Logarithms and Exponential Functions
x ln(x) exp(x) ln(exp(x))
-0.5 0.60653 -0.5
-0.4 0.67032 -0.4
-0.3 0.74082 -0.3
-0.2 0.81873 -0.2 6
-0.1 0.90484 -0.1
0 1 0 4
0.01 -4.60517 1.01005 0.01
0.05 -2.99573 1.05127 0.05 2
0.1 -2.30259 1.10517 0.1
y
0.2 -1.60944 1.2214 0.2 0
-1 -0.5 0
0.3 -1.20397 1.34986 0.3
-2
0.4 -0.91629 1.49182 0.4
0.5 -0.69315 1.64872 0.5
-4
0.6 -0.51083 1.82212 0.6
0.7 -0.35667 2.01375 0.7
-6
0.8 -0.22314 2.22554 0.8
0.9 -0.10536 2.4596 0.9
1 0 2.71828 1
1.1 0.09531 3.00417 1.1 Note: graph is created using
1.2 0.18232 3.32012 1.2 chart wizard. Chart type is xy
1.3 0.26236 3.6693 1.3 scatter.
1.4 0.33647 4.0552 1.4
1.5 0.40547 4.48169 1.5
1.6 0.47 4.95303 1.6
1.7 0.53063 5.47395 1.7
1.8 0.58779 6.04965 1.8
1.9 0.64185 6.68589 1.9
2 0.69315 7.38906 2
y = ln(x), y=exp(x)
2
ln(x)
y
0 exp(
x)
-1 -0.5 0 0.5 1 1.5 2 2.5
-2
-4
-6 x
Note: graph is created using
chart wizard. Chart type is xy
scatter.
Time Value of Money
Compounding
V R Frequency m Periodic Rate
$1,000.00 0.2 annual 1 20.00%
quarterly 4 5.00%
weekly 52 0.38%
daily 365 0.05%
continuously infinity 0.00%
FVnc $V e Rn
Value of V Effective
at end of Annual
year Rate m
$1,200.00 20.00% R
RA 1 1
$1,215.51 21.55% m
$1,220.93 22.09%
$1,221.34 22.13%
$1,221.40 22.14% eR 1
mn
FVnc $V e Rn R
FVn
m
$V 1
m
Ex 1
P(t-1) P(t) R(t) 1+R(t)
msft 85 90
Ex 2
P(t-2) P(t-1) P(t)
msft 80 85 90
R(t-1) R(t)
R2(t)
Ex 3
P(t-1) P(t) R(t) x(t) Rp(t)
msft 85 90 0.058824 0.25 0.014706
sbux 30 28 -0.06667 0.75 -0.05
-0.03529
Ex 3
P(t-1) P(t) R(t) r(t) x(t) rp(t) rp(t)
msft 85 90 0.058824 0.057158 0.25 -0.03593 0.01429
sbux 30 28 -0.06667 -0.06899 0.75 -0.05174
-0.03746
0
1
2
3
4
5
0
10
20
30
40
7/1/1992 7/1/1992 50
2/1/1993 2/1/1993
9/1/1993 9/1/1993
4/1/1994 4/1/1994
11/1/1994 11/1/1994
6/1/1995 6/1/1995
1/1/1996 1/1/1996
8/1/1996 8/1/1996
3/1/1997 3/1/1997
10/1/1997 10/1/1997
5/1/1998 5/1/1998
12/1/1998 12/1/1998
7/1/1999 7/1/1999
2/1/2000 2/1/2000
Log Price of SBUX
9/1/2000 9/1/2000
4/1/2001 4/1/2001
11/1/2001 11/1/2001
6/1/2002 6/1/2002
1/1/2003 1/1/2003
8/1/2003 8/1/2003
3/1/2004 3/1/2004
R(t)
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.1
0.2
0.3
0.4
0
8/1/1992
12/1/1992
4/1/1993
8/1/1993
12/1/1993
4/1/1994
8/1/1994
12/1/1994
4/1/1995
8/1/1995
12/1/1995
4/1/1996
8/1/1996
12/1/1996
4/1/1997
8/1/1997
12/1/1997
simple
4/1/1998
8/1/1998
Month
return
12/1/1998
4/1/1999
8/1/1999
Monthly simple return on SBUX
12/1/1999
cc return
4/1/2000
8/1/2000
12/1/2000
4/1/2001
8/1/2001
12/1/2001
4/1/2002
8/1/2002
12/1/2002
4/1/2003
8/1/2003
12/1/2003
4/1/2004
8/1/2004
Date Adj. Close*log price R(t) r(t) Rt(2) r2(t)=ln(P(t)/P(t-2))
Jul-92 1.66 0.506818
Aug-92 1.55 0.438255 -0.06627 -0.06856
Sep-92 1.89 0.636577 0.219355 0.198322 0.138554 0.129759
Oct-92 1.84 0.609766 -0.02646 -0.02681 0.187097 0.171511
Nov-92 2.36 0.858662 0.282609 0.248896 0.248677 0.222085
Dec-92 2.31 0.837248 -0.02119 -0.02141 0.255435 0.227482
Jan-93 2.41 0.879627 0.04329 0.042379 0.021186 0.020965
Feb-93 2.27 0.81978 -0.05809 -0.05985 -0.01732 -0.01747
Mar-93 2.38 0.8671 0.048458 0.047321 -0.01245 -0.01253
Apr-93 2.42 0.883768 0.016807 0.016667 0.066079 0.063988
May-93 3 1.098612 0.239669 0.214845 0.260504 0.231512
Jun-93 3.06 1.118415 0.02 0.019803 0.264463 0.234647
Jul-93 2.97 1.088562 -0.02941 -0.02985 -0.01 -0.01005
Aug-93 3.03 1.108563 0.020202 0.020001 -0.0098 -0.00985
Sep-93 3.42 1.229641 0.128713 0.121078 0.151515 0.141079
Oct-93 3.34 1.205971 -0.02339 -0.02367 0.10231 0.097408
Nov-93 2.78 1.022451 -0.16766 -0.18352 -0.18713 -0.20719
Dec-93 2.78 1.022451 0 0 -0.16766 -0.18352
Jan-94 3 1.098612 0.079137 0.076161 0.079137 0.076161
Feb-94 2.91 1.068153 -0.03 -0.03046 0.046763 0.045702
Mar-94 3.05 1.115142 0.04811 0.046989 0.016667 0.016529
Apr-94 3.72 1.313724 0.219672 0.198582 0.278351 0.245571
May-94 3.56 1.269761 -0.04301 -0.04396 0.167213 0.154619
Jun-94 3.16 1.150572 -0.11236 -0.11919 -0.15054 -0.16315
Jul-94 3.62 1.286474 0.14557 0.135902 0.016854 0.016713
Aug-94 3.53 1.261298 -0.02486 -0.02518 0.117089 0.110726
Sep-94 2.88 1.05779 -0.18414 -0.20351 -0.20442 -0.22868
Oct-94 3.39 1.22083 0.177083 0.16304 -0.03966 -0.04047
Nov-94 3.35 1.20896 -0.0118 -0.01187 0.163194 0.15117
Dec-94 3.44 1.235471 0.026866 0.026511 0.014749 0.014642
Jan-95 3 1.098612 -0.12791 -0.13686 -0.10448 -0.11035
Feb-95 2.98 1.091923 -0.00667 -0.00669 -0.13372 -0.14355
Mar-95 3 1.098612 0.006711 0.006689 0 0
Apr-95 2.94 1.07841 -0.02 -0.0202 -0.01342 -0.01351
May-95 3.63 1.289233 0.234694 0.210823 0.21 0.19062
Jun-95 4.45 1.492904 0.225895 0.203671 0.513605 0.414495
Jul-95 4.67 1.541159 0.049438 0.048255 0.286501 0.251926
Aug-95 5 1.609438 0.070664 0.068279 0.123596 0.116534
Sep-95 4.74 1.556037 -0.052 -0.0534 0.014989 0.014878
Oct-95 4.91 1.591274 0.035865 0.035237 -0.018 -0.01816
Nov-95 5.28 1.663926 0.075356 0.072652 0.113924 0.107889
Dec-95 5.25 1.658228 -0.00568 -0.0057 0.069246 0.066954
Jan-96 4.19 1.432701 -0.2019 -0.22553 -0.20644 -0.23123
Feb-96 4.41 1.483875 0.052506 0.051174 -0.16 -0.17435
Mar-96 5.83 1.763017 0.321995 0.279142 0.391408 0.330316
Apr-96 6.78 1.913977 0.16295 0.15096 0.537415 0.430102
May-96 6.78 1.913977 0 0 0.16295 0.15096
Jun-96 7.06 1.954445 0.041298 0.040468 0.041298 0.040468
Jul-96 6.5 1.871802 -0.07932 -0.08264 -0.0413 -0.04217
Aug-96 8.19 2.102914 0.26 0.231112 0.160057 0.148469
Sep-96 8.25 2.110213 0.007326 0.007299 0.269231 0.238411
Oct-96 8.12 2.09433 -0.01576 -0.01588 -0.00855 -0.00858
Nov-96 8.66 2.158715 0.066502 0.064385 0.049697 0.048502
Dec-96 7.16 1.96851 -0.17321 -0.1902 -0.11823 -0.12582
Jan-97 8.56 2.1471 0.195531 0.17859 -0.01155 -0.01161
Feb-97 8.41 2.129421 -0.01752 -0.01768 0.174581 0.160911
Mar-97 7.41 2.00283 -0.11891 -0.12659 -0.13435 -0.14427
Apr-97 7.47 2.010895 0.008097 0.008065 -0.11177 -0.11853
May-97 7.88 2.064328 0.054886 0.053433 0.063428 0.061497
Jun-97 9.73 2.275214 0.234772 0.210886 0.302544 0.264319
Jul-97 10.23 2.325325 0.051387 0.050111 0.298223 0.260997
Aug-97 10.25 2.327278 0.001955 0.001953 0.053443 0.052064
Sep-97 10.45 2.346602 0.019512 0.019324 0.021505 0.021277
Oct-97 8.25 2.110213 -0.21053 -0.23639 -0.19512 -0.21706
Nov-97 8.72 2.165619 0.05697 0.055406 -0.16555 -0.18098
Dec-97 9.6 2.261763 0.100917 0.096144 0.163636 0.15155
Jan-98 9.14 2.21266 -0.04792 -0.0491 0.048165 0.047041
Feb-98 9.89 2.291524 0.082057 0.078864 0.030208 0.029761
Mar-98 11.33 2.427454 0.145602 0.13593 0.239606 0.214794
Apr-98 12.03 2.487404 0.061783 0.059949 0.21638 0.195879
May-98 12 2.484907 -0.00249 -0.0025 0.059135 0.057453
Jun-98 13.36 2.592265 0.113333 0.107359 0.110557 0.104862
Jul-98 10.47 2.348514 -0.21632 -0.24375 -0.1275 -0.13639
Aug-98 7.89 2.065596 -0.24642 -0.28292 -0.40943 -0.52667
Sep-98 9.05 2.202765 0.147022 0.137169 -0.13563 -0.14575
Oct-98 10.85 2.384165 0.198895 0.1814 0.375158 0.318569
Nov-98 11.53 2.444952 0.062673 0.060787 0.274033 0.242188
Dec-98 14.03 2.641198 0.216826 0.196246 0.293088 0.257033
Jan-99 13.02 2.566487 -0.07199 -0.07471 0.129228 0.121534
Feb-99 13.22 2.581731 0.015361 0.015244 -0.05773 -0.05947
Mar-99 14.03 2.641198 0.061271 0.059467 0.077573 0.074711
Apr-99 18.47 2.916148 0.316465 0.27495 0.397126 0.334417
May-99 18.44 2.914522 -0.00162 -0.00163 0.314326 0.273324
Jun-99 18.78 2.932792 0.018438 0.01827 0.016784 0.016645
Jul-99 11.62 2.452728 -0.38126 -0.48006 -0.36985 -0.46179
Aug-99 11.44 2.437116 -0.01549 -0.01561 -0.39084 -0.49568
Sep-99 12.39 2.51689 0.083042 0.079774 0.066265 0.064162
Oct-99 13.6 2.61007 0.097659 0.09318 0.188811 0.172954
Nov-99 13.28 2.586259 -0.02353 -0.02381 0.071832 0.069369
Dec-99 12.12 2.494857 -0.08735 -0.0914 -0.10882 -0.11521
Jan-00 16 2.772589 0.320132 0.277732 0.204819 0.18633
Feb-00 17.57 2.866193 0.098125 0.093604 0.44967 0.371336
Mar-00 22.41 3.109507 0.27547 0.243314 0.400625 0.336919
Apr-00 15.11 2.715357 -0.32575 -0.39415 -0.14001 -0.15084
May-00 17 2.833213 0.125083 0.117857 -0.24141 -0.27629
Jun-00 19.09 2.949165 0.122941 0.115951 0.263402 0.233808
Jul-00 18.75 2.931194 -0.01781 -0.01797 0.102941 0.09798
Aug-00 18.32 2.907993 -0.02293 -0.0232 -0.04034 -0.04117
Sep-00 20.03 2.997231 0.093341 0.089238 0.068267 0.066037
Oct-00 22.34 3.106379 0.115327 0.109148 0.219432 0.198385
Nov-00 22.78 3.125883 0.019696 0.019504 0.137294 0.128652
Dec-00 22.12 3.096482 -0.02897 -0.0294 -0.00985 -0.0099
Jan-01 24.97 3.217675 0.128843 0.121193 0.096137 0.091792
Feb-01 23.82 3.170526 -0.04606 -0.04715 0.076854 0.074043
Mar-01 21.22 3.054944 -0.10915 -0.11558 -0.15018 -0.16273
Apr-01 19.35 2.962692 -0.08812 -0.09225 -0.18766 -0.20783
May-01 19.52 2.97144 0.008786 0.008747 -0.08011 -0.0835
Jun-01 23 3.135494 0.178279 0.164055 0.18863 0.172802
Jul-01 18.04 2.892592 -0.21565 -0.2429 -0.07582 -0.07885
Aug-01 16.87 2.825537 -0.06486 -0.06705 -0.26652 -0.30996
Sep-01 14.94 2.704042 -0.1144 -0.12149 -0.17184 -0.18855
Oct-01 17.12 2.840247 0.145917 0.136205 0.014819 0.01471
Nov-01 17.72 2.874694 0.035047 0.034447 0.186078 0.170652
Dec-01 19.05 2.947067 0.075056 0.072373 0.112734 0.10682
Jan-02 23.77 3.168424 0.247769 0.221357 0.341422 0.29373
Feb-02 23.01 3.135929 -0.03197 -0.0325 0.207874 0.188862
Mar-02 23.13 3.14113 0.005215 0.005202 -0.02692 -0.02729
Apr-02 22.82 3.127637 -0.0134 -0.01349 -0.00826 -0.00829
May-02 24.28 3.189653 0.063979 0.062016 0.049719 0.048522
Jun-02 24.85 3.212858 0.023476 0.023205 0.088957 0.08522
Jul-02 19.63 2.977059 -0.21006 -0.2358 -0.19152 -0.21259
Aug-02 20.1 3.00072 0.023943 0.023661 -0.19115 -0.21214
Sep-02 20.66 3.028199 0.027861 0.02748 0.052471 0.05114
Oct-02 23.84 3.171365 0.153921 0.143165 0.18607 0.170645
Nov-02 21.74 3.079154 -0.08809 -0.09221 0.052275 0.050954
Dec-02 20.38 3.014554 -0.06256 -0.0646 -0.14513 -0.15681
Jan-03 22.72 3.123246 0.114818 0.108692 0.045078 0.044092
Feb-03 23.45 3.15487 0.03213 0.031625 0.150638 0.140316
Mar-03 25.76 3.248823 0.098507 0.093952 0.133803 0.125577
Apr-03 23.51 3.157426 -0.08734 -0.0914 0.002559 0.002555
May-03 24.67 3.205588 0.049341 0.048162 -0.04231 -0.04323
Jun-03 24.55 3.200712 -0.00486 -0.00488 0.044236 0.043286
Jul-03 27.33 3.307985 0.113238 0.107273 0.107823 0.102397
Aug-03 28.39 3.346037 0.038785 0.038052 0.156415 0.145325
Sep-03 28.8 3.360375 0.014442 0.014338 0.053787 0.05239
Oct-03 31.6 3.453157 0.097222 0.092782 0.113068 0.10712
Nov-03 32.17 3.471034 0.018038 0.017877 0.117014 0.110659
Dec-03 33.16 3.501344 0.030774 0.03031 0.049367 0.048187
Jan-04 36.61 3.600321 0.104041 0.098977 0.138017 0.129287
Feb-04 37.4 3.621671 0.021579 0.021349 0.127865 0.120326
Mar-04 37.87 3.634159 0.012567 0.012489 0.034417 0.033838
Apr-04 38.92 3.661508 0.027726 0.027349 0.040642 0.039838
May-04 40.6 3.703768 0.043165 0.04226 0.072089 0.069609
Jun-04 43.49 3.772531 0.071182 0.068763 0.11742 0.111023
Jul-04 46.99 3.849935 0.080478 0.077404 0.157389 0.146167
Aug-04 43.24 3.766766 -0.0798 -0.08317 -0.00575 -0.00577
Sep-04 45.46 3.816833 0.051341 0.050067 -0.03256 -0.0331
Asset Return Calculations
I. Simple Returns
One Two Two
Period Period Period
month Price Return Return Return
t Pt Rt Rt(2) Rt(2)
1 80
2 85 0.0625
3 90 0.0588 0.1250 0.1250
Pt Pt 1
Rt Pt Pt 2 Rt (2) (1 Rt 1 )(1 Rt ) 1
Pt 1 Rt (2)
Pt 2
II. Continuously Compounded Returns
One One
Period Period Two Period
month Price ln(Price) Return Return Return
t Pt ln(Pt) rt rt rt(2)
1 80 4.38203
2 85 4.44265 0.0606 0.0606
3 90 4.49981 0.0572 0.0572 0.1178
rt (2) ln(1 Rt (2))
rt ln( Pt ) ln( Pt 1 ) rt ln(1 Rt )
rt (2) ln( Pt ) ln( Pt 2 )
Note: this formula is not
very practical for
computations.
1 )(1 Rt ) 1
Two Period Two Period
Return Return
rt(2) rt(2)
0.1178 0.1178
rt (2) ln(1 Rt (2))
rt (2) rt 1 rt
(2) ln( Pt ) ln( Pt 2 )
