Simple Returns
Simple returns (more precisely: simple rate of returns) are basically the percent change in market value of a certain assets over time.
Working with rates insted of market values assumes implicitly that the size of the asset is irrelevant. In an economic interpretation, this means that the investment "technology" exhibits constant-returns-to-scale and that the asset return is a complete and scale-free characteristics for all actual and prospective investors alike. This is correct in a perfectly competitive market where the size of the asset has no impact on price and therefore market value.
Rates are "flow variables" and represent a percentage change over a certain time horizon.
Usually, returns are calculated starting from a certain inception time to later points in time. This is usually what is done in the contex of a single investor's portfolio. But it is also possible to calculate returns from inception to a point in time "back" in the past. Such returns are very interesting in the context of pooled assets (funds) in which money is contributed and withdrawn from the same and/or different investors over time. Such holdings period returns express the rate of return for investors that contributed their money at a certain point of time in the past.
Modeling time in practical return measurement is far from trivial: Time issues like calendar versus trading days, trade versus settlement dates, closing versus intra-day time etc. make it difficult to model time consistently for one asset and even more so across several assets.
Time can be represented in a a discrete or continuous manner.
Discrete returns (more precisely: simple rate of returns measured in discerete time) assume that trading (and therefore market valuation) takes place at certain points in time. Mathematically speaking, a series of n returns R sampled over the time period 0 to T is said to be "discrete", if t = T/n is an integer for all R(t).
Continuous returns (more precisely: simple rate of returns measured in continuous time) assume that trading can take place anytime, and therefore that the market value of the portfolio grows"continuously" over time, and not in "discrete" steps. You can download a spreadsheet illustrating the effect of the number of trading points on the difference between the discrete and the continuous return here. Mathematically speaking, a series of n returns r sampled over the time period 0 to T is is said to be "continuous", if t = T/n is a real value for all r(t).
Leveraged funds can potentially have returns below -100%. As the ln[.] function for values below 0 yields complex numbers, continuous returns for leveraged funds can be difficult to interpret. Continuous returns are always smaller than their discrete counterparts. The smaller the return, the smaller the difference between discrete and continuous versions.
Investment managers typically think in terms of trading days, therefore working with continuous returns is appropriate. The client's perspective, on the other hand, is shaped by the reporting frequency, typically on a monthly basis. When communicating returns to existing or prospective clients, discrete figures are more appropriate.
A general shortcoming of returns is the fact that their calculation involves a division: If market value can take on the value of 0, returns can be undefined. For example, try to calculate returns for the following market value series {15, 5, -15, 10, -20, 25}.
Notation & Basic Definitions
To calculate returns, it is important to develop a consistent notation...
MV(t)... Market value a a point in time t
Single Period Rate of Return in Discrete Time: R(t-n,t) = (MV(t)-MV(t-n))/MV(t-n)=1+MV(t)/MV(t-n), n as integer, n=1, t = {0,...,n}
Multi Period Rate of Return in Discrete Time: R(t-n,t) = (MV(t)-MV(t-n))/MV(t-n), n>1
Holdings Period Rate of Return in Discrete Time: R(t-n,t) = (MV(t)-MV(t-1))/MV(t-n), n<0
Period Rate of Return in Continuous Time: r(t-n,t) = ln((MV(t)/MV(t-n)) = ln(MV(t))-ln(MV(t-n))
Some obvious relationships that immediatly follow from the above definitions...
1. Discrete & Continuous returns
r(t-n,t) = ln(1+R(t-n,t))
R(t-n,t) = exp(r(t-n,t)) -1
2. Single-Period & Multi-Period Discrete Returns
1+R(t-n,t) = MV(t)/MV(t-n) = MV(t)/MV(t-1) *MV(t-1)/MV(t-2) * ... * MV(1)/MV(0)
1+R(t-n,t) = (1+R(t-1,t))*(1+R(t-2,t-1))*...*(1+R(1,0)) = prod(1+R(t-i-1,t-i, i=1..n)
..this is the so-called "chain-linking" of single period discrete returns to multiperiod discrete returns: "discrete returns don't behave additive across time (but "geometric)".
3. Single-Period & Multi-Period Continuous Returns
r(t-n,t) = ln ( (1+R(t-1,t))*(1+R(t-2,t-1))*...*(1+R(1,0)) )
r(t-n,t) = ln ( 1+R(t-1,t)) + ln(1+R(t-2,t-1)) + ... + ln(1+R(1,0))
r(t-n,t) = r(t-1,t) + r(t-2,t-1) + ... + r(1,0) = sum(r(t-i-1,t-i, i=1..n)
...the multi-period continuous return is the sum of single period continuous returns: "continuous returns behave additive across time".
4. Aggregating Discrete Returns Accross Several Assets
i as integer = {1..m}, m as integer = number of assets
1+R(t-n,t,i) = MV(t,i)/MV(t-n,i) ...definition of asset return
MV(t-n,i)*(1+R(t-n,t,i)) = MV(t,i)
1+R(t-n,t) = sum(MV(t,i))/sum(MV(t-n,i)) ...definition of portfolio return
1+R(t-n,t) = sum((1+R(t-n,t,i))*MV(t-n,i))/sum(MV(t-n,i))
w(t,i) = MV(t,i)/sum(MV(t-n,i))
1+R(t-n,t) = sum(w(t-n,i) *(1+R(t-n,t,i)))
R(t-n,t) = sum(w(t-n,i) *R(t-n,t,i))
...portfolio return is the beginning-market-value weighted sum of asset returns: "discrete returns behave additive across assets".
5. Aggregating Continuous Returns Accross Several Assets
ln(1+R(t-n,t)) = ln(sum(w(t-n,i) *(1+R(t-n,t,i))))
r(t-n,t) = ln(sum(w(t-n,i) *exp(r(t-n,t,i)))) > sum(w(t-n,i) *r(t-n,t,i))
...continous portfolios cannot be expressed as a simple weighted average of their continous constituent returns: "continuous returns don't behave additive across assets".
Below some further relationships that do not follow immediatly from the initial difinitions, but require additional assumptions
1. General Approximation Formula for Continuous Returns
r(t-n,t) = ln(1+R(t-n,t)) ~ R(t-n,t) - 0.5 * R(t-n,t)^2
...for | R(t-n,t) | small relative to 1.
2. Taylor Series Approximation
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3. Distribution of Expected Returns when Applying the Central Limit Theorem
Independently of the distribution of continuous single-period returns, the CTL ensures that "average" (=expected) continuous returns are normally distributed and therefore discrete returns lognormally.
It is important to keep in mind that the distribution of expected returns is the result of a fundamental statistical relationship, and not caused by an distributional assumptions. On the other hand, the result is only valid within the assumptions of the CTL: a) finite variance and b) absence of serial correlation.
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Expected Returns
Relationship geometric mean and average single-period return and arithmetic mean and average continuous single-period return.
Relationship expectations operator and average mean.
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