413.7

7.Luke Skywalker is negotiating to buy a new Landspeeder. The dealer, Jabba, is offering it for $25,000. Luke can get a 4-year loan with monthly payments at a 9% APR, compounded monthly.

1. What would Luke’s loan payments be if the first payment is due one month from today? (6)

   Straight-up annuity. PV=25000, r=.09/12 (from the compounded monthly), n=48, CF=?

   \( \Large CF = \frac{25{,}000} { \left [ \frac{1}{\frac{.09}{12}} - \frac{1}{\frac{.09}{12} \left (1+\frac{.09}{12} \right )^{48}} \right ] } = 622.13 \)

2. Luke’s other option is to lease the Landspeeder. The Landspeeder has a residual value of $12,000 (meaning that at the end of the lease it is worth $12,000). This means that at the end of the lease you must either buy the Landspeeder for $12,000 or return it to the dealer. Assume a 4-year lease with monthly payments, the first payment is due next month and the rate is the same as for the loan. What would Luke’s lease payments be? (8)

   Annuity with a twist: There is the equivalent of a balloon payment at the end when Luke either returns the speeder (worth $12000) or pays the dealer $12000. Thus, the amount left over to finance is the price of the speeder minus the PV of the balloon payment:

   \( \Large 25000 - \frac{12000}{\left( 1 + \frac{.09}{12} \right)^{48}} = 25{,}000 - 8{,}383.37 = 16{,}616.63 \)

   So the PV of the lease payments must be 16{,}616.63. Calculate the payment:

   \( \Large CF = \frac{16{,}616.63} { \left [ \frac{1}{\frac{.09}{12}} - \frac{1}{\frac{.09}{12} \left (1+\frac{.09}{12} \right )^{48}} \right ] } = 413.51 \)

3. Assume Luke took the lease and each month invested the difference between his lease payment and what he would have paid on the loan. If he can earn 9% compounded monthly on these monthly investments, how much will he have in his investment account at the end of the lease? Will he be able to buy the Landspeeder at its residual value? (6)

The point of this question was to show you that if you save the difference between the purchase and the lease, you can buy the car at the end of the lease (effectively constructing a 4-yr purchase).

The difference is \(622.13 - 413.51 = 208.62 \)

The PV of monthly savings of 208.62 is:

\( \Large PV = 208.62 \left [ \frac{1}{\frac{.09}{12}} - \frac{1}{\frac{.09}{12} \left(1+\frac{.09}{12} \right )^{48} } \right ] = 8383.37 \).

The FV of this (what you'll have at the end of the lease) is:

\( \Large FV = 8383.37 \left( 1 + \frac{.09}{12} \right)^48 = 12000 \)

If you rounded the payment instead of carrying all significant digits, your answer will be slightly less than 12000, but that is only due to rounding. Conceptually, the only difference between the lease payments and the loan payments is coming from the residual value at the end.

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