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!
Best Answer
Annuity due of $n=8$ years with nominal rate $i=21\%$ compounded quaterly
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 $PV=166,708.94$.