# How does NPV assume re-investment at the Opportunity Cost?

Hey guys,

So I understand how the IRR assumes re-investment at the IRR perfectly and I’ve created multiple methods to display it. However, I cannot see how the NPV assumes re-investment at the market-determined discount rate.

For example:

CF0 = -100 CF1 = 40 CF2 = 30 CF3 = 80 NPV @ 8% = \$26.26 IRR = 20% I can prove this for IRR in the following manner: Period Beginning End before withdrawal Withdrawal Amount After withdrawal 1 -100 120 40 80 2 80 96 30 66 3 66 80 80 0

And I can also prove this by saving [100*(1+20%)^3 <=> 40*(1+20%)^2 + 30*(1+20%) + 80] But how am I going to construct this for NPV @8%?

• I tried to think about this logically. If there is a investment out there offering 8% return per annum, then that should be my opportunity cost (assuming that is the only investment out there for this case), and I was also offered a investment that offers these 3 CF’s (40, 30,80). So I solved NPV at 8% and NPV was \$26. So I attempted to derive the \$26 in this manner:
• What I could earn if I just deposited my \$100 in that alt investment @ 8%: \$100(1+8%)^3 => 126 If I took the project and earned its CF’s and deposited them @8%: 40(1.08)^2+30(1.08)+80 => 159 The difference however turned out to be \$33 and not \$26 (the NPV at 8%).

So where am I going wrong?

37.04*1.08+25.72*1.08^2+63.51*1.08^3=126.26

Seems fine to me.

Hello Nenorr …

Why would you assume that the first cash flow is worth 37.04 and not 40 (i.e: you discounted it and left it at year 1). Plus doesn’t that mean that the CF’s received by the project have been altered now?

• If rates were 8%, and I discount them at 8%, my PV would be \$126.26. Therefore, I would be indifferent between receiving \$126.26 today, or \$40 in Y1, \$30 in Y2, and \$80 in Y3 (assuming I reinvest all interim CF’s at 8%).
• What I’m paying however is \$100, (so I am earning more than a 8% return), I am earning an extra \$26 above the 8% return. I’m just trying to see how that works out through a schedule (like in my OP).

The CFs of the project are received in specific periods. To get NPV you need to discount all cash flows to their present values. Or like I showed above, the present values compounded at the discount rate become the future values (cash flows).

Okay, so I discounted all the interim cash flows at 8% and got \$126.26. I see the NPV is the difference (\$26.26) between the initial investment and the PV of the cash flows.

I am just trying to map out like I did in my OP where this marginal \$26.26 lands. Because if NPV assumes reinvestment at cost of capital, when I am re-investing the interim cash flows (or the lump sum pv of the interim cash flows) I get \$159.

When I invest my initial investment of \$100 at 8% for 3 years I get \$126.

Their difference is not \$26.26. When I do the same exercise and re-invest at IRR, the initial investment compounded at IRR by n periods has a FV that is equivalent to the interim cash flows compounded at IRR. So their difference is 0. I want to see the link between the \$26.26 and how interim cash flows are re-invested at 8%.

NPV doesn’t assume reinvesting. It is simply the sum of all discounted cash flows of a single project.

But the curriculum says that;

The NPV utilizes the external market determined discount rate and that reinvestment is then assumed to occur on that discount rate.

Look at the compounding calculation above. The CF3 is calculated as its PV compounded 3 times, CF2 - 2 times, at the discount rate, that’s their assumed reinvestment (even though there’s no actual investment in anything, your actual investment is just the 100 in the beginning). But nothing happens after you receive the cash flows.

Okay I have a follow up question:

If NPV does not assume re-investment as you’re saying, then why do we bother to create a NPV schedule at different opportunity costs of capital. For example:

If I have the following CF’s:

CF0 = -4329 CF1 = 1000 CF2 = 1000 CF3 = 1000 CF4 = 1000

The above project has an IRR of 5%. If the opportunity costs of capital are from (1-4%), my NPV would always be positive, but regardless what the opportunity cost of capital is in the market, if I pay 4329 and receive those cash flows, my rate of return will always be 5%, so NPV’s by discounting from (1-4%) give figures that we interpret as “value creation to shareholders” but when in reality, the only value creation they will get is the IRR.

So why bother going through 1%, 2%, 3%, and 4% discounting if the result is meaningless if I will always earn the IRR?

If my boss asks me to evaluate an investment, and I come up with the IRR (say 5%), why would I need to show him or her what the NPV is at from (1-4%) discount rate? How does that benefit his or her knowledge?

NPV is basically a decision making tool. You discount the cash flows at an appropriate rate and if positive, you will gain value by investing. If there are multiple alternatives, you choose the one with the higher NPV. Assuming regular cash flows, NPV is positive if the irr (which you earn) is higher than the discount rate (your cost). In your case, you don’t need to bother with all 4 calculations, NPV will be positive in each. But imagine alternative projects with different cash flows (all of them with 5% irr) and appropriate discount rates, how would you choose between them if you didn’t do the NPV calculation?

Regarding your edit: like I said, irr is what you earn, discount rate is the “cost”. If you showed your boss just the 5% irr, do you think he would have enough information whether to proceed with the project? If discount rate is <5 he may choose to proceed, if it is >5 he will not, as the project would lose value to the investors. If they can earn 20% elsewhere for a similar project for example, why would they risk their capital to earn 5% here? It just isn’t going to cut it.

Thank you. To answer the question.

I would choose the project with the highest NPV because the goal is to maximize shareholder value. Even though my return may be the same, but I am receiving a larger chunk of value relatively.

26.26 x 1.083 = 33.08

FV = PV * (1+i)n

Hmm

“What I could earn if I just deposited my \$100 in that alt investment @ 8%: \$100(1+8%)3 => 126”

You are investing 100 for 3 years @ 8%. 126 is correct. In this you don’t have cashflow. If I took the project and earned its CFs and deposited them @8%: 40(1.08)2+30(1.08)+80 => 159

For this you are not actually investing \$100. If you are going to invest \$150 for 3 periods and compare it with \$100 investment. you are bound to have differences.

So now, Let’s assume you borrow \$100 @ 8% (cost of capital) for 3 years to invest in this project and as you mentioned you are able to deploy the said CF @ 8%(That’s your reinvestment).

Given these, You will pay \$126 after 3 years for borrowing \$100 and plus interest for 3 years. And you would have made \$159 after reinvestment from the project. you would have earned \$33 after 3 years.

Now you can discount back the cashflow received by 8 % for 3 periods to T-0 and you will get 126.26.

In real world you will not receive reinvestment same as your borrowing cost so if you borrow at say 14% and reinvest at 8% the project decision won’t be same.

There are broadly two schools of thought on NPV and IRR:

• One school notes that we’re using cost-of-capital (NPV) or IRR (IRR) only as a discount rate to get a present value, so any positive cash flows are assumed to have been removed from the investment. For this school, there is no reinvestment rate assumption, because there is no reinvestment.
• The other school assumes that any positive cash flows are still considered to be part of the project until the project ends. This school then assumes that the reinvestment rate is the discount rate: cost-of-capital for NPV and IRR for IRR.

I believe that the former has the stronger case, but you have to know the latter for the CFA exams.

That’s why there’s modified IRR (MIRR).

Hello Again!

After reading your comments again and again, I’ve realized that I have missed out on a very important point, and that is that the IRR is unrealistic. Taking all what you said into consideration, I will construct one example which hopefully will be correct and logical.

• There is an investment which your company wants to undertake which offers the following CFs: CF0 = -100 CF1 = 40 CF2 = 30 CF3 = 80 The manager must borrow the \$100 from the bank at a 8% interest rate.
• The first thing I would do is observe if this project earns an 8% return or more, and I can do this by discounting each cash flow at 8%, and see what PV I get. Doing this I get \$126. Therefore, I need to deposit \$126 today in order to earn those cash flows and have those cash flows earn 8% per annum. The good thing is, this project is only costing me \$100. Now here’s where I need to throw an assumption, this project will only work if I can re-invest those cash flows at the NPV rate (unrealistic). So on a chart: I borrow \$100 at 8% and in three years I have to pay back: 100*(1+8%)3 => \$126 I take the project and re-invest all the CFs at 8%, I get: 40*(1+8%)2+30*(1+8%)+ 80 => \$159 Great, I made profit, digging further I need to realize important things: (1) You assumed that there was a financial institution that would give you 8% per annum on your CFs (hence re-investing at the NPV-rate). (2) The reason you made profit is because the cost of the project (\$100) is sufficient enough to realize profit if the cost of capital is 8%, and that is because, at 8% the PV of those cash flows is (\$126). If the cost of capital was greater than 8% (say it was 21%) then there would be no profit to be realized because you would end up paying more on the loan then on earning your CFs and reinvesting (assuming there’s a financial institution offering you the chance to re-invest your earnings at the IRR). (3) The IRR is the internal rate of return on a set of CFs which has an initial outflow, however, you only realize that IRR return if you can re-invest your earnings at IRR (which is very unrealistic) - and that is why the curriculum says it is more realistic to re-invest at the NPV. (4) THE MOST IMPORTANT POINT: Just because you look at a set of CFs (such as above) and hit the IRR formula (which turned out ot be 20% on the above CFs) - it does not necessarily mean you will earn 20% rate of return because just like in this example, there was no institution offering you 20% on your CFs! There was only an institution offering 8%. _ That being said, we still employ the IRR function to see at what rate will the re-investment of those CFs outweigh the hurdle of the initial investment _. (5) I started out this post thinking I would always earn the IRR [40*(1+20%)2 + 30*(1+20%) + 80], but now I realize, I earn what is available in the market only [40*(1+8%)2 + 30*(1+8%) + 80]. (6) When IRR > Cost of Capital, you accept the project, not because you are earning IRR , but because your current investment structure is tailored in a manner that covers your cost of capital.
• Thank you everyone. Magician - I will do more research on the MIRR! Haven’t crossed it yet on the curriculum.

That all looks good.

MIRR is _ not _ in the CFA curriculum, by the way.