Tag Archives: statistics

Linear vs. Logarithmic Returns

Definitions

Simple Returns are defined as
R^S_t = \frac{P_{t+1}}{P_t} - 1 = \frac{P_{t+1} - P_t}{P_t}
Logarithmic Returns are defined as
R^L_t = log (R^S_t + 1) = log (P_{t+1}) - log (P_t)

Main Properties

Simple Returns

Simple returns are additive accross securities. If (\omega_i)_{i=1 \ldots n} are the weights of different securities within a portfolio, with linear returns (R^S_{t,i})_{i=1 \ldots n}, the linear return of the overall portfolio is
R^S_{t} = \sum_{i = 1}^n \omega_i R^S_{t,i}

Logarithmic Returns

Logarithmic returns are additive over time. If the logarithmic returns of a security over time are (R^L_{t+i})_{i=0 \ldots T}, the logarithmic return over the whole [t, t+T] period is:
R^L_{t->t+T} = \sum_{i = 0}^T R^L_{t+i}
If a security follows a Brownian motion, its logarithmic returns are normally distributed.

Difference

The Taylor series decomposition of log is as follows:
log(1 + x) = \sum_{i=1}^\infty (-1)^{(n+1)}\frac{x^n}{n}, \qquad \forall x \in ]-1,1]
So the error of the log-return vs. the simple return is in o(|x|) for returns under 100%.
Typically, the error is less than 0.5% for returns below 10% in absolute value. However, it reaches several percentage points above 15% and more than 10% for returns above 50% or under -40%. The following 2 charts illustrates the extent of the error for small and large values.


Moments

The mean of the log return can be approximated as follows: \overline{R^L} \approx  \overline{R^S} - 0.5\sigma^2_S
The 2nd, 3rd and 4th moment are also affected.

Conclusion

Whenever the terminal wealth is assessed or returns can be higher than a few percentage points, log returns can lead to significant estimation errors.

References

  • Hudson, Robert, Comparing Security Returns is Harder than You Think: Problems with Logarithmic Returns (February 7, 2010). Available at SSRN: http://ssrn.com/abstract=1549328
  • Meucci, Attilio, Quant Nugget 2: Linear vs. Compounded Returns – Common Pitfalls in Portfolio Management (May 1, 2010). GARP Risk Professional, pp. 49-51, April 2010 . Available at SSRN: http://ssrn.com/abstract=1586656
  • Pini, Sergio, Approximations of Portfolio Returns: Are the Errors Really Small? (December 10, 2009). Available at SSRN: http://ssrn.com/abstract=1521442
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University of Michigan Confidence Index: a good indicator of the future performance of stock markets?

1 University of Michigan Survey of Consumer Confidence Sentiment

1.1 Definition

According to Bloomberg, the index is a survey of consumer attitudes concerning both the present situation as well as expectations regarding economic conditions conducted by the University of Michigan. For the preliminary release approximately three hundred consumers are surveyed while five hundred are interviewed for the final figure. The level of consumer sentiment is related to the strength of consumer spending.

1.2 Latest Announcement

The latest announcement on August 12th showed that the University of Michigan Confidence Index had dropped to 54.9 when the market was expecting 62.0. That sharp drop to a very low figure by historical standards has raised concerns over the economy. Bloomberg News analysed the situation in the following terms:
“Aug. 12 (Bloomberg) — Confidence among U.S. consumers plunged in August to the lowest level since May 1980, adding to concern that weak employment gains and volatility in the stock market will prompt households to retrench.”
What nobody is saying is whether that index is a good indicator of future stock market performance, which is the only thing that matters in the end.

1.3 Is the University of Michigan Confidence Index good at forecasting stock market performance?

Although there is no significant relationship between the University of Michigan Confidence Index and stock market returns over the next 6 months, the Index works as a good contrarian signal for longer period. Over the 3 years that follow very low levels of the University of Michigan Confidence Index, the stock market generally performs well, whereas when the Index reaches 100 or more the stock market performance over 3 years is on average flat to negative.

2 Historical Data

2.1 Data

Bloomberg has data about the University of Michigan Confidence Index since January 1978:

Interestingly, it seems that low readings on the University of Michigan Confidence Index have more often than not coincided with market bottoms. The latest example being the lows of Q1 2009.

2.2 Generalisation

2.2.1 Short Term impact

The main caveat of the previous chart is the limited number of occasions when the University of Michigan Confidence Index has been below 60 (8 months out of 398 since January 1978) and the significant dispersion of returns following these announcements. The chart gives a more granular view on the relationship and the main conclusion seems to be that there is no relation between the University of Michigan Confidence Index and the short term performance of the stock market.

2.2.2 Long Term Impact

Over a longer time horizon, extreme levels constitute a good contrarian signal: the market has almost always had a positive performance over 3 years when the University of Michigan Confidence Index has fallen below 72.

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