TVM solve for interest rate, revisited

[Albert Chan]

(04-11-2022 03:11 PM)Albert Chan Wrote:  This may be how I₀ = 1/P - P/N² comes from

Update: Perhaps formula designed also to match asymptote, when C is big.
When compounding factor C is big, literally all payments goes to paying interest.

C = N/P = I*N/(1-(1+I)^-N) ≈ N*I      → I = 1/P

Rate formula matched this behavior: I₀ = 1/P - P/N^2 = (1 - 1/C^2)/P ≈ 1/P

Another way to show rate edges, without complexity of doing tanh transformation (*)
Above quote assumed FV=0. But, we can fix it ...

I = loan_rate(N, PV, PMT, FV) = loan_rate(N, PV+FV, PMT-FV*I, 0)

From RHS, P = (PV+FV) / -(PMT-FV*I)

I = 1/P
PV*I + FV*I = -PMT + FV*I      → I = -PMT/PV

For asymptote edge I, value of FV does not matter.

To get the other edge, we use time symmetry, travelling "backward in time"
(N,PMT) sign flipped, (PV,FV) get swapped.

I = loan_rate(N, PV, PMT, FV) = loan_rate(-N, FV, -PMT, PV)

The other asymptote edge: I = PMT/FV

--

(*) tanh version, with A=(PV+FV)/2, B=(PV-FV)/2:

I = loan_rate(N, PV, PMT, FV) = loan_rate(N, A, PMT+B*I, A)

Numerically to show both forms equivalent:

XCAS> C := I*N/(1-(1+I)^-N); // compounding factor
XCAS> NPMT := C*PV + C(N=-N)*FV + N*PMT

XCAS> solve(NPMT(N=10, PV=50, PMT=-30, FV=100) = 0, I)      → [-0.28443599888, 0.582038296883]
XCAS> A, B := (50+100)/2, (50-100)/2
XCAS> solve(NPMT(N=10, PV=A, PMT=-30+B*I, FV=A)=0, I)      → [-0.28443599888, 0.582038296883]