NPV and discounted payback

ZMW1,000 in your hand today is worth more than ZMW1,000 promised in a year. You could put today’s money to work, and the promise might not be kept. Before you can add up cash flows that arrive in different years, you have to restate them all in today’s money.

That restatement is discounting, and the rate you discount at is the company’s cost of capital — what its investors demand for leaving their money in the business. Multiply a future cash flow by its discount factor and you have its present value.

At a cost of capital of 10%, a kwacha due in one year is worth 1 ÷ 1.10 = 0.909 today, and one due in two years is worth 1 ÷ 1.10² = 0.826. The factor falls every year, which is the arithmetic saying what everyone already knows: the further away the money, the less you should pay for it now.

Net present value is the sum of the present values of every cash flow a project causes, including the money you spend at the start. It answers one question: does this project make the owners of the business richer, and by how much in today’s money?

• A positive NPV means the project earns more than the cost of the money funding it. Accept it.
• A negative NPV means it earns less. The business would be better off leaving the money where it is.
• An NPV of zero means it earns exactly the required return — no better, no worse than the alternative.

The number itself is meaningful, not just its sign. An NPV of ZMW1,967 says the project is worth ZMW1,967 more than it costs, in today’s money, after paying every provider of capital what they demanded.

A project needs ZMW24,000 up front and will generate cash over five years. The cost of capital is 10%. Lay the years out in a column, discount each one, and add up.

The NPV is positive, so the project is worth doing: it returns the ZMW24,000, pays the 10% the providers of capital require on it, and leaves ZMW1,967 of value over.

Notice how the answer is built. The initial outlay sits at year 0 at a factor of 1.000, because it is spent today and needs no discounting. Every other line is a forecast cash flow multiplied by its factor. There is no arithmetic in an NPV harder than that — the difficulty in an exam is always in deciding which cash flows belong in the table at all.

The discounted payback period is how long the project takes to earn its money back in present-value terms — the point at which the discounted inflows have caught up with the initial cost. A project passes if it pays back within the company’s target, where the company has set one.

You find it by carrying the present values forward in a running total and watching for the year the balance turns positive.

To pin down where in that year, assume the year-5 cash arrives evenly and take the fraction you still needed.

Here that is 4 + (3,746 ÷ 5,713) = 4.66 years. Against a target of 4 years, the project fails — while its NPV said accept. The two tests disagree because they measure different things, and only one of them is about value.

Discounted payback is a real improvement on ordinary payback, because it counts the cost of the money while the project is waiting to break even. It still has the flaw that ordinary payback has: it stops counting the moment the money is back.

Two projects can pay back on exactly the same day, and one of them can go on earning for another decade while the other stops dead. Payback cannot see the difference; NPV can. That is why NPV is the primary test and payback is a secondary one — a check on risk and liquidity rather than on whether the project is worth doing.

For a Zambian business the liquidity question is not a technicality. A firm carrying kwacha debt through a period of high interest rates may genuinely need its money back inside three years, whatever the NPV says about year eight.