Chapter 4: Time Value of Money Objective Explain the concept of compounding and discounting and to provide examples of real life applications Copyright, 2000 Prentice Hall ©Author Nick Bagley, bdellaSoft, Inc Value of Investing $1 – Continuing in this manner you will find that the following amounts will be earnt: Year $1.1 Years $1.21 Years $1.331 Years $1.4641 Value of $5 Invested • More generally, with an investment of $5 at 10% we obtain Year $5*(1+0.10) $5.5 years $5.5*(1+0.10) $6.05 years $6.05*(1+0.10) $6.655 Years $6.655*(1+0.10) $7.3205 Future Value of a Lump Sum FV = PV * (1 + i ) n FV with growths from -6% to +6% Future Value of $1000 3,500 6% 3,000 2,500 4% 2,000 1,500 2% 1,000 0% -2% -4% -6% 500 0 10 Years 12 14 16 18 20 Example: Future Value of a Lump Sum • Your bank offers a CD with an interest rate of 3% for a year investments • You wish to invest $1,500 for years, how much will your investment be worth? FV = PV * (1 + i ) n = $1500 * (1 + 0.03) = $1738.1111145 n i PV FV Result 3% 1,500 ? 1738.911111 Present Value of a Lump Sum FV = PV * (1 + i ) n Divide both sides by (1 + i ) to obtain : FV −n PV = = FV * (1 + i ) n (1 + i ) n Example: Present Value of a Lump Sum • You have been offered $40,000 for your printing business, payable in years Given the risk, you require a return of 8% What is the present value of the offer? FV PV = (1 + i ) n 40,000 = (1 + 0.08) = 34293.55281 ≅ $34,293.55 today Solving Lump Sum Cash Flow for Interest Rate FV = PV * (1 + i ) n FV = (1 + i ) n PV FV n (1 + i ) = PV FV n i= −1 PV Example: Interest Rate on a Lump Sum Investment • If you invest $15,000 for ten years, you FV n i = −1 receive $30,000 PV What is your annual 30000 10 = − = 10 − = − return? 15000 10 = 0.071773463 = 7.18% (to the nearest basis point) Review of Logarithms • The basic properties of logarithms that are used by finance are: e = x, x > ln( x ) ln(e ) = x ln( x * y ) = ln( x) + ln( y ) x ln( x ) = y ln( x) y 10 FV Annuity Formula: Payment ( ) pmt n FV = * (1 + i ) − i FV * i pmt = n (1 + i ) − ( 27 ) FV Annuity Formula: Number of Payments ( ) pmt n FV = * (1 + i ) − i FV * i n 1+ = (1 + i ) pmt ( ) FV * i ln (1 + i ) = n * ln (1 + i ) = ln1 + pmt FV * i ln1 + pmt n= ln(1 + i ) n 28 Perpetual Annuities / Perpetuities • Recall the annuity formula: PV = pmt * 1 − n i (1 + i ) • Let n -> infinity with i > 0: pmt PV = i 29 Mortgage: The payment • We will examine this problem using a financial calculator • The first quantity to determine is the amount of the loan and the points Loan =$500000 * (1 −0.1) =$450,000 Points =$500000 * (1 −0.1) * 0.03 =$13,500 30 Calculator Solution n i PV FV PMT 360 0.5% 450,000 This is the monthly repayment 31 ? Result -2,697.98 Calculator Solution n i PV FV 360 5% 450,000 300 5% ? PMT Result ? -2,697.98 -2,697.98 Outstanding @ 60 Months 32 418,745 Summary of Payments • The family has made 60 payments = $2687.98*12*5 = $161,878.64 • Their mortgage repayment = 450,000 - 418,744.61 = $31,255.39 • Interest = payments - principle reduction = 161,878.64 - 31,255.39 = $130,623.25 33 Amortization of Principal 450000.00 Outstanding Balance 400000.00 350000.00 300000.00 250000.00 200000.00 150000.00 100000.00 50000.00 0.00 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 Months 34 After Tax Cash Flow $2,900 $2,700 Month $2,500 $2,300 $2,100 $1,900 $1,700 $1,500 24 48 72 96 120 144 168 192 216 Monthly Cash Flow 35 240 264 288 312 336 360 Percent of Interest and Principal 100% 90% 80% Percent 70% % Interest 60% 50% 40% % Principal 30% 20% 10% 0% 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 Months 36 10% Aditional Payments 500000 450000 Principal Outstanding 400000 350000 300000 250000 200000 150000 100000 50000 0 24 48 72 96 120 37 144 168 192 Months 216 240 264 288 312 336 360 Time U.S.A Japan 0.01 $/¥ $10,000 10% $/$ (direct) 1,000,000¥ 3% ¥ / ¥ ? $/¥ $11,000 ¥ 38 1,030,000¥ Time U.S.A Japan 0.01 $/¥ $10,000 10% $/$ (direct) 1,000,000¥ 3% ¥/¥ 0.0108 $/¥ $11,124 $11,000 ¥ 39 1,030,000¥ Time U.S.A Japan 0.01 $/¥ $10,000 10% $/$ (direct) 1,000,000¥ 3% ¥ / ¥ 0.0106 $/¥ $10,918 ¥ $11,000 ¥ 40 1,030,000¥ Time U.S.A Japan 0.01 $/¥ $10,000 10% $/$ (direct) 1,000,000¥ 3% ¥ / ¥ 0.01068 $/¥ $11,000 ¥ $11,000 ¥ 41 1,030,000¥ [...]... ) (1 + i ) 17 Derivation of PV of Annuity Formula: Algebra 2 of 5 1 1 PV = pmt *{ + + 1 2 (1 + i ) (1 + i ) 1 1 1 ++ + } 3 n −1 n (1 + i ) (1 + i ) (1 + i ) 18 Derivation of PV of Annuity Formula: Algebra 3 of 5 1 1 PV * (1 + i ) = pmt * (1 + i ) *{ + + 1 2 (1 + i ) (1 + i ) 1 1 1 ++ + } 3 n −1 n (1 + i ) (1 + i ) (1 + i ) 19 Derivation of PV of Annuity Formula: Algebra 4 of 5 1 1 + + 0 1 (1 + i )... (1 + i ) ln (1 + i ) 12 Effective Annual Rates of an APR of 18% Annual Percentage rate 18 Frequency of Annual Compounding Effective Rate 1 18.00 18 2 18.81 18 4 19.25 18 12 19.56 18 52 19.68 18 365 19.72 13 The Frequency of Compounding • Note that as the frequency of compounding increases, so does the annual effective rate • What occurs as the frequency of compounding rises to infinity? km m ... m 14 The Frequency of Compounding m k 1 + EFF = 1 + m m 1 km m 1+ = (1 + EFF ) m ( ) k m = m * (1 + EFF ) − 1 1 m 15 The Frequency of Compounding Annual Compounding Effective Rate Frequency 12 1 Annual Percentage Rate 12.00 12 2 11.66 12 4 11.49 12 12 11.39 12 52 11.35 12 365 11.33 12 Infinity 11.33 16 Derivation of PV of Annuity Formula: Algebra 1 of 5 pmt pmt PV = + + 1 2 (1 +... ) 2 (1 + i ) n−2 (1 + i ) n−1 (1 + i ) n (1 + i ) n PV * (1 + i ) = pmt *{ 20 Derivation of PV of Annuity Formula: Algebra 5 of 5 1 1 PV * (1 + i ) = pmt * + PV − pmt 0 n (1 + i ) (1 + i ) 1 PV * (1 + i ) + PV = pmt − pmt n (1 + i ) 1 pmt *{1 − } n pmt 1 ( 1+ i) PV = = * 1 − n i i (1 + i ) 21 PV of Annuity Formula pmt *{1 − PV = 1 } n (1 + i ) i pmt 1 = * 1 − n i (1 + i )...Review of Logarithms • The following properties are easy to prove from the last ones, and are useful in finance ln( x / y ) = ln( x) − ln( y ) ln( x * y * z ) = ln( x) + ln( y ) + ln( z ) ln( x + y ) ≠ ln( x) * ln( y ) 11 Solving Lump Sum Cash Flow for Number of Periods FV = PV * (1 + i ) n FV = (1 + i ) n PV FV n ( ) ln = ln (... Formula: Number of Payments ( ) pmt −n PV = * 1 − (1 + i ) ; i (1 + i ) −n (1 + i ) − n PV * i −n = 1 − (1 + i ) pmt PV * i PV * i = 1− ; − n * ln(1 + i ) = ln1 − pmt pmt PV * i ln1 − pmt PV * i = 1− ; n=− pmt ln (1 + i ) 24 Annuity Formula: PV Annuity Due PVdue = PVreg * (1 + i ) pmt −n *{1 − (1 + i ) } * (1 + i ) i pmt 1− n = *{(1 + i ) − (1 + i ) } i = 25 Derivation of FV of Annuity... to determine is the amount of the loan and the points Loan =$500000 * (1 −0.1) =$450,000 Points =$500000 * (1 −0.1) * 0.03 =$13,500 30 Calculator Solution n i PV FV PMT 360 0.5% 450,000 0 This is the monthly repayment 31 ? Result -2,697.98 Calculator Solution n i PV FV 360 5% 450,000 0 300 5% ? PMT Result ? -2,697.98 0 -2,697.98 Outstanding @ 60 Months 32 418,745 Summary of Payments • The family has... $130,623.25 33 Amortization of Principal 450000.00 Outstanding Balance 400000.00 350000.00 300000.00 250000.00 200000.00 150000.00 100000.00 50000.00 0.00 0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 Months 34 After Tax Cash Flow $2,900 $2,700 Month $2,500 $2,300 $2,100 $1,900 $1,700 $1,500 0 24 48 72 96 120 144 168 192 216 Monthly Cash Flow 35 240 264 288 312 336 360 Percent of Interest and Principal... 1 n * 1 − * ( 1 + i ) i (1 + i ) n ( ) pmt n = * (1 + i ) − 1 i 26 FV Annuity Formula: Payment ( ) pmt n FV = * (1 + i ) − 1 i FV * i pmt = n (1 + i ) − 1 ( 27 ) FV Annuity Formula: Number of Payments ( ) pmt n FV = * (1 + i ) − 1 i FV * i n 1+ = (1 + i ) pmt ( ) FV * i ln (1 + i ) = n * ln (1 + i ) = ln1 + pmt FV * i ln1 + pmt n= ln(1 + i ) n 28 Perpetual Annuities