# add helping materials 

pv_perpetuity<-function(c,r){
"Objective: estimate the present value of perpetuity
                                 c
   Formula:   pv(perpeturity) = ------
                                  r
    c  : cash flow with the 1st one at the end of 1st period
    r  : effective period rate

 Example #1: pv_perpetuity(10,0.08)
             [1] 125   

 Example #2: pv_perpetuity(r=0.10,c=100)
            [1] 1000

"
return(c/r)
}

# The second program below has the same codes. 
pv_perpetuity_due<-function(c,r){
" Objective: estimate the present value of a perpetuity
                              c
Formula:   pv(perpeturity) = ------
                              r
c: cash flow (1st at the end of 1st period
r: effective period discount rate
e.g.,
pv_perpetuity(10,0.08)
[1] 125   
"
return(c/r*1(1+r))
}