TVM solve for interest rate, revisited
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06-29-2024, 06:21 PM
(This post was last modified: 06-30-2024 12:19 AM by Albert Chan.)
Post: #56
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RE: TVM solve for interest rate, revisited
(06-26-2024 02:31 PM)Albert Chan Wrote: let Q = 1/q ≈ n*log(2) → (1+i)^Q ≈ (1+i0)^Q - 1 Previous post showed why ln(2) is there, because RHS is (s-1) If RHS is (s-2), then ln(3) pops out ... prove is not complete. Let's try again, assuming n is huge. (i is interest rate, not √(-1)) C = i*n / (1-(1+i)^-n) ≈ i*n / (1-e^(-i*n)) = 1 + (n*i)/2 + (n*i)²/12 - ... s = (1+i0)^Q = (1+C/n)^Q ≈ (1 + 1/n + i/2 + n/12*i² - ...)^Q (1+1/n)^Q ≈ ((1+1/n)^n) ^ ln(2) ≈ e ^ ln(2) = 2 We can pull out tiny 1/n term, as correction instead (note: n ≈ Q/ln(2)) RHS = s-1 ≈ 2 * (1 + i/2 + n/12*i² - ...)^Q - 1 ≈ 1 + Q*i + 0.490*(Q*i)² - ... LHS = (1+i)^Q = 1 + Q*i + Q*(Q-1)/2*i² + ... ≈ 1 + Q*i + 0.500*(Q*i)² + ... QED With Q=n*ln(2), RHS is still smaller. This explained Q = (n+0.5)*ln(2) is better. |
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