John Corp. can lease the machine for a 10 year period and pay $25,000 for the first payment due on the agreement date (11/30/2010). Maintenance and insurance cost $5,000 per year that would be paid by Peter Inc who leased the machine to John Corp. Current interest rate is 12%. The other option is that John Corp. can purchase the machine for $130,000 and sell it for $10,000 after 10 years. Out of the two options, which would be the most attractive to John Corp. Prepare formal schedule to support your decision

For the lease payment, am I suppose to use Present Value of Annuity due?

I got $25,000 x 6.32825 = $158,206 for the lease payment
John Corp. should purchase the machine instead of leasing it. Am I correct and what does it mean by a formal schedule? I get the feeling, I am missing something or using the wrong annuity formula.

You're correct on the computation of the NPV of the lease option (assuming that you're asked to use an annual compounding convention).

In this case, you want the option with the lowest NPV--in these situations I usually refer to the NPV as "Net Present Cost".

Now you need to compute the NPC of the purchase option. You've got an immediate cash outflow of 130K, annual maintenance and insurance costs of 5K per year, and an expected cash inflow of 10K ten years down the road. Calculate the NPC (NPV) of these cash flows, and compare to the lease option's NPC.

Can you take it from here?

Sorry, I think I worded the question wrong. The $5,000 maintenance and insurance cost is paid by the company that loaned the machine (Peter Inc.). It is part of the leasing option. In the purchase option, the company sells the machine for $10,000 after its useful life.

Sorry for the confusion or am I misunderstanding what you are saying? Do I ignore the $5,000 in this situation?

That's right... in the leasing option, the lessor (owner of the asset) pays the maintenance and insurance. So it's reasonable to assume that under the purchase option, John Corp--as the asset's owner--would incur those same costs.

Can you show me how to do this?

You've correctly figured the NPV of the leasing option in your first post--it's 158,206 (negative; that's why I sometimes call it Net Present Cost).

Step Two is to calculate the NPC of the purchase option. You'll then compare the two, and the optimal choice is the one with the lower NPC.

In calc'g the NPC of the purchase choice, you first draw up a schedule of the cash flows associated with that choice, along with the timing of each cash flow.

In this case, you have an immediate cash outflow of 130K; annual outflows of 5K at the end of years 1 - 10; and finally an inflow of 10K at the end of year 10. (You can net those two end-of-year-10 flows and simply use a single inflow of 5K at the end of 10 in your calcs. In other words, at the end of Y10 you've got 5K going out (the final maint and ins cost) and 10K coming in (the sale of the asset). In your cash flow schedule, just say that you've got one inflow of 5K at the end of Y10.)

Next, compute the PV of each of those cash flows, and add those PVs together. For any cash flow amount C, which occurs at end of year n, its PV is

.
For example, the cash flow occurring at the end of Y 4 (which is an outflow of 5K), has a PV of = 3,177.59. Remember that it's a negative PV, since it's a cash outflow; an inflow has a positive PV.

Do a similar computation on all of the cash flows of the purchase option. Add the results together to get the purchase option's overall NPV. Compare to the lease option's NPV, as you've already computed. Pick the lesser NPV.

Finally, an immediate cash flow has a NPV exactly equal to its amount--no computation required.

I think that'll give you sufficient punching power to KO the problem in the first round or two. Go get 'em, and check back in with your results.

5000/1.12 + 5000/1.12^2 + 5000/1.12^3 +... 5000/1.12^10

= 28251.10

130,000 + 28,251.10 = 158,251.10
158,251.10 - 10,000(sold) = 148,251.10

Purchase option = 148,251.10
Lease option = 158,206

Purchase option is better. Is my calculation corrected?

You're getting very close, amigo. Your calculation of the PV of the 5,000 per year maint and ins expenses is good (nice job).

Adding that result to the PV of the initial expenditure (130K) is also correct.

But to finish it up, you don't want to deduct the full amount of the expected sales price (10K) to arrive at the overall NPV of the purchase decision. Instead, you want to deduct the 10K's present value (10K / 1.12^10), since the sales proceeds don't roll in the door until 10 years from now.

Make that one change, and you're done. (The purchase decision still wins, but not by as much as your current answer would suggest.)

Oh, would it be something like this?

130,000 + 28,251.10 = 158,251.10

10,000/(1.12^10) = 3,219.73
158,251.10 - 3,219.73 = 155,031.37

Purchase option = 155,031.37
Lease option = 158,206

I have a question. Why is the interest rate for the purchase option is 12%? Isn't the 12% interest for the borrowing?

You got it. Final score: Purchase has negative NPV of 155,031; Lease has negative NPV of 158,206. Purchase wins--it is the "less expensive" way of obtaining the asset, in terms of NPV.

Your follow-up question hits at the foundation of a HUGE subject--that is, the determination of the appropriate risk-adjusted discount rate by which to discount cash flows in NPV calcs. Out in the world, as well as in academia, some folks devote entire careers to developing the theories associated with this topic.

That's why textbooks, in NPV problems like this, will simply give you the appropriate discount rate to use, as they did in your John Corp. problem. The idea is for you to focus on the procedure for determining NPV, without worrying too much about how the discount rate was originally obtained. The reason that it's phrased as it was--"...the current interest rate is 12%..."--is because interest rates (such as might be charged to John Corp by a bank) are frequently used as a simple proxy for the appropriate PV discount rate.

The underlying rationale has some merit: If John Corp's lenders have concluded, after a thorough analysis of the company, that 12% is an appropriate risk-adjusted rate to charge John Corp on its loans, then we can say that 12% should be a good approximation for the discount rate to use in discounting John Corp's cash flows in our NPV work.

If you pursue this topic further, you'll eventually see the logical link between "interest rates" and "discount rates". But for now, just be aware that textbooks will frequently use "interest rate" and "NPV discount rate" as if they were one and the same. And in a lot of cases, that approximation ain't too far off the mark.