Buy-Down

[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.