Question

You want to buy a new sports car from roy's cars for $51,800. the contract is in the form of a 48-month annuity due at an apr of7.8 percent, compounded monthly. what would be your monthly payment?

Answer 1

The present value of an annuity is given by

$PV= P\left(\frac{1-\left(1+ \frac{r}{t}\right)^{-nt} }{ \frac{r}{t} } \right)$

where: PV is the current value of the annuity, P is the periodic payment, r is the apr, t is the number of compounding in one year and n is the number of years.

Thus, given that PV = $51,800; r = 7.8% = 0.078; t = 12; n = 4.

$51,800= P\left(\frac{1-\left(1+ \frac{0.078}{12}\right)^{-4\times12} }{ \frac{0.078}{12} } \right) \\ \\ =P\left(\frac{1-(1+ 0.0065)^{-48} }{ 0.0065 } \right)=P\left(\frac{1-(1.0065)^{-48} }{ 0.0065 } \right) \\ \\ =P\left(\frac{1-0.7327}{ 0.0065 } \right)=P\left(\frac{0.2673}{ 0.0065 } \right)=41.12P \\ \\ \Rightarrow P= \frac{51,800}{41.12} =\ 1,259.73$

Therefore, the monthly payment is $1,259.73

Answer 2

Monthly payment is calculated as follows: This would only be solved using a BA II plus calculator, no educational institution would as for a manual evaultion, key strokes are as follows: N = 48 I/Y = 7.8/12 = .65 PV = -51800 FV = 0 CPT PMT Calculator display will read : $1,259.73. Monthly payment will be $1,259.73 monthly for 48 months.