Buy-Down

[SlideRule]

Given: " Assume that interest rates are high and that you are purchasing a $150,000 townhouse with $50,000 down. The balance, $100,000, is to be financed by a first trust deed loan for 30 years at 15 percent. The builder agrees to buy down the loan to 12 percent for 3 years. "

Needed: " Payments by the buyer to the lender will drop from $¤¤¤¤.¢¢ to $¤¤¤¤.¢¢. "

Answer: to follow (with equation ).

ENJOY.

BEST!
SlideRule

hint: thread subject

[EdS2]

You're looking for the total payments over the lifetime of the loan, or the monthly repayments for the first 3 years?

[SlideRule]

" buy-downs play an important role in financing of homes, particularly when interest rates are high. The expression buy-down, simply stated, means that the seller will buy down the relatively high interest rate that the lender demands to a lower rate. This is usually accomplished by the seller paying a lump sum to the lender requesting the discounted value of the interest rate for the term agreed. Generally the builder will buy down the loan for only a relatively short term, varying from one to seven years. "
ENJOY.

BEST!
SlideRule

ps: an equation would be helpful

[Albert Chan]

Note: based on post#8, interest is compounded monthly. Redo calculations.

Without builder's help: n = 30*12, i = 15%/12, pv = 100e3, fv = 0      --> pmt = -1264.44

With builder's help effective rate ≈ (12*3 + 15*27)/30 % = 14.7%
But, builder lowered rate is applied to *beginning* of loan, let's say rate = 14%

i = 14%/12      --> pmt = -1184.87

To get exact pmt, we break-up 30 years = 3 + 27, and solve for f = 0:

NFV(n,i,pv,pmt,fv) = pv*k + pmt*(k-1)/i + fv, where k = (1+i)^n

f = NFV(27*12, 0.15/12, NFV(3*12, 0.12/12, 100e3, pmt, 0), pmt, 0)

f(pmt = -1200) = −162343.41
f(pmt = -1100) = +518573.43

Interpolate for f=0, we have pmt = -1176.16

With builder's help, monthly payment $1264.44 → $1176.16, for 30 years

Or, equivalently, interest 15.000 % → 13.900 %
Or, equivalently, loan $100,000.00 → $93,017.96

---

f = 0, with NFV formula applied twice.

(pv*k1 + pmt*(k1-1)/i1 + 0) * k2 + pmt*(k2-1)/i2 + 0 = 0

Solve for pmt, we have:

lua> i1, n1, i2, n2 = 0.12/12, 3*12, 0.15/12, 27*12
lua> k1, k2 = (1+i1)^n1, (1+i2)^n2
lua> pv = 100e3
lua> fv = k1*k2*(-pv) -- if pmt=0
lua> pmt = fv / (k2*(k1-1)/i1+(k2-1)/i2)
lua> pmt
-1176.1581151520215

[Albert Chan]

(10-26-2023 03:48 PM)Albert Chan Wrote:  Note: based on post#8, interest is compounded monthly. Redo calculations.

Without builder's help: n = 30*12, i = 15%/12, pv = 100e3, fv = 0      --> pmt = -1264.44

I might have mis-understood how buy-down work. It might be much simpler.
Buyer pay above annual payment, but the builder helped out for a while.

First 3 years, without builder's help:
n=3*12, i=15%/12, pv=100e3, pmt=-1264.44      --> fv = -99348.53

First 3 years, buyer subsidized rate = 12%
n=3*12, i=12%/12, pv=100e3, fv=-99348.53      --> pmt = -1015.12

Builder fill-in the difference: 1264.44 - 1015.12 = 249.32

Loan rate were not really lowered (bank still get 15%), just seems so.

Simplest is lower loan amount! Why do they make this complicated?
Is it intentional, to get more fees ?

[Maximilian Hohmann]

Hello!

(10-26-2023 04:55 PM)Albert Chan Wrote:  Simplest is lower loan amount! Why do they make this complicated?
Is it intentional, to get more fees ?

As I understand it, it allows the bank to give a loan to someone who can't otherwise afford the full monthly payments by paying part of the interest in advance. Lowering the loan by the same amount as this pre-payment would result in a much lower reduction of the monthly payments only. Simply because that reduction is only for three years and not the full duration of the loan.
If I remember correctly, it was this kind of trick played by American banks (i.e. finding ways to give loans for a house to everybody and his dog no matter if they can afford it or not) that caused the global financial crisis of 2008.

Regards
Max

[SlideRule]

" Many homebuyers today are losing sleep over steeply rising interest rates, which have more or less doubled over the past year and reached their highest level in two decades. Now hovering in the __% range, these higher rates are adding hundreds, or even thousands, to the monthly housing costs of new buyers. Yet buyers who feel trapped into paying high interest should know that there is a way to lower that rate with a mortgage rate buy-down.
  True to its name, a mortgage rate buy-down is where money is paid upfront to buy down the interest rate on the loan for a certain period of time. This, in turn, can reduce the buyer’s monthly mortgage payments—at least temporarily—so the buyer can ease into the housing costs. Best of all? Buyers don’t pay for buy-downs. Rather, home sellers, builders, and sometimes even lenders front the costs in order to entice cash-strapped buyers to the closing table.
  While homebuyers might be happy to hear that there’s a free end run around high rates, they might also wonder: Is there a … guide to mortgage rate buy-downs, the pros and cons, and how to decide if getting one is right for you. "

  There is an equation to facilitate such a decision; to follow.

BEST!
SlideRule

source: to follow

[SlideRule]

" Payments by the buyer to the lender will drop from $1264.45 to $1028.62. The $235.83 difference is not simply multiplied by 36 months, or $8490. to arrive at the total buy-down amount. This is because of the present value of money concept; that is, money paid in a lump sum today will be put to productive use.
   Computation:   The formula that we employ to arrive at the present value of the buy-down is very similar to one used by investors who buy trust deed loans at discounts.

   Y = PR + B / n
             P - B
where
    Y = yield,
    P = principal loan amount,
    R = rate of interest on buy-down loan,
    B = buy-down amount,
    n = number of years of buy-down.

    Thus, since the seller agreed to pay down the 15 percent market rate on the $100,000 loan to 12 percent for 3 years, the values are substituted for the letters and computed as follows:

 .15 = $100,000(0.12) + B / 3
                $100,000 - B
 …

    B = $9,000
            1.45
    B = $6,206.90

    Thus we see that it would take a lump sum of $6206.90 to buy down the 15 percent rate by 3 points.
 …
    In effect, the seller in this case is paying 6.2 points on the buyer's behalf in order to obtain the loan at only 12 percent for the first three years. The number of points is calculated by dividing the dollar charge, $6,207, by the loan amount $100,000. Thus, $6,207 / $100,000 = 6.2 points, rounded to the nearest tenth of a percent.
    Mathematics is, at best, time-consuming albeit challenging in the computation of solutions to problems. Figure 13-8 describes how financial calculators can help in speeding up calculations. "

sources: California Real Estate Finance 6e, Prentice Hall Series in California Real Estate, pages 353-355

            Want To Lower Your Mortgage Interest Rate? You Can Actually Buy It Down, Lisa Marie Conklin Jan 10, 2023

BEST!
SlideRule

[Albert Chan]

(10-26-2023 12:46 AM)SlideRule Wrote:  The builder agrees to buy down the loan to 12 percent for 3 years.

Problem is this phrase has many interpretations!
Cost to builder depends heavily on loan balance distribution.

My 1st post assumed buyer payments are all the same.
--> pmt lowered, from 1264.44 to 1176.16, for 30 years
--> builder paid for buy-downs = $6982.04

My 2nd post assumed buyer payments jumped after 3 years.
--> pmt = 1015.12 for first 3 years, 1264.44 for 27 years.
--> builder paid for buy-downs = $7192.20

(10-27-2023 12:55 PM)SlideRule Wrote:  Payments by the buyer to the lender will drop from $1264.45 to $1028.62.

Above quoted number, $1028.62, is based on something else entirely.
Payments assumed 12% rate for 30 years! Very sneaky!

lua> tvm(30*12, .12/12, 100e3, nil, 0)
-1028.6125969255042

lua> tvm(3*12, .15/12, nil, (1264.44 - 1028.61), 0)
-6803.051065027908

--> builder paid for buy-downs = $6803.05

Finanical people pick the interpretation with least cost for them! What a surprise!
This is why I ask for *walk out* price. (I don't care who paid what ...)

[SlideRule]

THANKS ALL for your analysis/interpretations and attendant mathematics. I hope the post was edifying as well as enjoyable.

BEST!
SlideRule

ps: anyone care to write a 12C program as the 17B & 19B can use the equation as-is with solver.

[Albert Chan]

(10-27-2023 12:55 PM)SlideRule Wrote:  " Payments by the buyer to the lender will drop from $1264.45 to $1028.62. The $235.83 difference is not simply multiplied by 36 months, or $8490. to arrive at the total buy-down amount. This is because of the present value of money concept; that is, money paid in a lump sum today will be put to productive use.

(10-16-2020 04:02 PM)Albert Chan Wrote:  How to estimate car payments ?

XCas> C := I*N / (1 - (1+I)^-N)       // C = |N*PMT/PV|, "compounding factor"
XCas> series(C,I,polynom)

\(1
+\frac{I(N+1)}{2}
+\frac{I^2 (N^2-1)}{12}
+\frac{I^3 (-N^2+1)}{24}
+\frac{I^4 (-N^4+20N^2-19)}{720}
+\frac{I^5 (N^4-10N^2+9)}{480}\)

For car payments, N is usually not too big, we can use: \(\large C ≈ {(IN + 3)^2 + 3\over 12}\)

This estimate does not require converting annual interest rate to monthly rate.

Using C approximation, we can get very good buy-down estimate. (IN = I*N)

C ≈ ((IN+3)^2 + 3)/12 = 1 + (IN/2) * (1 + (IN/2)/3)

IN = 0.15 * 3 = 0.45
C ≈ 1 + 0.225 * (1 + 0.075) ≈ 1.242

Buy-down = N*PMT/C ≈ $8490. / 1.242 = $6836.

(10-17-2020 11:27 AM)Albert Chan Wrote:  C-1 ≈ IN*(6+IN)/12 + I/2

Monthly rate, I = 15%/12 is very high. We may need +I/2 correction.

C ≈ 1 + 0.225 * (1 + 0.075) + (0.15/12)/2 ≈ 1.248

Compare this with exact calculations:

C = I*N / (1-(1+I)^-N) = 0.45 / (1-(1+.15/12)^-(3*12)) ≈ 1.248

Buy-down = N*PMT/C ≈ $8490. / 1.248 = $6803. (matched my previous post)

[Albert Chan]

(10-27-2023 12:55 PM)SlideRule Wrote:     Computation:   The formula that we employ to arrive at the present value of the buy-down is very similar to one used by investors who buy trust deed loans at discounts.

   Y = PR + B / n
             P - B
where
    Y = yield,
    P = principal loan amount,
    R = rate of interest on buy-down loan,
    B = buy-down amount,
    n = number of years of buy-down.

We introduce 2 more variables, p = payments per year, N = loan (years).
Based from above quoted formula, solve for B

lua> P, p = 100e3, 12
lua> Y, N = 0.15, 30
lua> R, n = 0.12, 3
lua> B = n*P*(Y-R) / (1+Y*n)
lua> B
6206.896551724138

The problem is simple formula does not take into account of loan terms.
We don't know buy-down number is over-estimated or under-estimated.

Here is the exact formula, based from C(I,N) = I*N / (1-(1+I/p)^-(N*p))

lua> C = fn'I,N: I*N / -expm1(-N*p*log1p(I/p))' -- more accurate version
lua> B = n*P/C(Y,n) * (C(Y,N)-C(R,N))/N
lua> B
6803.092161985603

Exact calculations is expensive, however, requiring 3 TVM calculations.
For B estimate, we may assume N = ∞, which turns out same as N = 1

L = (C(Y,N)-C(R,N))/N = Y/(1-(1+Y)^-N) - R/(1-(1+R)^-N)      // assume p=1

L(N=∞) = Y/(1-0) - R/(1-0) = (Y-R)                                        // assume Y > R > 0
L(N=1)  =  (1+Y)  -  (1+R)  = (Y-R)

Quote:C ≈ ((IN+3)^2 + 3)/12 = 1 + (IN/2) * (1 + (IN/2)/3)

lua> n*P*(Y-R) / (1 + Y*n/2 * (1 + Y*n/6))
7247.106190236537
lua> n*P*(Y-R) / C(Y,n)
7211.816843730551

Formula over-shooted B(N = ∞), we may adjust down for B estimate.

lua> B = n*P*(Y-R) / exp(Y*n/2)
lua> B
7186.645968834393

Over-estimated buy-down may be better for the builder to plan things.

[dm319]

I used my 12c with:

PV:-100'000
FV:0
i: 15%/12
n: 30x12

solving for PMT: 1264.44

then adjust i:

i:12%/12, the rolldown and

solve for PMT: 1028.61

subtract: 235.83 and put this into PMT

n: 3x12
i: 15%/12

solve for PV: -6803.09

Not quite the same number so I'm not sure what's going on.

[SlideRule]

(10-28-2023 12:44 PM)dm319 Wrote:  I used my 12c with:
 …
solve for PV: -6803.09
 …
Not quite the same number so I'm not sure what's going on.

Thanks for the response.

BEST!
SlideRule