# Question on using the interest rate on a loan as the hurdle rate for a net present value calculation

I'm reading a real estate investing book that provided the following example.

1. You finance a property for \$100,000 using a HELOC with a 5% interest rate.
2. The property generates \$2,000 in cash flow each year for four years.
3. You sell in the fifth year for \$130,000.

To analyze this deal, the authors computed the net present value of the deal using a 5% discount rate (Equal to the interest rate on the loan.). They said you would do this to determine whether you can "...cover the cost of debt." This doesn't make sense to me. It seems to me that as long as you make a profit overall then you are covering your cost of debt.

Calculating net present value makes total sense to me when the discount rate represents the amount you expect to make in alternative investments, but it doesn't make sense to me when it's based off of the interest rate of a loan.

Am I missing something here? How is it that a discount rate could be used to determine whether you are going to break even on a financed deal?

UPDATE 10/11/2023

Since some people are asking for more context about how this problem appeared in the book, I am copy/pasting the exact text below.

...in the investing and finance world, we’re often dealing with projects that are financed through debt. When financing a deal, the question often isn’t “Do the returns on this investment beat out the returns on an alternative investment?” Instead, the question is often “Are the returns on this investment enough to surpass our cost of debt?”

Let’s say that we have the opportunity to invest \$100,000 in a deal that would return \$2,000 per year for four years, and then \$130,000 in Year 5.

Further, let’s say that we didn’t currently have the cash to make this investment, but we knew that we could take out a home equity line of credit (HELOC) on our personal residence to acquire the \$100,000 to invest... The interest rate on the HELOC in this example would be 5 percent.

Does it make sense to borrow money against our personal residence at 5 percent interest for this investment? NPV can give us that answer:

Our NPV in this scenario is positive. In other words, we would make money—in today’s dollars—by borrowing \$100,000 at 5 percent interest for five years to do this deal. Specifically, the return on this deal would surpass our hurdle (discount rate) of 5 percent interest on the loan we took out to fund the deal.

• Are you sure the 5% number was related to the loan rate or is it possible that it was just a coincidence? I'm inclined to agree that doesn't make sense. Oct 10, 2023 at 17:11
• Could you give some details on exactly how the authors apply this to the scenario described? I'm not even sure exactly what this means. What number(s) are being discounted by 5%, for example. Oct 10, 2023 at 20:22
• @JimmyJames I edited my post to include more context. Oct 11, 2023 at 19:40

It seems to me that as long as you make a profit overall then you are covering your cost of debt.

That's true, and NPV tells you that if you use the cost of borrowing as the discount rate.

If you ignore discounting (i.e. set the discount factor to 0), the NPV will tell you the net difference between all income and expenses, if you include the interest portion of the loan payment as an "expense" (plus all of the other ancillary expenses that come with renting a property like taxes, maintenance, etc.). If the NPV (income - expenses) is positive, you made a profit overall.

By using a 5% discount rate, the cost of debt is accounted for automatically. You just need to include the income and expenses of the property, and the interest expense is accounted for through the discount rate.

If you use a higher discount rate than the cost of debt, that means that your debt is "cheaper" that it should be, and interest payments should be discounted at a different rate to account for that, which would increase the profitability of the project.

Comparing "profitability" using discount/interest rates is a little more sophisticated that just calculating net profit including financing costs, but in the end they should be equivalent.

You are right that the "discount rate" is often described as the return you'd expect from other investments, but it can also be used to determine if a project financed through debt is making more than it costs to finance it. Meaning, you might use a higher discount rate if you were comparing it to a different investment, but using the cost of funds indicates whether the project is profitable on its own.

--EDIT--

After including the full context, notice that the analysis does not include the interest expense, just the P/L from the rental. The 5% discount factor effectively factors in the interest payments for you without having to calculate them. If you did an analysis (without discounting) of the net profit after interest, you'd find that it was positive as well.

• I don't follow this. The point on NPV is to discount future cash flows. Using it for account for debt costs seems like a needless complication and doesn't really address the cost of money. The mortgage rate has no relationship to the NPV of the \$2K per year, for example. Oct 10, 2023 at 19:31
• If you discount the cash flows from the rental at the rate at which you borrow the money to pay for the rental, then a positive NPV would tell you that your actual rate of return is higher than your cost of, hence it is profitable. Certainly there are other way to determine if the deal is profitable but using NPV is not unreasonable. Oct 10, 2023 at 20:05
• But the NPV is used to understand cash flows. Are we talking about using it in lieu of calculating the payment costs, the NPV of the cash flows, or both? Oct 10, 2023 at 20:12
• I believe the book is using NPV in lieu of breaking out the interest portion of the debt payments. The principal part of the debt payments would not go toward profitability since it would increase equity. Oct 10, 2023 at 20:24
• Another option would be to calculate the IRR of the cash flows and compare that to the borrowing rate, but that's a harder calculation than picking a discount rate and seeing if the NPV is positive. Oct 10, 2023 at 20:26
1. Remember to subtract the interest costs from your sales price when figuring profit.

2. Remember that with a standard loan, your earliest payments are mostly interest. You can get actual numbers by doing an amortization table; there are lots of free tools that can do this for you. So if you flip the house too quickly, you have built up almost no equity and most of what you get for it is going to be immediately eaten by paying off the mortgage.

3. There will be agent fees and closing costs. And again when you sell the place.

4. If you're living in it, you aren't paying rent. But you usually do have some costs early on for deferred maintenance (first thing I did was replace the furnace, \$6k then and presumably more now), plus ongoing maintenance and tools to perform that work. (I'm currently deciding whether to fix the ancient lawnmower again or buy a new one for about \$400.) You may also want to count at least one round of movers.

Now combine these. The house has to have appreciated enough and saved you enough in rent to make up for the interest and other costs just to break even.

Also remember that when you move out, you will likely be moving into another house whose costs have also gone up during that time period. So you may not have a real profit at all.

And unless you buy a wreck and put a lot of sweat equity into it, few renovations actually add more to sales price than they cost, so for most people that isn't a source of profit either (though it can make the house more enjoyable to live in).

Finally, when running the numbers, remember that the money you put into the closing costs and down payment and initial renovations could have been invested in something else and earning a profit there. So that has to be factored into the comparison too.

As a general rule: you can buy property and run it as an investment and business, or you can buy it to live in it, but trying to do both just doesn't work out mathematically. You wind up passing the money from one pocket to the other. There are exceptions, but not as many as folks want to believe.

This idea does seem very questionable to me. At a very simple level, to estimate the profitability in nominal terms, you would take the debt payments net income and subtract them from the sale amount net loan balance. That's all you need to know to determine whether you can 'cover the cost of debt'.

The discount rate, as you point out, is about whether the return profit you get when the property is sold was worth the debt payments during the years up to that sale. You could end up with a nominal profit that didn't beat a money-market fund's returns if you don't account for NPV. I gather that you know all this, I am just being clear for anyone else who might read this.

I'm not sure what book this. It's possible you are leaving out info or misinterpreting it, but what you describe seems pretty questionable at best. Your knowledge and judgement seem more sound to me based on the information here. The only thing I can come up with is that discount rates often are based on or incorporate the risk-free rate which is a factor in loan rates. Maybe they are using it a shortcut for the discount rate. Does this book say you should do this in lieu of account for the loan payments?

UPDATE:

Based on a better understanding of what this book is saying, I find this to be a really strange way to model this. Maybe it 'works' but it isn't obvious to me that it does. I might update later with a more detailed analysis of this 'trick'. I call it a trick because it doesn't really describe the reality of the situation. I think, if you are serious about putting your money into something, you should really understand what will actually happen. Please note that I am not accounting for things like fees, commissions, and taxes and other things that will absolutely matter in the real world. This is a very high-level analysis meant to align with the level of the example as given.

The first problem I have with the way the authors are modelling this is that they start with a negative cashflow of \$100K. At no point, if you did something like this (and all goes well) will you be committing \$100K. I've never used a HELOC but I gather that is being used here because it doesn't require a down payment. The \$100K used to buy the home is the bank's money. Assuming a 20-year loan (which is what the google told me is typical) compounded monthly, I get a monthly payment of \$659.96 based on this calculator. Since everything is stated in terms of years, I'm going to multiply that by 12 and round to \$8K yearly in payments for 4 years. You get \$2K from the property each year so you are putting in \$6K each year. That's what you are actually committing (in cash) to this deal. After 4 years the balance on the loan is \$87K when you sell it for \$130K which leaves you with \$43K assuming you pay off the loan. You paid in \$24K so you made \$19K in nominal terms.

So yes, this is profitable in nominal terms which agrees with the authors assertions. Are they correct for the right reasons? After thinking some more, the math is just a backwards way of determining if the interest costs exceed the payoff without considering the time-value of money.

This downside of this approach is that it doesn't actually help you understand the most important question: can you fund this? That is, you can have a profitable deal that you cannot execute successfully because you don't have the cashflow required. To that point, if you can't scrape together \$500 every month (really up to \$667) you shouldn't do this deal regardless of whether the payoffs cover the interest. And based on discussions with people who invest in real-estate, you probably want to have the rents at least cover the loan because you will almost surely have other costs to worry about. Speculating by assuming you can sell a property in a certain number of years for a given price is a risky proposition.

And, as you note, this doesn't actually tell you anything about the NPV of the deal. Spreadsheets are your friend. This isn't a drug-deal that you need to do in your head on a street-corner. I don't see the point of confusing things by bringing in NPV to solve for profit in nominal terms.

OP: How is it that a discount rate could be used to determine whether you are going to break even on a financed deal?

The discount rate is used to calculate the periodic repayment amount `d` by balancing the present value of the loan `s` with the sum of the repayments all discounted to present value. For example, (neglecting fees & taxes):-

``````s = principal
r = periodic rate
n = number of payments
d = payment amount
``````

$s=\sum_{k=1}^{n}\frac{d}{(1+r)^k}=\frac{d-d(r+1)^{-n}}{r}$

``````  s = (d - d (r + 1)^-n)/r
∴ d = r s/(1 - (1 + r)^-n)
``````

Assuming the example property is initially financed for a term of 20 years.

``````  s = 100000
r = 0.05/12
n = 20*12

∴ d = r s/(1 - (1 + r)^-n) = 659.96 per month
``````

Total payments over five years prior to sale:

``````paidout = 5*12*d = 39597.34
``````

Income over this time: `income = 4*2000`

Balance at the end of five years. (Example does not actually specify 'end'.)

``````x = 5*12
balance = (d + (1 + r)^x (r s - d))/r = 83454.86
``````

Sale price at the end of five years:

``````saleprice = 130000
``````

Profit margin:

``````saleprice - balance + income - paidout = 14947.79
``````

Note, if the term of the loan is only 5 years the profit is `24772.60`, i.e.

``````  s = 100000
r = 0.05/12
n = 5*12

∴ d = r s/(1 - (1 + r)^-n) = 1887.12 per month

paidout = 5*12*d = 113227.40

x = 5*12
balance = (d + (1 + r)^x (r s - d))/r = 0

saleprice - balance + income - paidout = 24772.60
``````

Plotting for various terms, arranging the shortest term is the most profitable because the payments are higher and the loan is paid-down quicker lessening the interest paid.

The present values of all cash flows are:

Loan: -100000

Yearly revenue: 2000 a_[4, 0.05] = 2000 (1 - 1.05^-4)/0.05 = 7091.90

Sale: 130000 (1.05)^-5 = 101858.40

Therefore, the net present value is 101858.40 + 7091.90 - 100000 = 8950.30 at the time of the issuance of the loan. Assumptions are that the interest rate on the loan is constant and reflects, irrespective of the schedule of payments, the effective annual rate of interest. Also, the revenue is paid at the end of each year, and the sale occurs at the end of five years.

There is no need to do any additional computations to account for the loan structure: the equation of value must balance for any fixed valuation point, and all cash flows are simply discounted or accrued to represent their present values at that time point.

I should also mention that this calculation does not depend on the term of the loan, as long as it is at least 5 years. This is because the proceeds from the resale of the property would be used to pay off the loan balance in full. Otherwise, additional unstated assumptions would come into play, such as the interest rate that you could receive by investing the proceeds of the resale.

If you wanted to calculate the NPV at other points in time, we would need additional assumptions about how payments on the loan are structured, as well as how the revenue from ownership of the property is received. If you wanted to calculate the net accumulated value at the time that the resale is made--that is to say, what is your profit at the conclusion of these transactions, simply accrue the above NPV for five years at 5%.