# Future Value and Present Value of a General Annuity Due

I understand that a general annuity due, the payments are made at the beginning of each payment period, and the compounding period is not equal to the payment period. Then to solve I need to transform compounding period to payment period or patment period to compounding period.

I'am using the next formulas:

Compounding Period $i_{eq}=(1+i)^{1/p}-1$ and $i_{eq}=(1+i)^{p}-1$

Payment Period $R_{eq}=\frac{Ri}{(1+i)^{p}-1}$ and $R_{eq}=\frac{Ri}{(1+i)^{1/p}-1}$

I will use it, in this example: An item was acquired, which is paid for 8 years with 3500 payments at the beginning of each month by applying an interest rate of 21% per annum compounded quarterly. What is the cash value?

Then if I want transform Compounding Period to Payment Period:

$i_{eq}=(1+0.0525)^{1/3}-1=0.0172$ I will calculate Present Value

$PV=3500*[1+\frac{1-(1+0.0172)^{-96+1}}{0.0172}]=166708.94$

Otherwise I transform Payment Period to Compounding Period:

$R_{eq}=\frac{3500*0.0525}{(1+0.0525)^{1/3}-1}=10681.66$ I will calculate Present Value

$PV=10681.66*[1+\frac{1-(1+0.0525)^{-32+1}}{0.0525}]=172493.86$

Why they are different??? I calculate future value, and there are different too. Please help me to understand. Thank you and good day!

Annuity due of $n=8$ years with nominal rate $i=21\%$ compounded quaterly
1. payment $P_m=3500$ at the beginning of each month
Transforming the compounding period, from quarterly to monthly (so that it's equal to the payment period), we have the quarterly effective interest rate $i_q=\frac{i}{4}=5.25\%$ and then for the effective molthly interest rate $$(1+i_q)=(1+i_m)^3\quad\Longrightarrow\quad i_m=(1+i_q)^{1/3}-1=1.72\%$$ and then for $12n=96$ months we have the present value $$PV=P_m\, \ddot a_{\overline{12n}|i_m}=P_m\,(1+i_m)\,a_{\overline{12n}|i_m}=P_m\,(1+i_m)\frac{1-(1+i_m)^{-12n}}{i_m}$$ that is $PV=166,708.94.$
Transforming the payment period, from monthly to quarterly, that is finding an equivalent payment $P_q$ equal to the present value of 3 consecutives monthly payments, that is $$P_q=P_m\, \ddot a_{\overline{3}|i_m}=P_m\,(1+i_m)\,a_{\overline{3}|i_m}=P_m\,(1+i_m)\frac{1-(1+i_m)^{-3}}{i_m}=10,323.43$$ and then the present value of the equivalent annuity due with quarterly payments $P_q$ for $4n=32$ quarters is $$PV=P_q\, \ddot a_{\overline{4n}|i_q}=P_q\,(1+i_q)\,a_{\overline{4n}|i_q}=P_q\,(1+i_q)\frac{1-(1+i_q)^{-4n}}{i_q}$$ that is $PV=166,708.94$.
• Hello @alexjo. If I want to transform the payment period to compounding period for ordinary annuity I will use $R_{eq}=\frac{Ri}{(1+i)^{p}-1}$ or $R_{eq}=\frac{Ri}{(1+i)^{1/p}-1}$... but they only works in ordinary annuity... Are there smallest formulas for annuity due? – Salvattore Apr 23 '16 at 1:32