## Discounts and Default Risk

We now take a look at cash discounts, default risk, and the relationship between the two. To get started, we define the following:

^ = Percentage of credit sales that go uncollected d = Percentage discount allowed for cash customers P' = Credit price (the no-discount price)

Notice that the cash price, P, is equal to the credit price, P', multiplied by (1 - d): P = P'(1 - d), or, equivalently, P' = P/(1 - d).

The situation at Locust is now a little more complicated. If a switch is made from the current policy of no credit, then the benefit from the switch will come from both the higher price (P') and, potentially, the increased quantity sold (Q').

Furthermore, in our previous case, it was reasonable to assume that all customers took the credit, because it was free. Now, not all customers will take the credit because a discount is offered. In addition, of the customers who do take the credit offered, a certain percentage will not pay.

To simplify the discussion that follows, we will assume that the quantity sold (Q) is not affected by the switch. This assumption isn't crucial, but it does cut down on the work (see Problem 5 at the end of the appendix). We will also assume that all customers take the credit terms. This assumption isn't crucial either. It actually doesn't matter what percentage of the customers take the offered credit.4

NPV of the Credit Decision Currently, Locust sells Q units at a price of P = \$49. Locust is considering a new policy that involves 30 days' credit and an increase in price to P' = \$50 on credit sales. The cash price will remain at \$49, so Locust is effectively allowing a discount of (\$50 - 49)/50 = 2% for cash.

What is the NPV to Locust of extending credit? To answer, note that Locust is already receiving (P - v)Q every month. With the new, higher price, this will rise to (P' - v)Q, assuming that everybody pays. However, because ^ percent of sales will not be collected, Locust will only collect on (1 - X P'Q; so net receipts will be [(1 - rn)P' - v] X Q.

The net effect of the switch for Locust is thus the difference between the cash flows under the new policy and those under the old policy:

Net incremental cash flow = [(1 - -n)P' - v] X Q - (P - v) X Q

4The reason is that all customers are offered the same terms. If the NPV of offering credit is \$100, assuming that all customers switch, then it will be \$50 if only 50 percent of our customers switch. The hidden assumption is that the default rate is a constant percentage of credit sales.

5To see this, note that the net incremental cash flow is:

Net incremental cash flow = [(1 - w)P' - v] X Q - (P - v) X Q = [(1 - w)P' - P] X Q

Net incremental cash flow = [(1 - w)P' - (1 - d)P'] X Q = P'Q X (d - w)

Ross et al.: Fundamentals of Corporate Finance, Sixth Edition, Alternate Edition

PART SEVEN Short-Term Financial Planning and Management

If Locust does make the switch, then the cost in terms of the investment in receivables is just P X Q since Q = Q'. The NPV of the switch is thus:

For example, suppose that, based on industry experience, the percentage of "deadbeats" is expected to be 1 percent. What is the NPV of changing credit terms for Locust? We can plug in the relevant numbers as follows:

= -\$49 X 100 + 50 X 100 X (.02 - .01)/.02 = -\$2,400

Because the NPV of the change is negative, Locust shouldn't switch.

In our expression for NPV, the key elements are the cash discount percentage (d) and the default rate (^). One thing we see immediately is that, if the percentage of sales that goes uncollected exceeds the discount percentage, then d - ^ is negative. Obviously, the NPV of the switch would then be negative as well. More generally, our result tells us that the decision to grant credit here is a trade-off between getting a higher price, thereby increasing sales revenues, and not collecting on some fraction of those sales.

With this in mind, note that P'Q X (d - is the increase in sales less the portion of that increase that won't be collected. This is the incremental cash inflow from the switch in credit policy. If d is 5 percent and ^ is 2 percent, for example, then, loosely speaking, revenues are increasing by 5 percent because of the higher price, but collections only rise by 3 percent because the default rate is 2 percent. Unless d > we will actually have a decrease in cash inflows from the switch.

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