### Quantifying the importance of 20% downpayment

Posted:

**Sun Nov 08, 2009 11:03 am**Introduction

There are many variables affecting the decision how much downpayment to use when buying a house. Some choices are rather subjective in nature (i.e. the importance one assigns to a size of a rainy-day fund vs importance of saving money in the long term), but it should be possible to quantify some of the considerations.

For example consider function TP that maps downpayment (d) to the total amount paid throughout the life of the loan (discounted into present value). For each value of downpayment, one can calculate how much an extra dollar (x) put toward downpayment lowers the total amount paid throughout the loan:

dollar utility function: DU(d) = lim(x->0): (TP(d) - TP(d+x)) / x

It should be obvious that DU(d) > 1 for all values of d - therefore it is *always* better to use a bigger downpayment (or pay cash altogether). Nevertheless one can hypothesize that DU(18%) is significantly higher then DU(22%). In particular - above certain threshold (th), DU(th) can be lower then sacrifices one has to make to save toward bigger downpayment (i.e. canceling 401k contributions means forfeiting employer match; or investing the extra funds into another investment vehicle might mean returns higher then DU(th); similarily happiness/benefits of home ownership might outweight DU(th)).

Problem

I more or less know how to use Excel to calculate TP(d) when d>=20%. The only variables are 1) mortgage intgerest rate and 2) discount rate for calculating present value of the total amount paid throughoyut the life of the laod.

I don't know how to calculate TP(d) when d<20%. PMI is one approach (and I have no idea how to calculate PMI). Another approach is assumming a secondary, piggy-back load with a different mortgage interest rate (and I have no idea how to reasonably estimate the secondary interest rate).

Question / Request

Did anybody else try to quantify the importance of 20% downpayment? Any chance to get somebody who knows his mortgage-fu to write a blog article on this subject?

Alternatively - could anyone share articles/literature related to the subject? How do I calculate/estimate PMI and other costs associated with <20% downpayment?

Thanks,

Lukasz

There are many variables affecting the decision how much downpayment to use when buying a house. Some choices are rather subjective in nature (i.e. the importance one assigns to a size of a rainy-day fund vs importance of saving money in the long term), but it should be possible to quantify some of the considerations.

For example consider function TP that maps downpayment (d) to the total amount paid throughout the life of the loan (discounted into present value). For each value of downpayment, one can calculate how much an extra dollar (x) put toward downpayment lowers the total amount paid throughout the loan:

dollar utility function: DU(d) = lim(x->0): (TP(d) - TP(d+x)) / x

It should be obvious that DU(d) > 1 for all values of d - therefore it is *always* better to use a bigger downpayment (or pay cash altogether). Nevertheless one can hypothesize that DU(18%) is significantly higher then DU(22%). In particular - above certain threshold (th), DU(th) can be lower then sacrifices one has to make to save toward bigger downpayment (i.e. canceling 401k contributions means forfeiting employer match; or investing the extra funds into another investment vehicle might mean returns higher then DU(th); similarily happiness/benefits of home ownership might outweight DU(th)).

Problem

I more or less know how to use Excel to calculate TP(d) when d>=20%. The only variables are 1) mortgage intgerest rate and 2) discount rate for calculating present value of the total amount paid throughoyut the life of the laod.

I don't know how to calculate TP(d) when d<20%. PMI is one approach (and I have no idea how to calculate PMI). Another approach is assumming a secondary, piggy-back load with a different mortgage interest rate (and I have no idea how to reasonably estimate the secondary interest rate).

Question / Request

Did anybody else try to quantify the importance of 20% downpayment? Any chance to get somebody who knows his mortgage-fu to write a blog article on this subject?

Alternatively - could anyone share articles/literature related to the subject? How do I calculate/estimate PMI and other costs associated with <20% downpayment?

Thanks,

Lukasz