Lets go ahead and start our strict capital budgeting example. What I’ve done here is I’ve created 1, 2, 3, 4 ,5 assets. A through E, and I’ve shown you the restrictive component that we have, which is their cost of the year that you’re supposed to purchase them. Now remember the assumptions for the strict capital budgeting are that you have restrictions on first year cash flow –there’s a limitation there – but not on any year that comes after. To do anything more complicated than that is a little beyond the scope of this course and you end up with kinds of constraint optimization problems that we just can’t cope with.

I’ve also associated with each one of these assets a present worth, it doesn’t matter which interest rate we use in order to calculate that present worth, it’s done for you and you can safely think of it as ten percent. So what we’ve done here is we have asset A which is cheap to get into, costs under two dollars, gives you a present worth of 2. Asset B, cost is 7, has a present worth of 3. Asset C cost you 10, has a present worth of 7. So on down the line. Now the first thing you will notice is Asset E costs you 20 dollars but has a present worth of minus 1. What I should point out is that this is never an Asset that you are actually going to try to use. It’s something that you would eliminate from any possible combination. You know it won’t be part of your winning combo.

Now, what you’re doing in order to find out the best choice for a capital budgeting problem is your taking all these Asset combinations and seeing which combinations fit within your capital budget. What I’m actually doing here is I’m saying in this problem we’ll have a capital budget of 20 dollars. The idea is that in time period zero you can buy as much as 20 dollars’ worth of stuff. So there’s a limitation not on how many assets you can buy but what you can fit into your twenty dollars. To make this clear, you can buy all those little singletons there so it makes sense to buy asset A by itself, asset B by itself, asset C by itself, asset D by itself. Each one of those fits under the twenty dollar heading. You can also buy combinations so for example Asset A and B together cost you only 9 so you can definitely buy that. Asset A and C together cost you only 12, you can buy that. Asset A and D cost you 17 so you can buy that. B and C cost you 17 you can buy that. B and D on the other hand cost you 22, that’s a combination you can’t quite get to, you can eliminate that. What I’m doing here is I’m looking at the combinations which are actually feasible, that you can fit into a 20 dollar budget. Please note C and D is also something that can be knocked out of that feasibility set.

Now there are also some three way combinations in there that are possible, we are looking at all combinations that you can fit into the twenty dollar set. So that A B C combination cost you 10 plus 7 plus 2 costs 19, definitely feasible. But the A B D combination is just below that – too expensive – can’t afford it. That B C D combination – also can’t afford it. In fact, all of these other 3-way combinations and 4-way combinations are not things that you can afford. What I’ve done here is I’ve blocked out everything that you either don’t want to get because it has a negative present worth or things that you can’t afford. What we’re trying to do here is find the combinations that get you the maximum present worth.

You can scan down on these that are still feasible and notice that this ABC combination gets you 12 dollars’ worth of present worth. Now what’s interesting about this A B C combination is if you add up these values, how much you spent, the 10 the 7 and the 2 – you find out that you didn’t spend all the money. Now for capital budgeting problems and this is actually quite common because that internal rate of return that you calculate isn’t exactly the opportunity cost because your opportunity cost includes these foregone other options. Probably the opportunity cost is say you giving up investing in asset D. So it’s very common to have a relatively small amount of money left over and it’s also relatively common to have some asset that has a high internal rate of return that gets left out for whatever reason.

This kind of combinatorial optimization problem is done all the time. If you want an interesting example of it you can take a look at the Multnomah county budgeting process what they do is they have these program offerings, each one of them has a certain amount of cost and you have benefits that are measured in some way. What happens is they try to gather together the right combination of these investments in order to maximize the benefits that are available. You can also by watching the Multnomah county budgeting process see how to best react to what happens when you are a small high valued project , in terms of internal rate of return who doesn’t get funded because it gets pushed out by other assets. The best thing to do is not to argue that your investment is better than somebody else’s investment – that tends not to work- what you do is argue to expand the amount of money being spent. And that’s actually what you see when you see people fighting over budgets in Multnomah county. They’re always talking about adding in some contingency funds or adding in one time funds in order to fund more projects so you’re not trying to go head to head with other projects. What you’re trying to do is make it so that the funds available are somewhat larger.

The final thing I want to warn you about with these optimization problems is that you get past about 5 assets and suddenly this becomes something that is really really really difficult to do. You are required to have a computer in order to handle all of these combinations and after a certain point computers can’t handle it. You may have something like an inventory list or a maintenance list that has a 100 items on it. Trying to find the best combinations of a 100 items to fit within a capital budget can be extremely difficult. In fact the number within that exceeds the number of elementary particles in the universe. It’s not something you can just throw more computer power and find the best solution. The best you can usually hope for is something which is approximately correct. And finally, these solutions all predicate on you being able to take the remaining funds and actually throw it into a minimum acceptable rate of return and receive benefit from it. If you’re in a situation where if you’re given a capital budget and you don’t use it and you actually lose it, again, that minimum acceptable rate of return doesn’t show the opportunity cost. In fact in those kinds of situations it’s quite common for people to actually just spend their entire capital budget, desperately finding something which would provide some benefit for the money, not necessarily maximizing the benefit, taking into account that there is this opportunity cost.