You have an opportunity to invest $1,000,000 today in a business that will pay $200,000 in the first year, $400,000 in second year, $600,000 in the third year and $800,000 in the fourth year. You can earn 12% per year compounded annually on an investment in a mutual fund that has similar risk. Should you undertake the project?

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The question abstracts away from what the project actually involves, so basically we are really only asking one question: Which choice makes more money? If you just add up the amounts of cash that you get from doing the project, it obviously looks like more than $1,000,000. But be careful:...

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The question abstracts away from what the project actually involves, so basically we are really only asking one question: Which choice makes more money?

If you just add up the amounts of cash that you get from doing the project, it obviously looks like more than $1,000,000. But be careful: You don't get those cashflows right away. Time is money, and the delayed cashflows are not worth as much in real terms as they would be if you got them immediately.

How much less? For that, we need a discount rate. It could depend on a lot of things---what the inflation rate is, what else you could do with the money, how patient you are. But for this problem we really only have one thing to go on: the rate of return. We are only concerned about whether we'll have more actual dollars in the bank at the end, and our dollars grow each year by a 12% annual rate of return.

Using that 12% annual return (which is huge, by the way---a typical return is about 5-7%) as our discount rate, $1 today is worth the same as (1+0.12)*$1 a year from now, and (1+0.12)^2*$1 two years from now, etc.

Thus, this means that we have as our two options:

1. Don't invest:
Start with $1,000,000
After 4 years, we have $1,000,000*(1.12)^4 = $1,573,519.36

2. Invest:
Start with $0
After 1 year, get $200,000; this will grow for 3 years and become $200,000*(1.12)^3 = $280,985.60
After 2 years, get $400,000; this will grow for 2 years and become $400,000*(1.12)^2 = $501,760.00
After 3 years, get $600,000; this will grow for 1 year and become $600,000*(1.12) = $672,000
After 4 years, get $800,000; this will not grow at all, so it's just $800,000.
Total value after 4 years: $2,254,745.60

So, we definitely should invest.

You can also compute a slightly different way as a net present value, the amount of money you'd have to have right now in order to end up with the same amount at the end simply from leaving it in the bank.

The net present value of not investing and keeping the $1,000,000 we already have is simply $1,000,000.

The net present value of investing is not that $2,254,745.60; instead we must divide it by the amount it could grow if it had been there for 4 years, which is (1.12)^4. So the actual amount we'd need to have right now to end up with the same is $1,432,931.59. This is the net present value. (Sometimes you'll also see net present value computed a bit differently, dividing the first year's cashflow by 1.12, the second by 1.12^2, etc.; it's a good exercise to think for a little while about how those two methods are actually equivalent.)

Since the net present value is higher for investing, we should invest.

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