BUSI 103 - Introduction to Business (Chapter 7)
August 3, 2026
You have enough money for one investment and two ways to spend it - one cheaper, one bigger, both paying back on different clocks. How do you decide which is the better move?
Read time ~25 min - ~3,686 words - problems ~40 min
Why this matters. Every big move a business makes - new equipment, a second location, a product line - spends money today to get money back later. The methods in this chapter are how managers decide whether later is worth it - and they are the toolkit the case work ahead will ask you to bring.
What you’ll be able to do.
The big ideas.
Key terms. time value, interest, compounding, present value, discount rate, discount factor, net present value (NPV), hurdle rate, internal rate of return (IRR), payback period, aggregate return.
Two entrepreneurs run an event photography service - college events, weddings, graduations. They started with a $40,000 investment in basic camera equipment, lighting, and editing tools, and the business has grown on word-of-mouth and social media ever since: $25,000 of revenue in year 1, on track for nearly $90,000 by year 8. Now they are planning a second bet. In year 3 they want to put $30,000 more into new equipment and additional staff to handle the demand - which means that in year 3, the business will swing back to burning cash instead of producing it.
Is the expansion a sound investment, or are they pouring money into a business that never pays them back? Hold that question - by the end of the chapter, you will have five different ways to answer it, and a view on which answers to trust.
Big undertakings spend money now to get money later - so before you can evaluate any of them, you have to know what waiting costs.
As we talked about before with fixed costs and operating leverage, big undertakings require big upfront investments. Scale matters, and you have to anticipate and reason backwards. You have to think about what you’re trying to get to and then figure out: if we make those investments now, will we feel good about this in the future? Think about what the right date is that we want for this decision, and then reason backwards.
One of the things that we’re trying to figure out here that’s really important is time value. Value now is not the same thing as value later. If we have resources right now, they are more valuable, because later we have to wait and also later we’re not sure what’s going to happen - the world could look very different. There’s a difference when we get benefits: if they happen today or if they happen later. We have to find some way to equate that, since so much of what we’re making decisions about actually occurs in the future. So why is money now more valuable than money later, all things equal?
People are impatient. They want things now rather than later; enjoying things now is better than enjoying things later. People are risk-averse: anything can happen between now and tomorrow, and perhaps tomorrow we’ll be dead, or tomorrow the situation will really have changed. Having the certainty, not having to worry about tomorrow - people are risk-averse. People have alternatives: we could be doing things now, or we can make other choices, rather than just waiting and being in the waiting place. And people are creative. We are a creative species. You can imagine and create alternatives: what could the future look like? What kind of future do I want to live in, and how are the choices that I make right now going to affect the kind of future I could live in? For all these reasons, being able to take actions now for things that we can enjoy currently or in the future is more valuable than if we have to wait until later.
Bottom line: every method in this chapter is a different answer to the same problem - how to compare money that moves at different times.
Interest is the price of using money now - and running it backwards tells you what any future dollar is worth today.
Let me also talk about a more intuitive, more mechanical way of thinking about time value of money, and that is thinking about something that we’re all probably really intuitive about: interest. If you give money to a bank, they put that money in a deposit, but they can use that money for other things. Because you’re doing that, they kind of have to pay you to rent your money that you’re not using right now, and that often comes in the form of interest.
This is easier to understand when we think about it this way: I give you $100, and that’s 10% interest in one year. In one year, you would give me $10 - that’s 10% of $100. Now I would have $110, because I gave you this money that you could use. The $10 is the interest, the fee for using it right now. What’s going to happen in two years? It’s not going to be another 10% on the original, so it’s not $110 plus another $10 to be $120. It’s actually going to be $121. Where does that extra $1 come from?
What happens now is we’re computing the 10% interest not on the $100, but on what’s in there now, which is $110. This is what we call the magic of compounding. We’re making interest on the interest that we accrued from the previous period. In the first period, we just get $10, so over $100, it’s $110. In the second period, we go from $110 to $121 - we’re adding $11, because that is what 10% of $110 is. As you move forward, whatever’s in there compounds, because we’re always making interest on the interest, and then the interest on the interest after that. That is how money grows when we invest it upfront.
Now, if you want to talk about how much something in the future is worth today, this is backwards. How much is $110 in one year worth today? At a 10% interest rate, if we work backwards, it would be $100 - because in our previous example, we took $100 and locked it up for one year, and in one year we had $110. But here’s another way of thinking about it: what would it cost me to create a situation where I would get $110 in a year? If a bank paid me 10% interest, I could take $100 and put it in the bank right now, and in one year I’d have $110. That’s exactly what I’d be getting if you told me I have to wait for $110. That backwards number - what a future cash flow is worth right now - is its present value.
Bottom line: compounding grows money forward; present value runs the same math in reverse, and that reverse gear is what lets us compare cash flows that arrive in different years.
Real investment choices trade off cost, speed, and size all at once - which is exactly what makes them hard.
Here is the dilemma we will carry through the chapter. Imagine you’re running a successful cafe, you have funding for exactly one strategic upgrade, and two candidates are on the table.
Project A is a long-term growth bet. You initially put in $100,000, and in years 1 through 3, you get nothing back. In years 4 through 10, you get $25,000 back per year. When you add up the total revenue, it is $175,000, and the total profit is $175,000 - $100,000, which is $75,000.
Project B is quick gains but smaller upside. We invest a little bit less, $80,000. In years 1 through 3, it’s already paying out $15,000 a year. In years 4 through 6, it pays out $10,000 a year, and in years 7 through 10, it pays out something less, $5,000 a year. The total revenue is $95,000, and the total profit is $15,000.
Now, which one of these is better?
Cheaper tells you what you can afford - it tells you nothing about what you get back.
We start first with just a naive cost comparison. If we evaluate these two, which project costs less to undertake? Project A is $100,000; Project B is $80,000. If we wanted to minimize cost, Project B wins. That’s what we should do.
The problem is that cost doesn’t tell you the total value of the investment, because to do that, we need to look at both the cost and the benefits. Believe it or not, when it comes to projects, a lot of managers make this very mistake. They just try to figure out what they want to spend. They say, “What can I afford?” We don’t say, “Well, I would like to spend less now, so whatever I have spent less on is better.” You need to take into account the other side of the equation: profits, revenue minus costs. The question is whether this cost is attractive or not, depending not on the comparison to another cost, but on the whole picture: what is the revenue I can get for this cost?
Bottom line: cost comparison answers “what can we afford?” - it cannot answer “which is worth more?”
Adding up total profit gets both sides of the ledger - but it treats a dollar in year 10 like a dollar in year 1.
That brings us to evaluation method two: aggregate return, total profit. Now we’ve got both sides right. For Project A versus Project B, we take the total amount that we’re going to get minus what it costs:
\[ \text{Project A: } \$175{,}000 - \$100{,}000 = \$75{,}000 \]
\[ \text{Project B: } \$95{,}000 - \$80{,}000 = \$15{,}000 \]
In this particular case, Project A wins, because we’re now looking at profits in a more holistic picture. We take all the revenue minus costs, and it’s clearly winning here.
But we’ve got a problem here. We’re not talking about the time value. Remember, with Project A, we get more profits, but we’ve got to wait longer - and for all the reasons we talked about before: people are impatient, and waiting means something bad could happen in the meantime. Something could go wrong. Time matters, and this whole idea of just looking at total profit doesn’t take it into account.
Bottom line: aggregate return sees the size of the payoff but is blind to its timing.
Payback period asks one question - how fast do I get out of the hole - and ignores everything that happens after.
We have to think about: well, how about payback period? How fast do I actually get my money back? It’s very common for managers to use this. What I’m going to show is two tables, both built the same way: we take the years and talk about the cash flow that comes in or out, and the cumulative column is the running total.
| Year | A cash flow | A cumulative | B cash flow | B cumulative |
|---|---|---|---|---|
| 0 | -100,000 | -100,000 | -80,000 | -80,000 |
| 1 | 0 | -100,000 | 15,000 | -65,000 |
| 2 | 0 | -100,000 | 15,000 | -50,000 |
| 3 | 0 | -100,000 | 15,000 | -35,000 |
| 4 | 25,000 | -75,000 | 10,000 | -25,000 |
| 5 | 25,000 | -50,000 | 10,000 | -15,000 |
| 6 | 25,000 | -25,000 | 10,000 | -5,000 |
| 7 | 25,000 | 0 | 5,000 | 0 |
| 8 | 25,000 | 25,000 | 5,000 | 5,000 |
| 9 | 25,000 | 50,000 | 5,000 | 10,000 |
| 10 | 25,000 | 75,000 | 5,000 | 15,000 |
For Project A: year 0, right now, $100,000 goes out, so we’re $100,000 in the hole. For the first three years, we get no positive cash flow, so we’re still down the whole $100,000. In years 4 through 6, we start getting $25,000, so we start digging ourselves out, and the $25,000 payment in year 7 brings us to zero. We finally dug ourselves out of the hole. From that point on, years 8 through 10, we’re making money, $25,000 each year.
Project B looks different. First, we’re down $80,000, which is less, but we’re still in the hole - and we dig ourselves out right away, starting year 1, $15,000 at a time all the way through year 3. In years 4 through 6 we are digging ourselves out just $10,000 at a time, and by year 7, we finally broke even. From that point on, we make $5,000 a year through year 10. That’s how you read these tables.
What can we say for this? In this case, it’s a bit of a tie. How long does it take to get us out of the hole is the question, and the answer is, for both of them, it takes seven years - even though what the two projects deliver after year 7 could not be more different.
Bottom line: payback rewards speed, which matters when cash is tight - but it says nothing about what a project earns after it pays you back.
NPV translates every cash flow into what it is worth today, so timing and scale finally show up in one number.
Now, the final piece of machinery is what we call net present value. What we ask is: what are the cash flows worth today, if we adjust for time and risk? We’re going to need something here - a number that we haven’t had to have up until now - and that is a discount rate, which is like the interest rate. Here it’s 10%, reflecting what we think our money could earn elsewhere: our opportunity cost of capital. When NPV comes back negative against that rate, we say the project fails to clear our hurdle rate - the return a project has to beat before it deserves our money.
Worked example: what is Project A worth today? Here’s how I like to think about it. I like to set things up in four rows. The first row is the period: how much time has passed, 0 through 10 - period 0 is right now, 1 means one year has passed, and so on. In the next row, I put cash flow: negative for money that’s going out of my hands, and positive for money that’s coming back.
Step 1 - build the discount factor. The discount factor row is the most important part. You take this formula:
\[ \text{Discount factor} = \frac{1}{(1 + r)^t} \]
where \(r\) is the discount rate and \(t\) is the period. It’s always getting smaller as you go out, because mathematically, the further we go, the less a dollar out there is worth today.
Step 2 - discount each cash flow. What we’re asking is: how much is that cash we’re going to get in the future worth right now? You multiply the cash flow times the discount factor. Take year 4: what is that $25,000 worth?
\[ \$25{,}000 \times 0.683 = \$17{,}075 \]
It’s actually worth $17,075 - that’s how much it would be worth if you could have that money today. We’re translating every future cash flow into what it’s worth today.
One intuitive thing I want to bring up front is what happens in period zero, for the $100,000 that goes out. We discount that right at time zero, which means it’s raised to the zero power - and everything raised to the zero power is 1. No time has passed, so what something is worth right now is just itself.
Step 3 - add it all up. We add up all these present values of cash, and that’s the NPV. For Project A, the discounted inflows total $91,450 against the $100,000 that went out today:
\[ \text{NPV}_A = \$91{,}450 - \$100{,}000 = -\$8{,}550 \]
Running the same table for Project B gives discounted inflows of $64,920 against $80,000:
\[ \text{NPV}_B = \$64{,}920 - \$80{,}000 = -\$15{,}080 \]
You’ll see that both are negative, and what that means is: at this discount rate, this is the value of the investment. You should invest in NPV-positive projects - a positive NPV means it’s making you money. You should not do negative NPV projects; those things are destroying value. Both of these projects, because they’re negative NPVs, destroy value at the 10% hurdle rate. For both of these cases, we do not want to do either one. And when we are comparing, the verdict is: pick the one with the highest NPV, or the least negative NPV - here, Project A.
The other thing that’s good about NPV is you can compare all kinds of projects: big projects or small projects, projects that pay off up front, later, or a lot later, different time horizons. As long as you’ve got the inputs, you can compare directly on this single statistic. Whatever has a higher NPV is projected to create more value than whatever has a lower NPV.
Bottom line: NPV is the one method that prices both timing and scale - one number, comparable across any set of projects, with a built-in decision rule at zero.
IRR works NPV backwards - it finds the break-even return a project delivers, which you then compare to what capital costs you.
The final way of thinking about projects is something called IRR - internal rate of return. It’s very related; it’s like a close cousin to NPV. You’ll notice we need something to compute NPV: we need to know the discount rate. Sometimes we’re not really sure what that rate is. So we might say: let’s work backwards. Let’s assume we’re going to break even - we’re not going to create value, and we’re not going to destroy value. What would the rate have to be to make that math work?
Then you compare that number to how much it costs to get money in the first place. I’m going to finance some projects, and it costs me, let’s just say, 10% to borrow money. I want my projects to at least be able to create value at 10%. You back into what rate it would have to be just to break even, and then compare that to the rate it would cost to actually secure this money, either from investors or from a loan. What you’re looking to do is find projects that offer a higher level of return - a higher IRR - versus what it costs you to secure the capital in the first place.
On my side, we can just play with these numbers and figure out when we’re going to get an NPV close to zero - try 9%, try 8%, refine from there. For Project A, that IRR is about 8.5%, and for Project B, it’s 4.2%. Now, if the hurdle rate is 8%, then Project A is worthwhile, but Project B is not. What happens if we can borrow money for 4%? In that case, both of these projects will be worthwhile, because both of them return over 4%. And at the 10% we assumed in the last section, neither clears the bar - which is exactly what the negative NPVs were telling us.
Now, IRR can be a little misleading, because it’s misleading for projects of different sizes. A rate of return on a $1 million investment versus a $100 million investment are two very different things. It assumes there’s a single rate of return - that all future cash flows are reinvested at the IRR rate itself, when in fact reinvestment might occur at higher or lower rates. It assumes projects have one initial outflow followed by a stream of inflows; a lot of projects work that way, but not always. You might put out a big outflow of cash, get some inflows, followed by another outflow - and when you have more complicated investments like that, the math can produce multiple IRRs. And IRR is scale-neutral: returning 5% on a $100 million investment versus a $1 million investment are very different levels, and IRR doesn’t get into that.
Bottom line: IRR tells you the maximum cost of capital at which a project still makes sense - useful, but check the scale before you let it pick between projects.
Each method answered a different question - and only NPV answered the whole one.
Look back at what the five methods said about the same two projects. Cost comparison said B. Aggregate return said A. Payback period called it a tie. NPV said neither, if your money can earn 10% elsewhere - and A over B if you must pick. IRR said A returns about 8.5% and B about 4.2%, so your hurdle rate decides. If you just do subtractions, you can do evaluations based on cost differences or aggregate profit differences, but these will be incomplete. In order to do this completely, at the NPV level, you’re going to need to assume a rate - and depending on that rate, it can be different which one of these projects is worth more.
In the end, NPV is the standard, and we don’t have a lot of these complications. If you can find a good discount rate and calculate NPV, that’s a great way of scoring and prioritizing projects.
Bottom line: the cheap methods are fine for a first look; when the decision matters, discount the cash flows.
So should the photographers make their year-3 bet? Run the toolkit against their eight-year spreadsheet. Aggregate return says yes: $412,478 of total revenue against $358,263 of total costs leaves about $54,215 on the table. Payback says be patient: the cumulative cash flow does not climb out of the hole until partway through year 7 - about 6.2 years after the first camera was bought. But the business holds itself to a 15% hurdle rate, and that is where the picture turns. Discounting every cash flow at 15% gives an NPV of about -$3,000, and working backwards, the IRR comes out just under 14% - close, but under the bar. The project makes money; it does not make enough money to justify 15% capital. The entrepreneurs shouldn’t abandon the business - they should change the project until it clears the hurdle: grow revenue faster, phase the equipment purchases, or finance them rather than paying cash in one lump.
7-1 New Product Launch. [LO3] A new product line requires an initial investment of $60,000 and is expected to generate net cash flows of $10,000, $15,000, $20,000, $20,000, and $15,000 over the next five years. You evaluate projects at a 10% discount rate. (a) Calculate the present value of each year’s cash flow. (b) Sum them and subtract the initial investment to compute NPV. (c) Would you recommend moving forward?
7-2 Timing Makes the Difference. [LO1, LO2] Projects X and Y each require $10,000 upfront. X returns $6,000, $4,000, and $5,000 over years 1-3; Y returns $4,000, $6,000, and $5,000. Assume a 10% discount rate. (a) Find each project’s payback period - which looks better on that basis? (b) Compute each NPV - which is more valuable now? (c) Explain why the two answers agree or disagree.
7-3 The Photographers’ Books. [LO2, LO3, LO4] Using the Event Photography exhibit (net cash flows -40,000; 4,000; 5,850; -21,773; 11,262; 15,111; 19,969; 26,075; 33,719 for years 0-8): (a) Compute total revenues, total costs, and the aggregate benefit. (b) Find the payback period from the cumulative cash flows. (c) Compute the NPV at a 15% discount rate - is it positive or negative, and what does that indicate? (d) The IRR is approximately 13.6%; interpret it against the 15% hurdle, and state what the entrepreneurs should do.
© 2026 Eric Lin. All rights reserved. This chapter is provided for students in BUSI 103 - please do not repost or redistribute without permission.