# am,gm,ccm and leverage

## am,gm,ccm and leverage

Arithmetic, Geometric, and Continuously Compounded Returns

Arithmetic Mean Rate of Return Geometric Mean Rate of Return Continuously Compounded Mean
Rate of Return

A. Period Analysis A. Period Analysis Indirect Computation A. Indirect Computation
time index return ma (x-m)2 time index price relative time index cont. return mc (x-m)2
0 100 0 100 0 100
1 130 0.300 0.191 0.012 1 130 1.300 1 130 1.300 0.262 0.167 0.009
2 150 0.154 0.191 0.001 2 150 1.154 2 150 1.154 0.143 0.167 0.001
3 202 0.347 0.191 0.024 3 202 1.347 3 202 1.347 0.298 0.167 0.017
4 189 -0.064 0.191 0.065 4 189 0.936 4 189 0.936 -0.067 0.167 0.054
5 230 0.217 0.191 0.001 5 230 1.217 5 230 1.217 0.196 0.167 0.001
0.953 0.103 2.300 0.833 0.082

ma 0.191 rc 0.167
variance 0.021 variance 0.016
stdev 0.144 stdev 0.128

rg 0.181
xxxx FV = PV*(1+ma)n FV = PV * 2.718 (mc*n)
xxxx 239.26 =100*(1+.191)5 230 =100*2.718(.167*5)

B. Horizon Analysis Direct Computation B. Horizon Analysis Direct Computation
FV = PV*(1+mg)n

Probability Analysis rg= (FV/PV)(1/n)-1 mc = ln(FV/PV)/n
x= 0.000 rg = (230/100)(1/5)-1 0.181260188 mc = ln(230/100)/5= 0.166581825 0.166581825
ra 0.191
sd= 0.144 230 =100*(1+.181)5
z = -1.326 C. mg from ma Approximation C. mc from mg Exact
px 0.908 rg = ma-s2/2 PVemn = PV(1+mg)n
rg=.191-.1442/2 = 0.180 mc = ln(1+mg)
ln(1+.181)= 0.167