Seeking a high-level explanation to explain the relationships among cap rate, IRR, discount rate and NPV in commercial real estate in a conversation, assuming you are explaining to an entry level real estate analyst.

To kick off the discussion, there is one particular confusing concept is that IRR is defined as the annual rate of return that generates NPV of zero. The formula of net present value is NPV=Present Value - Initial Costs. If IRR is discounting the future value to present to break even, what does the return matrix of UIRR =12%,/15%,/18%/20% mean? I thought the NPV tells how much net value the investment is expected to create.

It seems like IRR can be synonymous to discount rate in certain context, but has a different meaning when discussing return (IRR is effectively the blended rate of the speed of money coming back to you; IRR needs to be greater or equal to cost of capital).

It will be great if you can share your thoughts on the above! All thoughts and comments are appreciated.

## Comments (24)

Just google for like five seconds for fucks sake...

https://www.google.com/search?q=npv+and+Irr&oq=npv...

https://www.google.com/search?q=npv+and+Irr&oq=npv...

Lower discount rate > Higher npv > lower cap rate

Higher npv > higher IRR

Assuming rent is held constant here.

Thank you for the explanation. Please let me know if I understand you correctly.

Lower discount rate > Higher NPV> lower cap rate

Basically, you are saying a low discount rate represents low risk on the future cash flow resulting in higher resale value so the cap rate is low.

Higher NPV> higher IRR

Basically, you are saying the higher net value of the investment created, the higher IRR.

Conclusion I drew from your response is

When Discount rate = IRR, NPV=0

When Discount rate < IRR, NPV>0

When Discount rate > IRR, NPV<0

Interpretation in the first half is correct.

Second half is correct too, and is based on finance concepts. I think you understand it.

Higher NPV doesn't necessarily equal higher IRR if you are considering more than one project and comparing the IRRs of the two.

I come from down in the Valley, where Mr. when you're young, they bring you up to do like your daddy done.

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I am confused on what you are asking -- it seems like you are struggling with the basic definitions, or am I missing something in the question? I don't mean to come off as rude.

No worries. Appreciate you being candid. I have trouble understanding the relationship among IRR, discount rate and NPV. If IRR is the % that generates NPV=0, what is the e.g. a 20% return in a doesn't seem to be relevant to discount rate. NPV = 0 means no value is being created, so the fact that IRR = 20% seems to contradict with its own definition "annual rate of return that generates NPV of zero."

Let's assume the below

Year 0: $(10,000,000)

Year 1: $2,000,000

Year 2: $2,080,800

Year 3: $2,208,162

Year 4: $2,390,185

Year 5: $11,000,000

IRR = 20%

See my narrative below. But to break it down even simpler, the IRR is objective--when you invest $10M at time 0 and your cash flows are presented as such, the IRR is the internal rate of return that is calculated. It's a mathematical fact, so-to-speak. 20% is the IRR that sets your NPV to 0. It's just a statement of fact.

Your discount rate, however, is a subjective number. If you subjectively determined that you only need a 10% IRR for the project, then when you discount your cash flows (as presented in your example) at 10% your project runs a positive NPV. Any project that has a positive NPV means the project is more profitable than your minimum requirement.

I, too, am a bit confused about the question, but here are my thoughts using a homebuilding example.

Your discount rate is the IRR that YOU, specifically, need to obtain for the project to make sense TO YOU (and ONLY you--no one else). In theory, your discount rate could literally be any real number fathomable, from 0% to no upper limit--it's whatever number that you/your company/your investors require to balance the risk/reward of a particular project. However, in practice, your discount rate--YOUR required internal rate of return--will be dictated by market fundamentals. For example, for new home subdivisions in the Washington, D.C. exurbs, large homebuilders are valuing land utilizing a discount rate of at least 25%. To continue with the homebuilder example, let's say Pulte projects cash flows on a piece of exurban land--those cash flows may be discounted annually using a 25% Pulte discount rate; using that discount rate, Pulte can determine the implied mathematical value of those cash flows (the discounted cash flows), which is the maximum price point at which PULTE is willing to purchase the land. So Pulte used its OWN subjective discount rate to determine the value of the piece of land to Pulte. When the "market" of homebuilders (let's say 6-12 homebuilders) values the land using their own subjective discount rates (and other inputs), a land "market value" is determined, which is the highest price one of the, say, 12 homebuilders ran the land DCF value to. In other words, the land value (DCF value) to each individual investor is subjective, but the competitive market helps guide the discount rates used by the homebuilders (i.e. you could use a 100% discount rate, but if your competitors are all <30% then you'll never win a bid and you'll go out of business).

Ok, so let's continue in the example. Pulte used a 25% discount rate and ran an implied land value of $13,000,000--again, what was used was the subjective, Pulte-specific discount rate of 25% to find the Pulte NPV (land value), which is $13M. Now, when Pulte goes back to the DCF and plugs in $-13M into the DCF analysis at time 0 for the land acquisition price, the NPV should be $0 when using the 25% discount rate. The 25% discount rate that solves for the land value becomes a 25% IRR when you are solving for your project's return at a $13M land purchase price. In other words, your discount rate (of 25%) is the IRR that sets your NPV to 0 (when you acquire land at time 0 for $13M).

Does that make sense?

A deal's IRR is the discount rate which sets the NPV = 0

If a deal's IRR is the discount rate which sets NPV to 0, wouldn't that mean this particular investment is projected to breakeven?

No, because NPV is the DISCOUNTED cash flow. The actual cash flows are still positive. "Break-even" is a measure of un-discounted cash flows. $10 out and $15 in produces positive cash flow of $5, but when you discount those cash flows at a desired rate of return (your discount rate) it may calculate to an NPV of $0 (if your discount rate is = to the IRR).

Lets say I am an investor and am entirely funded by one investor, who says she wont do a deal unless she gets a 20% irr. So that is my cost of capital.

If I buy a duplex for $100, and it cashflows $20/year for 5 years, and then I sell it for $100, my cash flows are.

Y0: -100

Y1: 20

Y2: 20

Y3: 20

Y4: 20

Y5: 120

Lets also assume for simplicity sake, that my NOI = my cashflows (no debt on the project, no capex, etc.) With these assumptions, the following is true:

The cap rate is 20%, The irr of this is 20%. The present value of this at a 20% discount rate is 100, the net present value is 0. But you do not only break even, you make 2x your money.

Now lets assume that this duplex actually costs $500, it still cashflows for $20/year and I sell it for $500 in 5 years. so the cashflows are as follows:

Y0: -500

Y1: 20

Y2: 20

Y3: 20

Y4: 20

Y5: 520

Now, the cap rate is 4%, the IRR is 4%, but if i discount the cashflows w/ a 20% discount rate, the present value is $261 and the net present value is -$239. If I were to discount the cashflows at 4%, I would get a present value of 500, and a net present value of 0. The present value of $261 tells me that if I want to get an IRR of 20% on the project, I am going to have to pay a lot less than $500-- paying $261 will get me an IRR of 20%. It also tells me that if I need a rate of rate of 20%, I should pay no more than 261.

Finally, lets assume the following cash flows:

Y0: -500

Y1: 5

Y2: 0

Y3: 100

Y4: 105

Y5: 2000

Lets say this is a distressed office building that is 5% occupied at purchase, I then kick them out, and lease full lease up, and then sell the project. Once again, assuming my NOI = my cash flows, then my cap rate here is 1%. My project IRR is 36.23%. If I discount the cashflows at 20%, the present value is 916.43 and the net present value is 416. This means that I would be getting a great deal at 500, and could afford to spend 916 and still get the 20% irr that I need.

Ym = Yo + Ye

IRR just generalizes a simple rate of return to multiple periods, taking into account the time value of money. Rate of return=payoff/investment -1 -> NPV=C_0+C_1/(1+discount rate)=0. These are equivalent, and the discount rate that makes NPV=0 is the rate of return (IRR for multiple periods). The IRR rule is to accept projects if the opportunity cost of capital (hurdle rate) is less than IRR.

This is equivalent to the NPV rule under some regularity conditions - we are effectively asking whether our project has positive NPV. Because the discount rate used in calculating NPV is just the opportunity cost of capital (return you could get in other projects with similar risk).

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