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Understanding Present and Future Value Formulas with Examples

This resource breaks down the calculations for present and future values in finance, focusing on the formulas used to determine loan payments, account balances, and investment returns. It clearly defines key variables such as payment (P), interest rate (i), number of periods (n), and loan amount (A). Utilizing practical examples, including a scenario with a $46,075 investment and 4% interest rate, it demonstrates how to calculate monthly payments over two years. Tailored for students and finance enthusiasts, this guide simplifies complex financial concepts for better understanding.

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Understanding Present and Future Value Formulas with Examples

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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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