Question

When you purchase a car, you may consider buying a brand-new car or
a used one. A fundamental trade-off in this case is whether you pay
repair bills (uncertain at the time you buy the car) or make loan payments
that are certain.
Consider two cars, a new one that costs $15,000 and a used one with
75,000 miles for $5,500. Let us assume that your current car’s value and
your available cash amount to $5,500, so you could purchase the used car
outright or make a down payment of $5,500 on the new car. Your credit
union is willing to give you a five-year, 10% loan on the $9,500 difference
if you buy the new car; this loan will require monthly payments of $201.85
per month for 5 years. Maintenance costs are expected to be $100 for the
first year and $300 per year for the second and third years.
After taking the used car to your mechanic for an evaluation, you
learn the following. First, the car needs some minor repairs within the
next few months, including a new battery, work on the suspension and
steering mechanism, and replacement of the belt that drives the water
pump. Your mechanic has estimated that these repairs will cost $150.
Considering the amount you drive, the tires will last another year but will
have to be replaced next year for about $200. Beyond that, the mechanic
warns you that the cooling system (radiator and hoses) may need to be
repaired or replaced this year or next and that the brake system may need
work. These and other repairs that an older car may require could lead
you to pay anywhere from $500 to $2,500 in each of the next 3 years. If
you are lucky, the repair bills will be low or will come later. But you
could end up paying a lot of money when you least expect it.

Draw a decision tree for this problem. To simplify it, look at the situation
on a yearly basis for 3 years. If you buy the new car, you can
anticipate cash outflows of 12 × $201.85 = $2,422.20 plus maintenance
costs. For the used car, some of the repair costs are known (immediate
repairs this year, tires next year), but we must model the uncertainty
associated with the rest. In addition to the known repairs, assume that in
each year there is a 20% chance that these uncertain repairs will be $500,
a 20% chance they will be $2,500, and a 60% chance they will be
$1,500. (Hint: You need three chance nodes: one for each year!)
To even the comparison of the two cars, we must also consider their
values after 3 years. If you buy the new car, it will be worth approximately
$8,000, and you will still owe $4,374. Thus, its net salvage value
will be $3,626. On the other hand, you would own the used car free
and clear (assuming you can keep up with the repair bills!), and it
would be worth approximately $2,000.
Include all of the probabilities and cash flows (outflows until the
last branch, then an inflow to represent the car’s salvage value) in your
decision tree. Calculate the net values at the ends of the branches.

Answer 1