## Info

Used Car

New Car

Initial cost

\$ 3000

\$ 8000

Maintenance costs/year

\$ 1500

\$ 1000

Fuel costs/mile

\$ 0.20/ mile

\$ 0.05/mile

4 years

5 years

Assume that you drive 5000 miles a year and that you have an opportunity cost of 15%. This choice can be analyzed with replication:

Assume that you drive 5000 miles a year and that you have an opportunity cost of 15%. This choice can be analyzed with replication:

Step 1: Replicate the projects until they have the same lifetime; in this case, that would mean buying used cars five consecutive times and new cars four consecutive times. A. Buy a used car every 4 years for 20 years.

Maintenance costs: \$ 1500 every year for 20 years Fuel costs: \$ 1000 every year for 20 years ( 5000 miles at 20 cents a mile). B. Buy a new car every 5 years for 20 years

Maintenance costs: \$1000 every year for 20 years Fuel costs: \$ 250 every year for 20 years (5000 miles at 5 cents a mile) Step 2: Compute the NPV of each stream. NPV of replicating used cars for 20 years = -22225.61 NPV of replicating new cars for 20 years = -22762.21 The net present value of the costs incurred by buying a used car every 4 years is less negative than the net present value of the costs incurred by buying a new car every 5 years, given that the cars will be driven 5000 miles every year. As the mileage driven increases, however, the relative benefits of owning and driving the more efficient new car will also increase.

### Equivalent Annuities

We can compare projects with different lives by converting their net present values into equivalent annuities. In this method, we convert the net present values into annuities. Since the NPV is annualized, it can be compared legitimately across projects with different lives. The net present value of any project can be converted into an annuity using the following calculation.

Equivalent Annuity = Net Present Value * [A(PV,r,n)]

where r = Project discount rate, n = Project lifetime

A(PV,r,n) = annuity factor, with a discount rate of r and an annuity of n years Note that the net present value of each project is converted into an annuity using that project's life and discount rate. Thus, this approach is flexible enough to use on projects with different discount rates and life times. Consider again the example of the 5-year and 10-year projects given in the previous section. The net present values of these projects can be converted into annuities as follows:

Equivalent Annuity for 5-year project = \$442 * PV(A,12%,5 years) = \$ 122.62 Equivalent Annuity for 10-year project = \$478 * PV(A,12%,10 years) = \$ 84.60 The net present value of the 5-year project is lower than the net present value of the 10-year project, but using equivalent annuities, the 5-year project yields \$37.98 more per year than the 10-year project.

While this approach does not explicitly make an assumption of project replication, it does so implicitly. Consequently, it will always lead to the same decision rules as the replication method. The advantage is that the equivalent annuity method is less tedious and will continue to work even in the presence of projects with infinite lives. §2»

^ eqann.xls: This spreadsheet allows you to compare projects with different lives, using the equivalent annuity approach.

Illustration 6.4: Equivalent Annuities To Choose Between Projects With Different Lives

Consider again the choice between a new car and a used car described in illustration 12.4. The equivalent annuities can be estimated for the two options as follows:

Step 1: Compute the net present value of each project individually (without replication) Net present value of buying a used car = - \$3,000 - \$ 2,500 * PV(A,15%,4 years)

Net present value of buying a new car = - \$8,000 - \$ 1,250 * PV(A,15%,5 years)

Step 2: Convert the net present values into equivalent annuities Equivalent annuity of buying a used car = -\$10,137 * (A(PV,15%, 4 years))

Equivalent annuity of buying a new car = -12,190 * (A(PV,15%, 5 years))

Based on the equivalent annuities of the two options, buying a used car is more economical than buying a new car.

### Calculating Break-even

When an investment that costs more initially but is more efficient and economical on an annual basis is compared with a less expensive and less efficient investment, the choice between the two will depend on how much the investments get used. For instance, in illustration 6.4, the less expensive used car is the more economical choice if the mileage driven is less than 5000 miles. The more efficient new car will be the better choice if the car is driven more than 5000 miles. The break-even is the number of miles at which the two alternatives provide the same equivalent annual cost, as is illustrated in Figure 6.6.

Figure 6.6: Equivalent Annual Costs as a function of Miles Driven

Figure 6.6: Equivalent Annual Costs as a function of Miles Driven  