# calculate equivalent yearly interst rate on house purchased with a mortgage

I've got 60k and I'm looking at buying a house for 145k. The bank offered me a 20 year loan of 100k at ~1.8% interest rate. After 20 years I would have repaid the bank around 16k, according to their loan offer.

I'm looking to calculate the equivalent return-of-investment per year I'd need to have if I wanted to make the same amount of money with a different investment, assuming I keep the house for X years and then sell it. The initial numbers don't matter too much, I'm more interested in the calculation.

I've seen several related questions but haven't found one that shows this calculation directly. Please excuse me if it's a duplicate.

Here's what I've got:

Buying the house costs around 160k (145k + 15k for closing costs). Property values typically go up ~2% a year. So at year X the house is worth 145*(1.02^X).

The rent from the house covers the loan, property taxes and regular maintenance, with a negligable amount left over. So after X years, I would have paid back 116*X/20 of the loan, therefore I'd have 116*(1-X/20) of it left to pay. That means after paying back the loan, I have

``````145*(1.02^X) - 116*(1-X/20)
``````

left. My initial investment was 60k, so I'd have

``````[145*(1.02)^X - 116*(1-X/20)]/60
``````

times my initial investment. Assuming I want the same behavior from another investment (compounded yearly), I'd need

``````{[145*(1.02)^X - 116*(1-X/20)]/60 } ^ (1/X)
``````

roi per year. I get these values for X = 5, 10, 15 and 20 years:

``````X = 5 => 1.04 => 4% yearly ROI
x = 10 => 1.07 => 7% yearly ROI
x = 15 => 1.07 => 7% yearly ROI
x = 20 => 1.066 => 6.6% yearly ROI
``````

Is this calculation correct? What does it leave out that should be considered?

Edit: replaced "notary fees" with "closing costs" as per Ben Voigt's comment

• Can you clarify your loan data? Do you just pay interest on the loan or do you also pay down the principal? If yes, how much ? Your statement around "repaid the bank around 16k," is unclear. Are you saying that your principal after 20 years is 84k (100k-16k) ? – Hilmar Oct 18 '19 at 13:09
• what is "15k that goes on notary fees"? I would certainly not pay 10% off the bat to buy a house. Is this points to get the interest rate to 1.8%? – D Stanley Oct 18 '19 at 14:29
• @Hilmar sorry about the amiguity, what I meant is that after 20 years the sum of my payments to the bank would be 116k – Amos Joshua Oct 18 '19 at 16:05
• @DStanley this is in France, notary fees for the purchase of such an asset are around 10%, payable by the buyer – Amos Joshua Oct 18 '19 at 16:06
• @AmosJoshua: Might those be taxes, with the notary serving as collection agent, and not "notary fees"? – Ben Voigt Oct 18 '19 at 16:30

You have one minor flaw:

So after X years, I would have paid back 116*X/20 of the loan,

Loan principal does not decrease linearly - you pay back very little principal at the beginning (since most of your payment in interest) and it accelerates as you pay it down.

Plus you don't "owe" 116k out of the gate. You only owe 100, so using 116*X/20 would be overstating the amount you owe.

You can use the excel function `CUMPRINC` to calculate how much principal you would have paid down in 5, 15, 15, and 20 years and see how much of the loan you've paid down.

For a hint, a 100k loan at 1.8% interest should have the following principal due at the end of each year:

`````` 5: 78k
10: 54k
15: 28k
20:  0k
``````

As you can see, the principal does not decrease linearly but is slower at the beginning (only 4,200 the first year) and accelerates (5,900 the last year). With a higher interest rate the difference is much more dramatic.

• Assuming all goes according to plan, the OP puts \$60k down, and will have a house worth \$215k after 20 years, with all the mortgage, interest, taxes, and maintenance paid by renters in between. That's almost a 7% annual return, which isn't bad, but the real estate market can be volatile and it requires a tenant for 240 months straight. There are probably easier and less risky investments, like the stock market, that could get a similar return. Being a landlord is a lot more work than being a shareholder. – Nuclear Wang Oct 18 '19 at 14:51
• I've removed my comments about the viability of the investment since I am not an expert on French real estate. – D Stanley Oct 18 '19 at 16:18
• @DStanley I appreciated your comments on the viability, IMO they were very much to the point. There are other dimensions to this investment (it's a tourist spot, etc) that I didn't mention, but beyond that my main interest was to know how many holes were in the back-of-the-envelope calcuation. Thanks for pointing them out – Amos Joshua Oct 18 '19 at 18:04

Here is how I would approach this. Taking the rates, 1.8% and 2%, as effective annual rates.

``````deposit  = 60000
fees     = 15000
house    = 145000
loan     = house + fees - deposit = 100000
loanrate = 1.8/100 = 0.018
monthlyloanrate = (1 + loanrate)^(1/12) - 1 = 0.00148777
numberofmonths  = 20*12 = 240

s = loan
r = monthlyloanrate
n = numberofmonths
d = monthlypayment
``````

Standard loan equation

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

$\therefore d=rs(\frac{1}{(1+r)^n-1}+1)$

``````d = r s (1/((1 + r)^n - 1) + 1) = 495.779

monthlypayment = 495.78

totalinterest = monthlypayment*numberofmonths - loan = 18987.20
``````

This is obviously different from the 16k mentioned by the OP. It appears either the 16k is wrong or the rate is wrong because it's difficult to square 16k with 1.8% on 100k. So continuing with the calculation as shown.

``````appreciation = 2.0/100 = 0.02
``````

This is what has been paid out after 5 years

``````paidout = deposit = 60000
``````

What follows is the amount of the house paid for. (The bank still owns the rest.)

First, the accumulated principal paid down by month `n` is given by `accpr(n)` (detailed here)

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

e.g. after 20 years (`n = 240`) the full balance has been paid down.

``````accpr(20*12) = 100000
``````

The amount paid out after 5 years is

``````paidforsofar = 45000 + accpr(5*12) = 66760.78
``````

The value of this amount after appreciation

``````value = paidforsofar (1 + appreciation)^5 = 73709.30
``````

The ROI for 5 years

``````roi = value/paidout = 1.22849

annualroi = roi^(1/5) - 1 = 0.0420157
``````

Now for 20 years

``````paidout deposit = 60000
paidforsofar = 45000 + accpr(20*12)          = 145000
value = paidforsofar (1 + appreciation)^20   = 215462.37
roi = value/paidout        = 3.59105
annualroi = roi^(1/20) - 1 = 0.0660094
``````

And 40 years, but this includes monthly rent payments added to the appreciated house value.

``````paidout = deposit = 60000
paidforsofar = 45000 + accpr(20*12)          = 145000
value = paidforsofar (1 + appreciation)^40   = 320165.75
roi = (value + 20*12*monthlypayment)/paidout = 7.31923
annualroi = roi^(1/40) - 1 = 0.0510216
``````

Plotting over 50 years

Interestingly there is a sweet spot after 10 years for the annualised ROI.

This calculation doesn't include the effects of inflation, which would not be negligible. Rental income would also increase of course, and would elevate the ROI after the term of the mortgage.

• that's a strikingly different answer and, assuming this more rigorous derivation is closer to reality, it raises serious doubts. Thank you for taking the time. I will have to work through this on my own and give it a hard long thought. Thank you! – Amos Joshua Oct 19 '19 at 8:02
• I've gone through the math and I see a few things I would calculate differently. The first is that it appears that you counted the fees twice: the initial cash sum of 60k splits into 45k for the house and 15k for the fees, so the amount spent at day 0 is 60k not 75k as implied in the calculation of `paidout = deposit + fees`. – Amos Joshua Oct 19 '19 at 17:23
• The second is that you count the mortgage payments as part of the investment, whereas it is intended that the rent cover the mortgage payments. It’s optimistic to assume the rent cover all mortgage payments, so let’s say (conservatively) that the rent covers 80% of mortgage payments, then we can calculate paidout as `60k + 0.2*monthlypayment*5*12 =~ 66k`. Already the roi after 5 years becomes positive. – Amos Joshua Oct 19 '19 at 17:24
• Finally, and I just might have this wrong, but you calculate that after 5 years: value = paidforsofar (1 + appreciation)^5 = 73709.30. This seems to me to assume that, if I sell the house, I only get the fraction from the sale for what I’ve paid for so far. Is that how it works? Because to me the value of the investment after 5 years is the sell-value at that time (145k*(1.02)^5 ~=160k) minus whatever I still owe the bank (~78k if I’m using your formula correctly), therefore ~82k. So the roi after 5 years is 82k/66k ~1.24 and the annual roi is (1.24)^(1/5) =~ 1.04 so 4%, which is not bad. – Amos Joshua Oct 19 '19 at 17:26
• Hi, I have amended my answer in line with your first two points. As regards the third point, I am just thinking of the proportion of the asset you have paid for, and added appreciation to that. I believe if you defaulted and the bank had to repossess they would sell the house expediently, maybe not for the best price, and there would be additional charges. I wasn't trying to factor that in though. Hope you find the calculations helpful. – Chris Degnen Oct 19 '19 at 18:46

I guess a realistic cash flow for home purchase should also include:

• Annual maintenance costs
• Depreciation due to deterioration
• Inflation
• Risks associated with bad tenants