Assume that you have two assets, A and B, follow log-normal distributioin respectively. Asset A has an annual return of ra and voltility of va; Asset B has an annual return of rb and volotility of vb;

To calculate return and volotility for horizon of n years: n*ra n^0.5*va for Asset A n*rb n^0.5*vb for Asset B

Let’s visualize them.

For 1 and 2 year’s horizon:

## Risk and return of A and B
ra<- 0.05   
va<- 0.1
rb<- 0.15
vb<- 0.3



par(mfrow=c(1,2))
# Plot 1 year
n<-1
x <- seq(-1,1,length.out=100)
ra <- dnorm(x,n*ra,n^0.5*va)
rb <- dnorm(x,n*rb,n^0.5*vb)
plot(n*ra+n,x,col="red",type="l",xlim=c(0,5),xlab="1 Year")
lines(n*rb+n,x,col="green",typle="l")
## Warning in plot.xy(xy.coords(x, y), type = type, ...): "typle" is not a
## graphical parameter
# Plot 2 years
n<-2
x <- seq(-1,1,length.out=100)
ra <- dnorm(x,n*ra,n^0.5*va)
rb <- dnorm(x,n*rb,n^0.5*vb)
plot(n*ra+n,x,col="red",type="l",xlim=c(0,8),xlab="2 Years")
lines(n*rb+n,x,col="green",type="l")