TVM solve for interest rate, revisited

[Albert Chan]

(05-13-2022 09:31 PM)Albert Chan Wrote:  Let t = tanh(n/2*log1p(x)), put Ce back to NPMT:

f = A*x/t + B*x + pmt

f = 0 → x = -pmt / (A/t + B)

We can also solve f=0, for t:

tanh(n/2*log1p(x)) = -A*x/(pmt+B*x)

There is a trivial solution, when x=0. We divide both side by x, to remove it.

XCAS> t1 := tanh(n/2*log1p(x))/x
XCAS> t2 := -A/(pmt+B*x)

We approximate t1 by pade approximation.
To keep things simple, we wanted degree(numerator)=1, degree(denominator)=2.
This allow solving for guess rate x, with simple quadratic.

XCAS> p1 := pade(t1,x,3,2)

(6*n+3*n*x) / (12+12*x+2*x^2+n^2*x^2)

Cross multiply, to get quadratic coefs.

XCAS> P := e2r(numer(p1)*denom(t2) - denom(p1)*numer(t2))

[2*A+3*B*n+A*n^2, 12*A+6*B*n+3*n*pmt, 12*A+6*n*pmt]

Let's try previous post example

XCAS> P2 := P(A=(pv+fv)/2, B=(pv-fv)/2)
XCAS> proot(P2(n=10,pv=50,pmt=-30,fv=100))

[-0.268464164628, 0.485855468976]

This is not much better than using edges for rate guesses.
The example is hard. Most cases, rate approximation is good.

Car lease examples, from Fun math algorithms

proot(P2(n=36,pv=30000,pmt=-550,fv=-15000)) .* 12

[-4.89392664913, 0.0696603112801]      // True APR = 6.96608738330 %

Just as previously done with guess_i(), we avoid using square roots.
(also, we only keep the "small" root)

Code:

function guess_i2(n, pv, pmt, fv)
    pmt, fv = pmt or -1, fv or 0
    local a = (pv+fv)/n
    local b = fv-pv-pmt-2*a
    local c = (a+pmt)/b
    b = ((n*n+2)*a+3*(pv-fv))*c/b   -- I coef = [b/c/3, -2, 4c]
    return c * (b-3)/(b-1.5)        -- pade(4c/(1+sqrt(1-4b/3)),b,2,2)
end

lua> guess_i2(10,50,-30,100) -- the "hard" example
-0.3460122699386503

lua> guess_i2(36,30000,-550,-15000) * 12 -- car lease example
0.06966051503474308