Quickly Estimate Leverage of Conventional Mortgage

Here, we introduce the conclusion first. Compared to cash investment, one can leverage it using debt. With a down payment percentage = D and the interest rate = APR, the annualized multi-year profit can be estimated as 2*(p-(1-D)*APR)/(1+D), when the fair-market percentage annual profit (or loss if negative) of the investment is p. You can also get the leverage ratio by dividing the annualized profit by p. The proof is shown below, neglecting any tax benefit. 

Let us assume the home price is $1 now, and it will become (1+p)^N dollars N years later. Since you pay D as down payment at present, the loan is (1-D) which produces potential leverage. Over the course of N years, you will have to pay back the loan principle ideally uniformly, which is equivalent to paying half of the principle (1-D)/2 on day one, and the rest (1-D)/2 at the end of N years. This way, we simplified the second part that costs us (1-D)/2 either at present or N years later. However, the part of loan you equivalently pay up front, (1-D)/2, would potentially worth more if you invested it elsewhere, making that part act just like a down payment. We can summarize this finding as follows. 

Price now = $ 1 has three parts T1~T3: 

T1 = D down 

T2 = (1-D)/2 loan just like down 

T3 = (1-D)/2 loan equally costly today or later

Price N years later = $ (1+p)^N has five parts T4~T8: 

T4 = T1*(1+p)^N = Fluctuation of original D, due to market and inflation 

T5 = T2*(1+p)^N = Fluctuation of (1-D)/2 just like down, due to market and inflation 

T6 = Leverage of investment gain due to market and inflation 

T7 = (1-D)*((1+APR)^N-1) = Bank interests to pay in addition to principle

T8 = (1-D)/2 loan equally costly today or later

Since T4+T5+T6+T7+T8=(1+p)^N, we can solve for T6 and get our investment leverage as T6/T3. Here, we simplify our calculation again by approximating compount interest (1+p)^N with single interest (1+p*N). 

Leverage ratio = T6/T3 ≈ (1+p*N-1/2+D/2-APR*N-(1+D)/2+APR*D*N) / (p*N*(1+D)/2) = 2*(p-(1-D)*APR)/(1+D)/p

Below are some example numbers you can quickly refer to when applying this equation. 

p	D	APR	==> 	p*leverage
7%	20%	4%	==> 	6.3%
7%	20%	2%	==> 	9.0%
15%	20%	2%	==> 	22.3%
p	0%	0%	==> 	2*p (any p)