Reverse mortgage math?

[curiousgeorge408]

I am curious about the math of reverse mortgages. I have done a
google search and looked at some sites, but I cannot find anything
that makes it clear to me.

Disclaimer: I am __not__ considering a reverse mortgage. So I do not
need advice. This is really just an academic exercise for me.

From what I can tell, it appears that there is a lot of smoke and
mirrors that lenders employ to determine the parameters of a reverse
mortgage. Let's not go there.

Assume that the lender has determined that the home equity is
$500,000, the applicable life expectancy(s) is 30 years, and the
lender is willing to pay interest at 5%.

Would the nominal monthly annuity (paid by the lender to me) be
simply:

=pmt(5%/12, 30*12, -500000)

And if I dispose of the property (or die) in 15 years, would the
equity to be paid to the lender be:

=500000 - fv(5%/12, 15*12, pmt, -500000)

In any case, how is the lump sum payment (to me) from the lender
determined?

I presume it's the PV of something. But what?

I have a lot of other questions that go beyond the math. For example,
it's not clear to me why a lender would offer a reverse mortgage. How
is the lender making money in the meantime before the sale of the
property?

Presumably, that "cost of capital" is factored into how the lender
chooses that the interest rate that the lender offers. But even that
math is less clear to me for reverse mortgages than for so-called
"forward" mortgages. I stumbled across an excellent book once that
explained all this. But, sigh, I don't remember the book title.

Anyway, these questions make me suspect that I have the wrong model
for the reverse mortgage, in the first place.

Any help with understanding the mechanics of reverse mortages from
both sides would be appreciated.

 

[Fred Smith]

If the lender feels you have $500,000 of collateral, and is happy with a 5%
return, and your life expectancy is 30 years, then, yes, your monthly
payment would be =pmt(5%/12,30*12,-500000) = 2684.11

At the end of 15 years, the lender simply wants its money back. You
calculate this with:
=fv(5%/12,15*12,-2684.11) =717,432.

how is the lump sum payment (to me) from the lender determined?

Differenct lenders have different rules, but typically, they limit the loan
to 40% of the value of the house. That's because they are taking the risk
that the eventual house sale might not cover the repayment of the loan.

it's not clear to me why a lender would offer a reverse mortgage

Because they like the rate of return. Typical interest rates for reverse
mortgages are 2 to 5 percentage points above convential mortgages. So
there's lots of room for the lender to take the risk on the eventual value
of the house. Of course, that was before the current real estate value
meltdown. I don't expect they'll be as willing to do these deals going
forward.

How is the lender making money in the meantime before the sale of the
property?

They accrue the interest income. It's no different from any other loan with
a balloon payment.

Regards,
Fred.

 

[curiousgeorge408]

If the lender feels you have $500,000 of collateral, and is happy with a 5%
return, and your life expectancy is 30 years, then, yes, your monthly
payment would be =pmt(5%/12,30*12,-500000) = 2684.11

At the end of 15 years, the lender simply wants its money back.
You calculate this with:
=fv(5%/12,15*12,-2684.11) =717,432

Thanks. That makes good sense. I see now that I misread an example
on a web page ( http://www.goldengateway.com/reverse/math.do ),
which contributed to my misunderstanding.

Using the more realistic example there: suppose we borrow $50,000
over 10 years at 6%. The web page states: "If you borrow a total of
$50,000 over ten years at an interest rate of 6%, you will owe $40,969
if you take all of the money up front, but only $18,283 if you take
the money in monthly installments".

Using your formula, in the lump sum case, the borrower must pay FV(6%/
12,10*12,0,-50000). That is about $90,969. Note that we get $40,969
in interest by subtracting the principal ($50,000).

In the monthly payment case, the web page is missing details. Based
on your formula, I would conclude that the monthly payment must be
about $416.67. Then, FV(6%/12,10*12,-416.67) is about $68,283.
Subtracting the principal, that's about $18,283 in interest.

Presumably, the monthly payment is based on about 184 months life
expectancy (15 1/3 yr). Again, using your formula, PMT(6%/
12,184,-50000) is about $416.27.

Differenct lenders have different rules, but typically, they limit the loan
to 40% of the value of the house.

That, too, makes sense. I know that I saw that 40% limit somewhere,
for US HUD/FHA-backed loans at least. There is also a limit of about
$417,000 -- at least pre-bail-out .

But for US HUD/FHA-backed, the limit seems to also be determined by
life expectancy. According to the aforementioned web page: "The HUD/
FHA amortization table basically subtracts your current age [at least
62] from 100 years, and divides the maximum loan amount by your
expected life span. The other reverse lenders also factor your age in
the same way, although each one has a slightly different forecast of
your expected life span".

I cannot find the HUD/FHA amortization table online. And in any case,
I do not understand "divides the maximum loan amount by your expected
life space". I don't believe it gibes with the example numbers below,
not literally, especially if life expectancy is determined simply by
"subtracts your current age from 100 years" (surprise!).

The example numbers are based on "an appraised home value" of $350,000
(really the appraised value less other factors). In that case, the
web page says you can borrow the following amounts at the various
ages:

Age 70: can borrow $197,000
Age 80: can borrow $236,000
Age 90: can borrow $237,000

Do you have any insight into how those numbers were derived?

I know that the derivation can be made difficult (impossible!) to
predict if lender costs are factored in. I don't know if they are.

I know there are numerous online reverse-mortgage calculators, some of
which provide this number. I'm just curious about the computation
behind them.

They accrue the interest income. It's no different from any other
loan with a balloon payment.

I beg to differ, and this was my point. With normal mortgage, the
interest and principal is partially repaid over time. The lender has
a periodic revenue stream. With a reverse mortgage, the lender does
not receive any money until the end of the loan term or when the
borrower dies (or the house is sold). Yes, the amount received
accounts for the time value of money, determined by the lender. But
there is no __periodic__ revenue stream.

Well, perhaps the answer to my question is simply: the lender must
rely on other sources of income in the meantime. And arguably, if a
lender carries enough reverse mortgages, there is some statistical
probability of a periodic revenue stream as borrowers die (or sell
their house). Hmm, I wonder how many junk food producers are in the
reverse mortgage business .

Anyway, I didn't intend to debate the issue. I only asked because the
question made me doubt my understanding of the reverse mortgage model.

Thanks again for clarifying the reverse mortgage math.

 

[Fred Smith]

Age 70: can borrow $197,000

Age 80: can borrow $236,000
Age 90: can borrow $237,000

Do you have any insight into how those numbers were derived?

The lender would start with their normal coverage ratio. Let's say it's 75%
(ie, they are willing to lend 75% of the asset's value). They would then
take the present value of this amount for the client's life expectancy to
determine how much they are willing to loan. The complicating fact is that,
up until now, lenders typically assume some appreciation of the asset (ie,
house prices go up, on average, by the rate of inflation). Given the recent
real estate market meltdown, this may change.

How is the lender making money in the meantime before the sale of the
property?

Perhaps a better example on my part would have been a zero-coupon bond.
There are lots of people (and instituions) who are happy to invest money
today with no income stream for a future value down the road. A good example
would be pension plans. Reverse mortgages work well for them because they
need their money down the road.

No debate intended. Your insight into the product is better than most.

Regards,
Fred.

If the lender feels you have $500,000 of collateral, and is happy with a
5%
return, and your life expectancy is 30 years, then, yes, your monthly
payment would be =pmt(5%/12,30*12,-500000) = 2684.11

At the end of 15 years, the lender simply wants its money back.
You calculate this with:
=fv(5%/12,15*12,-2684.11) =717,432

Thanks. That makes good sense. I see now that I misread an example
on a web page ( http://www.goldengateway.com/reverse/math.do ),
which contributed to my misunderstanding.

Using the more realistic example there: suppose we borrow $50,000
over 10 years at 6%. The web page states: "If you borrow a total of
$50,000 over ten years at an interest rate of 6%, you will owe $40,969
if you take all of the money up front, but only $18,283 if you take
the money in monthly installments".

Using your formula, in the lump sum case, the borrower must pay FV(6%/
12,10*12,0,-50000). That is about $90,969. Note that we get $40,969
in interest by subtracting the principal ($50,000).

In the monthly payment case, the web page is missing details. Based
on your formula, I would conclude that the monthly payment must be
about $416.67. Then, FV(6%/12,10*12,-416.67) is about $68,283.
Subtracting the principal, that's about $18,283 in interest.

Presumably, the monthly payment is based on about 184 months life
expectancy (15 1/3 yr). Again, using your formula, PMT(6%/
12,184,-50000) is about $416.27.

Differenct lenders have different rules, but typically, they limit the
loan
to 40% of the value of the house.

That, too, makes sense. I know that I saw that 40% limit somewhere,
for US HUD/FHA-backed loans at least. There is also a limit of about
$417,000 -- at least pre-bail-out .

But for US HUD/FHA-backed, the limit seems to also be determined by
life expectancy. According to the aforementioned web page: "The HUD/
FHA amortization table basically subtracts your current age [at least
62] from 100 years, and divides the maximum loan amount by your
expected life span. The other reverse lenders also factor your age in
the same way, although each one has a slightly different forecast of
your expected life span".

I cannot find the HUD/FHA amortization table online. And in any case,
I do not understand "divides the maximum loan amount by your expected
life space". I don't believe it gibes with the example numbers below,
not literally, especially if life expectancy is determined simply by
"subtracts your current age from 100 years" (surprise!).

The example numbers are based on "an appraised home value" of $350,000
(really the appraised value less other factors). In that case, the
web page says you can borrow the following amounts at the various
ages:

Age 70: can borrow $197,000
Age 80: can borrow $236,000
Age 90: can borrow $237,000

Do you have any insight into how those numbers were derived?

I know that the derivation can be made difficult (impossible!) to
predict if lender costs are factored in. I don't know if they are.

I know there are numerous online reverse-mortgage calculators, some of
which provide this number. I'm just curious about the computation
behind them.

They accrue the interest income. It's no different from any other
loan with a balloon payment.

I beg to differ, and this was my point. With normal mortgage, the
interest and principal is partially repaid over time. The lender has
a periodic revenue stream. With a reverse mortgage, the lender does
not receive any money until the end of the loan term or when the
borrower dies (or the house is sold). Yes, the amount received
accounts for the time value of money, determined by the lender. But
there is no __periodic__ revenue stream.

Well, perhaps the answer to my question is simply: the lender must
rely on other sources of income in the meantime. And arguably, if a
lender carries enough reverse mortgages, there is some statistical
probability of a periodic revenue stream as borrowers die (or sell
their house). Hmm, I wonder how many junk food producers are in the
reverse mortgage business .

Anyway, I didn't intend to debate the issue. I only asked because the
question made me doubt my understanding of the reverse mortgage model.

Thanks again for clarifying the reverse mortgage math.