All Forum Posts by: Kim Hopkins

 Hi @Immanuel Sibero! I've been working on applications of this concept all week. I have some good stuff to share but I need to iron out the kinks first. 

In terms of your option of explanations "in words"... I'm going to add one to the mix... 

It's really just a general rule about ratios and proportionality: 

If A/B = C/D then (A-C)/(B-D) = A/B = C/D. 

In words: 

if A is proportional to B, and C is equally proportional to D   then the difference of A and B is equally proportional to the difference of C and D.

So it's really just about understanding WHY this general statement is true. 

I tried to discuss it with ChatGPT, but I swear that guy is not as smart as everyone says he is...I still don't intuitively see it. 

But nonetheless, the application is if the cap rate (NOI/PP) is equal to the mortgage constant (DS/LA), then the differences are equally proportional, and that's the cash on cash (NOI-DS)/(PP-LA) by definition.

By the way, the same is true for a regular loan, not just interest only! 

Now I'm trying to get a list of relationships like this between the cap rate and interest rate that effect the cash on cash, so that I can easily analyze any property using these simple inputs .... stay tuned ... 

Someone on my team figured it out! This is called the mortgage constant or the mortgage capitalization rate. More to come... there's a lot this little guy tells you about a deal...

Here's more info on it, in case anyone finds this thread...

https://www.investopedia.com/terms/m/mortgageconstant.asp

@Immanuel Sibero Awesome! Yes, there are several ways to prove it, but yours I think is the cleanest, great!

Now back to explaining this "in words", I love the direction you were going with that, but when you conclude: 

"
In this scenario, IF the debt holder demands 10% INTEREST can you see that you, as the owner, would also get 10% COC? The reason for this is because the property happens to be paying 10% CAP Rate, so everybody gets 10%!"

I believe you (of course, since we already proved it's true), but I don't see why if the cap rate is 10% and the debt holder demands 10% in interest, then we get 10% in COC. Does that make sense to you?

Hi @Immanuel Sibero and @David M. , I'm excited for more math brains to discuss this with! Let's understand this very simple example first: 

Suppose it's an interest only loan. Define the variables: interest rate i, annual debt service DS, loan amount LA, purchase price PP, net operating income NOI, cash on cash COC, and cap rate C.

Since it's an interest only loan, 

i x LA = DS (interest rate times loan amount equals annual debt service).

Now suppose the interest rate equals the cap rate: i = C.

Then I claim the COC = i = C. In other words, the COC is equal to both the interest rate and cap rate.

Here's the proof: 

Since i = C, DS/LA = NOI/PP.

That's the same as saying NOI/DS = PP/LA. Call this ratio r.

So r = NOI/DS = PP/LA.

Then NOI - DS = r*DS - DS = (r-1) * DS

and PP - LA = r*LA - LA = (r-1)*LA.

So COC = (NOI - DS)/(PP-LA) (by definition)

which is = (r-1)*DS/(r-1)*LA = DS/LA = i = C. 

I'm trying to understand this in WORDS though. If the rate of payment (interest rate i) equals the rate of return (cap rate C), then WHY is the return on investment (COC) also equal to these rates? I can understand it mathematically, but not in practice.

@Immanuel Siberoundefined

 Good point. I shouldn't call it the "effective interest rate". That name is already taken. It should be called the "effective payment rate" or something like that because it gives you the total annual debt payment (including principal). This isn't the same as the effective annual interest rate you mentioned above which is only giving you interest. 

I have a math question for you! I'm trying to analyze our portfolio and have gotten into some pretty deep math calculations regarding interest rates and debt payments.

I want to know: is there a name for the percentage, call it "R", that is your total annual loan payment P divided by your total loan amount PV? 

I would call it the "effective interest rate" or something like that because R x PV = P. (The effective interest rate times your loan amount equals your annual loan payment).

To say it another way, in the classic loan amount formula below, is there a name for the "rate" R := r/(1-(1+r)^(-n)) circled below in red?

 Hey @David M., wow did this get me in a rabbit hole! I figured it out. I think I need to explain it in a fresh thread because of my variety of typos above, but here's the basic idea.

It is not actually true that the COC (Cash on Cash) is monotonically decreasing with increasing debt.

The simplest example is an interest only loan.

If you have an interest only loan, and your interest rate is less than the cap rate, then COC is INREASING as a function of debt. In other words, the more debt you take on, the higher your COC.

If your interest rate is higher than your cap rate, then COC is DECREASING as a function of debt. In other words, the more debt you take on, the worse your COC gets.

Lastly, if your interest rate is equal to your cap rate, then your COC is actually equal to your interest rate and your cap rate.

A similar phenomenon is true if it's not an interest only loan, but it's more complicated. 

I've written it all out with actual math proofs for myself. I can't believe this doesn't exist anywhere. Can't find it anywhere online. It's incredibly helpful for analyzing an existing portfolio or a new acquisition since you can very quickly understand what adding debt to a property will or will not do to it's COC.

After over a decade of doing this, I still don't understand how the debt terms directly affect the Cash on Cash return of a deal. 

For simplicity, define CAP Rate (:= NOI/Purchase Price) and COC (:= Cash Flow/Total Cash Invested).

Assume Cash Flow := NOI - Debt Service, and Purchase Price := Total Cash Invested + Debt Service.

• Say you are considering buying Property A. If you buy Property A WITHOUT debt, then CAP Rate = COC because Debt Service = $0 so Cash Flow = NOI and Purchase Price = Total Cash Invested.

• If you choose to add debt to Property A, then Total Cash Invested goes down (since you're using debt), so the denominator of COC goes down, so COC could potentially go up. HOWEVER, Cash Flow also goes down, because now you have to pay Debt Service, so the numerator of COC also goes down, so your COC could potentially go down as well.

The "math" question is what is an easy way to determine if the COC is going to go up or down, based on the terms of the debt?

For example, if the interest rate for the debt is less than the CAP rate, does that mean the COC will always go up? I don't think so. I think it depends on the LTV and possibly other factors.

Is there any simple relationship here? 

 @Allan Smith Hi, sorry for the delay. I'm back to this project now and still stuck! 

You said you were calculating ROE as: 

Profit / (FMV - Loan Balance).

Agree with the denominator. It's the numerator I'm questioning. 

Do you define Profit as: 

Profit := NOI - One-Time-Expenses (Capex, Lease Commissions etc) - Debt Service?

If you're comparing ROE of an existing asset to COC of a new asset, then Profit should definitely be after debt service, since that's accounted for in the definition of COC.

But for one-time expenses, if you are only looking at the ROE of an existing asset for one year (say the prior year) and you put on a new roof, you don't want to conclude the property has a terrible ROE from this one time expense. 

That's why I'm saying "Profit" in ROE should be defined as: 

Profit := NOI - Debt Service.

But this definition doesn't exist anywhere. So...still confused.

 @Chris Seveney Thanks so much for the reply! Sorry for the delay in responding. I'm returning to this project now and still very interested. 

- Great point about including "one-off" expenses - if a property has a major problem every year, it's not a great asset, as you said. 

- I agree with you that depreciation should not be included in measuring performance of different assets.

- I think that debt service DOES need to be included but I think there should be TWO different measurements - one inclusive and one exclusive of debt service. The reason I think it's important to consider debt service is b/c a property may look like it's performing well compared to another, but if the debt terms on it suck, it's not performing from a cash flow perspective. Conversely if a property is not performing well and is currently debt free, it might perform better if a good debt instrument is added to it. 

- I don't use IRR on my existing portfolio because we've held some properties for 20 years and some for 2 years. As I understand IRR, the calculation "punishes" a long term hold, and we are long term hold investors.

- For ROI vs ROE - I consider the "I" to be your original investment (e.g. down payment) into the deal. I consider the "E" to be your CURRENT equity in the deal (Current Fair Market Value - Debt Owed). Those are two entirely different numbers in my book. Again if you've held a property for a long time, or if you don't have debt on it, the ROE will be correctly "punished" compared to a property with healthy debt and not too much equity sitting in it.

Would love to hear your thoughts! I'm still stuck on this.