Lesson Present value of an ordinary annuity cumulative saving plan

Present value of an ordinary annuity cumulative saving plan

Problem 1

The question can be equivalently reformulated as follows:


    If you have enough amount of money Y, what is better for you:
       
         to deposit this amount Y "all in one time" today to this bank at 12% compounded quarterly
                 or to buy today an asset X for Y dollars, from which you will deposit 
                     $100,000 every 3 months to the bank at 12% compounded quarterly?



First option generates future value of A =  =  = 3.262037792*Y  dollars in 10 year.



Second option works as an Ordinary Annuity saving plan and generates the amount 

    B =  =  = $7,540,125.97  (rounded) in 10 year.


Therefore, the reasonable value/price to buy the asset X for 10 years is no more than 

    Y =  = $2,311,477.20  dollars.


Thus, we calculated the reasonable (~equivalent) value/price to buy the asset X for 10 year.

Present value of an ordinary annuity cumulative saving plan, as in this post,
is the amount, which, deposited "at once" in the bank at the given annual rate and compounding conditions,
will generate the same future value after given time interval as this ordinary annuity will generate 
with its regular deposits at the end of this time interval.



The general formula for the present value of an accumulative ordinary annuity saving plan, via 
the parameters of the corresponding ordinary annuity, is


    PV = ,


where FV is the future value after n compounding, r is the effective rate per the compounding period, 
n is the number of compounding.  It is the same as


    PV =  = ,


where D is the regular depositing amount.