51.3

3. You're planning to lease a 4Runner. Assume your lease payments are $434 per month. You're thinking that if you like the 4Runner, then after 2 years of leasing it, you will take-out a loan, and buy-out the lease. Assume that, in two years, you will have to borrow $25,500 to buy-out the lease. Your loan will be a 7-year loan (84 monthly payments). What will your loan payment be? What is the present value today, of all of your payments for the 4Runner (lease and loan payments)? Assume a discount rate of 6.25%, compounded monthly. [HINT: Set-up a cash flow timeline. Your first lease payment is one month from today, your last lease payment Is 24 months from today and your first loan payment is 25 months from today.]

First, figure-out your loan payment. PV=25,500, r=0.0625/12, n=84, P3=?

\( \large P3 = \frac{PV} { \left [ \frac{1}{r} - \frac{1}{r \left (1+r \right )^{n}} \right ] } = \frac{25{,}500} { \left [ \frac{1}{\frac{.0625}{12}} - \frac{1}{\frac{.0625}{12} \left (1+\frac{.0625}{12} \right )^{84}} \right ] } = $375.58 \)

Draw a CF timeline:

You know that in month 24, the value of the 84 month annuity of $375.58 per month, starting one month later is $25,500.

The PV of that 25,500 24 months from now is

\( \large PV = \frac{25{,}500}{ \left ( 1 + \frac{.0625}{12} \right)^{24}} = 22{,}510.97 \)

The PV of the 24 lease payments is:

\( \large PV = P3 \left [ \frac{1}{r} - \frac{1}{r \left(1+r \right )^{n} } \right ] = 434 \left [ \frac{1}{\frac{.0625}{12}} - \frac{1}{\frac{.0625}{12} \left(1+\frac{.0625}{12} \right )^{24} } \right ] = 9{,}767.44 \).

Thus, the total PV is 9767.44+22510.97=32278.41

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