Annuities and perpetuities with continuous compounding

With continuous compounding, present value formulas and future value formulas are different from those with discrete compounding,  PV = FV / (1+r)^t    or  FV = PV * (1+r)^t  .

The present value (PV) and future value (FV) of a cash flow with continuous  compounding is given by:

PV = FV*e^(-rt)         or FV = PV*e^(rt)

where r is the continuously compounded interest or discount rate which is different from the discrete compounded or discount rate.

As the present value formulas are different, so are the formulas for the net present values of annuities, growing annuities, perpetuity and growing perpetuity with continuously compounded rates. However, we will make use of the geometric progression analysis available within this blog at the link (click link) . The summary results are the sum of the first n terms of a geometric progression and the sum of the infinite terms of a geometric progression.

Sn = [A/(1-R)]*(1- R^n)  
(formula for sum of the first n terms)

S∞ =A / (1-R)   
(formula for sum of infinite terms of the GP)
where A is the first term and R is the constant ratio or constant multiple.

We start with the cash flows and the associated present values, assuming the cash flows grow by a factor g (also a continuously compounded rate). The cash flows follow the following sequence:

C , C*e^g , C*e^2g, ..., C*e^(n-1)g, ...    (note: nth term has an index of (n-1)g

The present value of each cash flow, bearing in mind the formula PV = FV*e^(-rt) follow the following sequence:

Ce^(-r), C*e^(g-2r) , C*e^(2g-3r), ..., C*e^[(n-1)g-nr], ... 

On closer inspection, these terms are a geometric progression with first term, A = Ce^(-r) and a constant ratio R = e^(g-r). These two terms can be used to find the formulas for the annuity, growing annuity, perpetuity and growing perpetuity. Let's start with the growing annuity.

Growing Annuity and Annuity with continuous compounding

using Sn = [A/(1-R)]*(1- R^n)    (formula for sum of the first n terms)
then PV (GA) = [Ce^(-r)/(1-e^(g-r))]*[1- e^(g-r)n]

multiplying the numerator and denominator by e^(r-g) gives
PV (GA) = [{Ce^(-r)}*e^(r-g)/[{1-e^(g-r)}*e^(r-g)]]*[1- e^(g-r)n]  
or 
PV (GA) =[Ce^(-g)/(e^(r-g)-1)]*[1- e^(-(r-g)n)] 

the present value of an annuity with no growth (g=o) directly results as:

PV (A) =[C/(e^r-1)]*[1- e^(-rn)] 

Growing Perpetuity and Perpetuity with continuous compounding

Using S∞ =A / (1-R)    (formula for sum of infinite terms of the GP)
then PV (GP) = [Ce^(-r)/(1-e^(g-r))]

multiplying the numerator and denominator by e^(r-g) gives
PV (GP) = [{Ce^(-r)}*e^(r-g)]/[{1-e^(g-r)}*e^(r-g)]
or 
PV (GP) =[Ce^(-g)/(e^(r-g)-1)]

the present value of a perpetuity with no growth (g=o) directly results as:

PV (P) =C/({e^r}-1)


Summary results

Continuous compounding formulas

PV (A) =[C/(e^r-1)]*[1- e^(-rn)]    - Present value of an annuity
PV (GA) =[Ce^(-g)/(e^(r-g)-1)]*[1- e^(-(r-g)n)]  - Present value of a growing annuity
PV (P) =C/({e^r}-1) - Present value of a perpetuity
PV (GP) =[Ce^(-g)/(e^(r-g)-1)] - Present value of a growing perpetuity


Discrete compounding formulas

PV(A) = [C/r]* [1-{1/(1+r)^n}]    - Present value of an annuity
PV(GA) = [C/(r-g)]* [1-{(1+g)/(1+r)}^n]   - Present value of a growing annuity
PV(P) =  C/r    - Present value of a perpetuity
PV(GP) =  C/(r -g) - Present value of a growing perpetuity