12% so its contribution equals 20% x 12% = .024 = 2.4%
+Adding together these percentage contributions gives 4% + 3.2% + 2.4% = 9.6%,
-*for the mining stock, its weighting is 40% and its rate of return is 10% so it equals 40% x 10% = .04 = 4%
+resulting in a rate of return on this portfolio of 9.6%.
-*for the child care centre, its weighting is 40% and its rate of return is 8% so it equals 40% x 8% = .032 = 3.2%
-*for the fishing company, its weighting is 20% and its rate of return is 12% so it equals 20% x 12% = .024 = 2.4%
-Adding these percentages gives 4% + 3.2% + 2.4% = 9.6%
-therefore the rate of return on this portfolio = 9.6%
+===Discrepencies===
-Mathematically this whole process may be written as
+If there are any external flows or other transactions on the assets in the portfolio during the period of measurement, and also depending on the methodology used for calculating the returns and weights, discrepencies may arise between the direct measurement of the rate of return on a portfolio, and indirect measurement (described above).
-''r<sub>p</sub>'' = ''A<sub>m</sub>r<sub>m</sub> + A<sub>c</sub>r<sub>c</sub> + A<sub>f</sub>r<sub>f</sub>'' where ''r<sub>p</sub>'' equals the rate of return on the portfolio.
 ==References==