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Deriving the Present and Future Values Formulas

Deriving the Present and Future Values Formulas. Caroline Gallant MAT 123. Present Value Formula. Let. P = payment i = period interest rate n = number of periods or payments to pay off the loan and A = loan amount. To calculate A k = the balance after k payments, start with A 0 = A.

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Deriving the Present and Future Values Formulas

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  1. Deriving the Present and Future Values Formulas Caroline Gallant MAT 123

  2. Present Value Formula Let • P = payment • i = period interest rate • n = number of periods or payments to pay off the loan • and A = loan amount To calculate Ak = the balance after k payments, start with A0 = A

  3. A1 = A(1+i)-P • A2 = A1(1+i)-P • A2 = (A(1+i)-P)(1+i)-P • A2 = A(1+i)2-P(1+i)-P • A3 = A2(1+i)-P • A3 = (A(1+i)-P)(1+i)-P(1+i)-P • A3 = A(1+i)3-P(1+i)2-P (1+i)-P • Ak = A(1+i)k-P(1+i)(k-1)-P (1+i)(k-2)…-P(1+i)-P

  4. Factor out the P Ak = A(1+i)k-P((1+i)(k-1)+(1+i)(k-2)+…+(1+i)2+(1+i)+1) Then total (1+i)(k-1)+(1+i)(k-2)+…+(1+i)2+(1+i)+1 using the geometric series formula (with x = 1+i) We get (1+i)(k-1)+(1+i)(k-2)+…+(1+i)2+(1+i)+1 = (1-(1+i)k)/(1-(1+i)) = ((1+i)k-1)/i

  5. So • Ak = A(1+i)k-P((1+i)k-1)/i When the loan is paid off, An=0 0 = A(1+i)n-P((1+i)n-1)/i Solve for A A = P((1+i)n-1)/(i(1+i)n) PV = Pmt((1-(1+i)-n))/i

  6. Example using the Present Value Formula • Susan and Bob have $46,075 to put into an account for a trip. How much can they get per month over the next two years with 4% interest? • i = 4/12% = .0033% • n = 24 • A = 46,075

  7. A0 = 46,075 • A1 = 46,075(1+0.0033)-P • A2 = 46,075(1+0.0033)2-P(1+0.0033)-P • A3 = 46,075(1+0.0033(1+i)3-P(1+0.0033)2-P (1+i)-P • A = P ((1-1.0033-24)/0.0033) • A = $2,000 per month

  8. Future Value Formula Again let • P = payment • i = period interest rate • n = number of periods or payments to pay off the loan • and A = loan amount To calculate An = the amount in the account after making n payments, start with A1 = P

  9. A1 = P • A2 = P(1+i)+P • A3 = A2(1+i)+P • A3 =P(1+i)+P((1+i)+P) • An = P(1+i)(n-1)+P(1+i)(n-2)+…+P(1+i)+P Factor out P • An = P(1+i)(n-1)+P((1+i)(n-2)(1+i)n-3)+…+(1+i)+1

  10. Then total • (1+i)n-1+(1+i)n-2+…+(1+i)+1 using the geometric series formula (with x – 1+i) • ((1+i)n-1)/i We have • An = A(1+i)n-1-P((1+i)n-2-1)/i • FV = Pmt(((1+i)n-1)/i)

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