213.3

3.You graduate and land that big job. It's time to treat yourself to a nice car! You are negotiating with the dealer and you are trying to decide whether to buy or lease. He presents you with a 4 year lease: interest rate is 7% compounded monthly, purchase price is $40,000, buyout price is $25,000. What this means is that you are essentially borrowing $40,000, which you will repay in 48 equal monthly installments plus one lump payment at the end equal to $25,000 (or you just return the car, which they figure will be worth $25,000). He claims that based on these figures, your lease payment will be $530.

1. Is he telling the truth or is he overcharging you? EXPLAIN. (8 pts.)

   Based on the lump payment at the end, what the lease is essentially covering is:

   \( \Large 40{,}000 - \frac{25{,}000}{\left( 1 + \frac{.07}{12} \right)^{48}} = 21{,}090.029 \)

   which is the PV of your lease annuity. Calculating the payment:

   \( \Large P3 = \frac{21{,}090.029} { \left [ \frac{1}{\frac{.07}{12}} - \frac{1}{\frac{.07}{12} \left (1+\frac{.07}{12} \right )^{48}} \right ] } = 505.027 \)

   Thus, he is overcharging you since the lease payment he's offering is higher than it should be.

2. Assuming that the purchase price and the buyout are correct, is he actually charging you a higher or lower interest rate? How can you tell? (5 pts.)

He's charging you a higher interest rate. If the $40,000 and the $25,000 are correct, then the only way to get a higher payment is to charge a higher rate. A higher rate will do two things: first it will reduce the present value of your lump sum payoff, leaving more to be covered by the lease payments; second, it will increase the payments for any given PV of your annuity.

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