Compound Interest Quarterly 2

Question

You want to buy a car that costs $250,000$ USD. To finance the outstanding debt you wan to take out a loan, which you have to pay back by quarterly payments over $7$ years. The first payment is due one quarter after you took out the loan. The interest is $6$ % p.a. compounded quarterly.

Calculate now the quarterly payment

My answer:

$FV=PV*(1+\frac{r}{n})^nt= 250 000*(1+\frac{6\%}{4})^4*7 = 250 000*(1,015)^28= 379 305$ --> pays back

Just divide it by $28$ to find the quarterly payment

$\frac{379 305}{28}=13 546.6$

Is it right? Would apreciate any help

Answer 1

The debt gets compounded then the rate subtracted, $$ D_{n+1}=D_n(1+\tfrac r4)-R $$ which is a simple linear recursion with general formula $$ D_n-\tfrac{4R}r=(D_0-\tfrac{4R}r)(1+\tfrac r4)^n $$ To get $D_{28}=0$ you will need $$ 0=D_0-(1-(1+\tfrac r4)^{-28})\tfrac{4R}r $$ resulting in the perhaps more familiar rate formula $$ R = \frac{\frac r4D_0}{1-(1+\tfrac r4)^{-28}}=11000.269118957289 $$

Answer 2

The effective quarterly rate is $i=\frac{i^{(4)}}{4}=\frac{6\%}{4}=1.5\%$. The number of payments is $n=4\times 7=28$ and the loan is $L=\$\,250,000$.

Then the quarter payment is $$ \boxed{ P=\frac{L}{a_{\overline{n}|i}}=\frac{\$\, 250,000}{22.73}\approx \$\,11,000.27 } $$ where $a_{\overline{n}|i}=\frac{1-(1+i)^{-n}}{i}$.