(11C) TVM for HP-11C

[Albert Chan]

For getting interest rate for the general case, npmt() is very nice formula.

TVM equation, for "net payment", NPMT = NFV * i/(K-1):

Code:

require 'mathx'
expm1, log1p = mathx.expm1, mathx.log1p

function npmt(n,i,pv,pmt,fv)        -- npmt = (pv*K+fv) * i/(K-1) + pmt
    return ((fv+pv) / expm1(log1p(i)*n) + pv)*i + pmt
end

lua> n, pv, pmt, fv = 36, 30000, -550, -15000
lua> i1 = guess_i(n, pv, pmt, fv)
lua> p1 = npmt(n, i1, pv, pmt, fv)
lua> i1, p1
0.005804795434882038      -0.006442911074373114
lua> i2 = 1.001*i1                    -- p1<0, i1 is a bit low
lua> p2 = npmt(n, i2, pv, pmt, fv)
lua> i2, p2
0.005810600230316919      0.12838859778003098
lua> i1 - p1/(p2-p1) * (i2-i1)     -- secant's method
0.005805072816490459

For comparison, with same i1, i2, and secant's method, for other TVM formula:

0.005805072843740277           -- NPV formula
0.005805072788863714           -- NFV formula

Solve npmt()=0, using Newton's method, we have: (guess_i code here)

Code:

function loan_rate(n, pv, pmt, fv, i)
    i = i or guess_i(n, pv, pmt, fv)
    return function()
        local x = expm1(log1p(i)*n)
        local y = (fv+pv)/x
        local z = (1+1/x)*i/(1+i)*n - 1
        i = i - ((pv+y)*i+pmt) / (pv-y*z)   -- Newton's method
        return i
    end
end

lua> g = loan_rate(n, pv, pmt, fv)
lua> g(), g()
0.005805072819567153      0.005805072819420132

lua> g = loan_rate(-n, fv, -pmt, pv) -- time symmetric, as expected
lua> g(), g()
0.005805072819567159      0.005805072819420132

Update: numbers adjusted with updated guess_i()