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In engineering (in particular, in fiber optics), the logarithmic form of representing numbers is often used. It is convenient if the value sequentially passes through a series of points with input and output. And at each point, the output value is equal to the input value multiplied by a certain coefficient. For example, a laser diode has a power of -5dBm, a fiber has an attenuation factor of 0.3dB/km, and connectors have an attenuation factor of 0.15dB. Therefore, at the output of a 10-kilometer fiber, the radiation power will be (-5)-0.15-(0.3x10)-0.15=-8.3dBm. It would be much more unpleasant to consider this in linear terms. Therefore, transmitter powers and receiver sensitivities are often given directly in logarithmic terms (dBm) rather than linear values ​​(mW).

I think that different schemes for bank deposits and other investments may also predispose in some cases to the use of logarithmic values. Is it so? Are logarithmic values ​​used in finance?

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  • $\begingroup$ A good example is doubling-time which is used extensively in finance. $\endgroup$
    – JimmyJames
    Jun 27, 2023 at 15:33
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    $\begingroup$ Or try the confusing continuously compounding rate where if you start with $\$100$ and a year later have $\$110$ and a year after that $\$121$, this is is not described as $10\%$ compound annual interest but as $\log_e(1.10)\approx 0.09531$. On the other hand, places where logarithms might be useful, such as elasticity calculations, they are less used $\endgroup$
    – Henry
    Jun 27, 2023 at 17:03
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    $\begingroup$ There are places where logarithms should be used but usually are not taught that way at least initially, such as elasticity: for example if prices doubling lead to demand halving (or the other way round) then it might be better to say this suggests an elasticity of $-1$, but you only get that result using logarithms. $\endgroup$
    – Henry
    Jun 28, 2023 at 10:45
  • $\begingroup$ The use of natural logarithm is common place in finance, not so much log10. $\endgroup$ Jun 28, 2023 at 12:54

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Logarithmic values are used very frequently.

  • financial modelling and statistics often use logarithmic values for its properties
  • logarithmic scales are used to display long term (exponential) growth (e.g. SPX since inception) => reason 3 in the link above
  • as pointed out by @bajun65537, many financial models like the Black Scholes option pricing model assume a lognormal distribution of returns (meaning log returns are normally distributed) => reason 5 in the link above
  • logarithmic values are even traded directly: a var swaps can be created by replicating a log contract position.
  • historical volatility is usually computed as standard deviation of log returns
  • ...

Edit I know nothing about fiber optics but I fail to see what would make dBd any different from using logarithms in finance? On the contrary, finance almost always uses the natural logarithm, which is the only logarithm where the base is not some arbitrary number and the most well behaved (in terms of calculus). The reason dBd uses the base 10 logarithm is (according to my rudimentary understanding) simply that it is convenient for human beings because it makes (manual) calculations in the decimal system a lot easier to grasp.

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  • $\begingroup$ As I understand it, the logarithms themselves are actively used in finance, but there is no specialized value like the decibel-dollar (dBd). Is it so? $\endgroup$
    – Arseniy
    Jun 28, 2023 at 11:58
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    $\begingroup$ @Arseniy I use something we call log returns at least once an hour in my finance job and that us a specialized value although it doesn't have a special name other than its description $\endgroup$
    – MD-Tech
    Jun 29, 2023 at 7:29
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Logarithmic values are used in stock price analysis and volatility measurements as mentioned by @AKdemy. Some other examples include:

  1. Black–Scholes model posits that instruments, such as stock shares or futures contracts, will have a lognormal distribution of prices following a random walk with constant drift and volatility. Well, the distribution does not need to be lognormal, but it's often the case. The logarithmic returns are quite essential for the underlying assumptions and calculations in these models.

  2. Logarithmic returns are extensively used in portfolio optimization, where investors aim to construct portfolios that maximize returns or minimize risks. Essentially you want to take advantage of the log properties like additivity, stability (log returns are less affected by outliers than arithmetic returns) or their interpretability. One needs to be careful when using log returns though. As mentioned here:

It is unfortunately common to see people that purely work with log-returns by linearly aggregating across trades and time. Implicitly they are assuming that $r^{(i)}_d≈R^{(i)}_d$, which is true for small returns. However, when this is not the case, or when summing many small returns e.g. return over large horizons, this might cause large errors.

  1. Logarithmic returns are also used in risk management, for example to calculate the value-at-risk (VaR). They are used to estimate the historical volatility of portfolio returns.
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  • $\begingroup$ Regarding point 2, log-returns are additive over time but not across assets, so their are not particularly convenient for portfolio optimization. I think this point is often overlooked. $\endgroup$ Jun 28, 2023 at 13:54
  • $\begingroup$ One version of logarithmic returns is this blog post about log points (which is $100 \ln x$): a-loonie-saved.blogspot.com/2008/08/dogbert-and-log-points.html $\endgroup$
    – Nayuki
    Jun 28, 2023 at 14:59
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There is a term "force of interest", which is basically continuously compounding interest. In compounding in general, you have $A_f = A_0*(1+\frac i n)^{Yn}$, where $A_f$ is the final amount, $A_0$ is the starting amount, $i$ is the interest rate, $Y$ is the number of years, and $n$ is the number of periods per year. If you take the limit as $n$ goes to infinity, you get $A_f = A_0*e^{Yi}$. We can also write this as $A_f = A_0*(e^i)^Y$. Thus if we want to write it as $A_f = A_0*(1+i')^Y$ for some $i'$, we find that $i = \ln(1+i')$. So the rate for continuously compounding interest is the log of one plus the interest rate that would give equal returns if compounded each year.

If you have an investment that has had different returns over several years, and you want to find the equivalent constant return (that is, the return that, had you had that return every year, you would have ended up with the same amount of money), then you can take the geometric mean of the growth factors, but you also have that the log of the overall growth factor is the arithmetic mean of the logs of the individual growth factors.

The log is also important in Kelly Criterion, as the Kelly Criterion is found by maximizing the expected value of the log of the final bankroll. It turns out that this also maximizes the median return.

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