Continuously compounded returns
Rt = (pt − pt−1) / pt−1 × 100% (1.1)
rt = 100% × ln(pt/ pt−1) (1.2)
In many of the problems of interest in finance, the starting point is a time series of prices -- for example, the prices of shares in Ford, taken at 4p.m. each day for 200 days. For a number of statistical reasons, it is preferable not to work directly with the price series, so that raw price series are usually converted into series of returns. Additionally, returns have the added benefit that they are unit-free. So, for example, if an annualized return were 10%, then investors know that they would have got back ￡110 for a ￡100 investment, or ￡1,100 for a ￡1,000 investment, and so on. There are two methods used to calculate returns from a series of prices, and these involve the formation of simple returns, and continuously compounded returns, which are achieved as follows:
If the asset under consideration is a stock or portfolio of stocks, the total return to holding it is the sum of the capital gain and any dividends paid during the holding period. However, researchers often ignore any dividend payments. This is unfortunate, and will lead to an underestimation of the total returns that accrue to investors. This is likely to be negligible for very short holding periods, but will have a severe impact on cumulative returns over investment horizons of several years. Ignoring dividends will also have a distortionary effect on the crosssection of stock returns. For example, ignoring dividends will imply that ‘growth’ stocks, with large capital gains will be inappropriately favoured over income stocks (e.g. utilities and mature industries) that pay high dividends.where: Rt denotes the simple return at time t, rt denotes the continuously compounded return at time t, pt denotes the asset price at time t, and ln denotes the natural logarithm.
(1) Log-returns have the nice property that they can be interpreted as continuously compounded returns – so that the frequency of compounding of the return does not matter and thus returns across assets can more easily be compared.
(2) Continuously compounded returns are time-additive. For example, suppose that a weekly returns series is required and daily log returns have been calculated for five days, numbered 1 to 5, representing the returns on Monday through Friday. It is valid to simply add up the five daily returns to obtain the return for the whole week:
Monday return r1 = ln (p1/p0) = ln p1 − ln p0
Tuesday return r2 = ln (p2/p1) = ln p2 − ln p1
Wednesday return r3 = ln (p3/p2) = ln p3 − ln p2
Thursday return r4 = ln (p4/p3) = ln p4 − ln p3
Friday return r5 = ln (p5/p4) = ln p5 − ln p4
Return over the week ln p5 − ln p0 = ln (p5/p0)
Alternatively, it is possible to adjust a stock price time series so that the dividends are added back to generate a total return index. If pt were a total return index, returns generated using either of the two formulae presented above thus provide a measure of the total return that would accrue to a holder of the asset during time t.
The academic finance literature generally employs the log-return formulation (also known as log-price relatives since they are the log of the ratio of this period’s price to the previous period’s price). Box 1 shows two key reasons for this. There is, however, also a disadvantage of using the log-returns. The simple return on a portfolio of assets is a weighted average of the simple returns on the individual assets. But this does not work for the continuously compounded returns, so that they are not additive across a portfolio. The fundamental reason why this is the case is that the log of a sum is not the same as the sum of a log, since the operation of taking a log constitutes a non-linear transformation. Calculating portfolio returns in this context must be conducted by first estimating the value of the portfolio at each time period and then determining the returns from the aggregate portfolio values. Or alternatively, if we assume that the asset is purchased at time t − K for price Pt−K and then sold K periods later at price Pt , then if we calculate simple returns for each period, Rt , Rt+1, . . . ,RK, the aggregate return over all K periods is
RKt = (Pt − Pt−K)/Pt−K = Pt/Pt−K − 1 = (Pt/Pt−1 X Pt−1/Pt−2 X . . . X Pt−K+1/Pt−K) – 1= [(1 + Rt )(1 + Rt−1) . . . (1 + Rt−K+1)] – 1
In the limit, as the frequency of the sampling of the data is increased so that they are measured over a smaller and smaller time interval, the simple and continuously compounded returns will be identical.