Log Transformations in Finance

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Log Transformations in Finance

• International Finance
• Dick Sweeney

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Log Transformations in Finance

• Reasons to use log transformations in finance
• On levels If level has changed greatly over
  time, logs give better view of what has happened
• Log levels slope in percentage rate of change
• For changes Changes in log values give a better
  view than just changes in levels.
• Changes in log levels are percentage rates of
  change
• Changes in log levels are more likely to be
  homskedastistic than changes in levels.

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NYSE Level, Log of NYSE LevelSame Scale
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NYSE Value, Log of NYSE Value,Separate Scales
Aug. 29 77,962,348 Feb. 33 15,770,881 Fall
79.77
Fall 16.27
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Change in NYSE, Change in Log of NYSE
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CPI and Log of CPI Same Scale
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Change in CPI, Change in Log of CPI, Separate
Scales
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Change in Wealth from 30-day Bills,Change in Log
of Wealth
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Logs of NYSE and SP500
NYSE, SP500 Indices tell pretty Much the same
story
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Compounding ("The Magic of Compounding")

• Suppose the simple interest rate is 5/year. 1
  deposited today results in 1.05 in one year 1
  (1.05) 1.05.
• But suppose the interest is compounded 10 times
  per year and gives the same 1.05.
• Call the rate for each of the ten sub-periods
  R10.
• Then, (1 R10)10 1.05, or (1 R10)10/10 (1
  R10) (1.05)1/10 1.0048909 (1 R10) and
  R10 0.0048909.
• Multiply by 10 to give 0.048909 for year and
  multiply by 100 to put in percentage terms,
  giving the compound rate 4.8909.
• How do we find R10? Divide 4.8909 by 10 and then
  by 100. 4.89095 per year compounded ten times
  per year is "as good as" simple interest of 5
  per year.

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Compounding (cont.)

• Now suppose the interest is compounded 100 times
  per year and gives the same 1.05.
• Call the rate for each of the hundred periods
  R100.
• Then, (1 R100)100 1.05, or
• (1 R100)100/100 (1 R100) (1.05)1/100
• 1.00048791 (1 R100) and R100 0.00048791.
  
• Multiply by 100 to give 0.048791 for the year and
  multiply by 100 to put in percentage terms,
  giving the compound rate 4.8791.

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Compounding (cont.)

• Call the number of times you discount per year N.
  Then, (1 RN)N (1.05), (1.05)1/N (1 RN),
• and as N ? ?, you get the continuously compounded
  rate 0.0487902, or 4.87902.
• You can find the continuously compounded rate by
  using (natural logs) the continuously compounded
  rate is ln(1.05) 0.0487902, or 4.87902.
• Just so you know, 0.0487902 can be converted back
  to 1.05 by finding e0.0487902 1.05, where e is
  the "natural" number, e 2.718281828

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Continuously Compounded Rates of Return in
Financial Markets

• Suppose that P0 100, P1 105, div1 5. Then,
  the simple rate of return is
• R1 ?P1 div1 / P0
• (P1 - P0) div1 / P0
• (P1 / P0) - 1 (div1 / P0)
• 0.10 ? 10.00,
• where div1 / P0 is the dividend yield.
• In log terms, or continuously compounded terms,
• ln(1 R1) ln(1.1) 0.0953102 ?
• 9.53102 lt 10.00.

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Continuously Compounded Rates of Return (cont.)

• Suppose div1 0.00 so div1 / P0 0.00. Then,
• R1 ?P1 / P0 (P1 - P0) / P0 (P1 / P0) - 1
  ? 5
• In log terms, or continuously compounded terms,
• ln(1 R1) ln(1.05) 0.0487902 ?
• 4.87902 lt 5.00.
• Note that ln(1 R1) ln1 (P1 / P0) - 1
• lnP1 / P0 lnP1 - lnP0 ? lnP1.
• ? lnP1 is the continuously compounded rate of
  change of P.
• Note ? lnP1 lt (?P1 / P0).
• For small (?P1 / P0), ? lnP1 ? (?P1 / P0). If
  (?P1 / P0) 0.01, then ? lnP1 ln(1.01)
  0.0099503 ? 0.01.

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Practice 1 Figure
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Practice 1 (same data)
Change in log levels

• Dependent Variable PROFIT_RATE
• Sample (adjusted) 2 100
• Variable Coefficient Std.
  Error t-Statistic Prob
• C -0.173938 0.110005 -1.581179
  0.1171
• PROFIT_RATE(-1) -0.011498 0.101046 -0.113792
  0.9096
• R-squared 0.000133     
• Durbin-Watson stat 2.013712

Urg!
R2 ? 0.0133 (!!)
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Practice 1 (cont.same data)
Slope approximately equal to one.

• Dependent Variable LEVEL
• Sample (adjusted) 2 100
• Included observations 99 after adjustments
• Variable Coefficient Std. Error t-Statistic
  Prob.  
• C -0.281801 0.188601 -1.494159 0.1384
• LEVEL(-1) 0.988304 0.016429 60.15428
  0.0000
• R-squared 0.973893     
• Durbin-Watson stat 2.019364     

R2 ? 97.39
Wowbut what does it mean?
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Practice 1 (cont.)
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Practice 1 (cont.)
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Practice 2 Figure
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Practice 2 (cont.)

• Dependent Variable PROFIT_RATE
• Sample (adjusted) 2 100
• Variable Coefficient Std. Error t-Statistic
  Prob.  
• C -0.038100 0.104134 -0.365876 0.7153
• PROFIT_RATE(-1) 0.103083 0.100919 1.021445
  0.3096
• R-squared 0.010642     
• Durbin-Watson stat 1.943564     

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Practice 2 (cont.)

• Dependent Variable LEVEL
• Sample (adjusted) 2 100
• Variable Coefficient Std. Error t-Statistic
  Prob.  
• C -0.575786 0.187474 -3.071280 0.0028
• LEVEL(-1) 0.815729 0.054805 14.88426 0.0000
• R-squared 0.695487     
• Durbin-Watson stat 1.667700     

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Practice 2 (cont.)
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Practice 2 (cont.)