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The last part of QTM1300 at Babson College. Topic: Time Value of Money for business mathematics.

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- 1. . Time Value of Money. QTM1300: Quantitative Methods for Business Dr. Ji Li Babson College November - December, Fall 2010 . . . . . .
- 2. Simple Interest and Compound Interest Sinking Funds, Annuities, and Bonds Simple Interest More on Finance Compound Interest Notations and Formulas. Time Value of Money 1. Simple Interest and Compound Interest Simple Interest Compound Interest 2. Sinking Funds, Annuities, and Bonds Sinking Funds Annuities Amortization Schedule More Examples 3. More on Finance Sinking Funds and Annuities: New Formulas Perpetuities Net Present Value 4. Notations and Formulas Notations Simple Interest and Compound Interest Sinking Funds, Annuities, and Perpetuities Net Present Value . . . . . . Dr. Ji Li Time Value of Money
- 3. Simple Interest and Compound Interest Sinking Funds, Annuities, and Bonds Simple Interest More on Finance Compound Interest Notations and Formulas. Simple Interest An investment of PV dollars growing with simple interest rate of r after t years is worth FV dollars: FV = PV (1 + r t). . . . . . . Dr. Ji Li Time Value of Money
- 4. Simple Interest and Compound Interest Sinking Funds, Annuities, and Bonds Simple Interest More on Finance Compound Interest Notations and Formulas. Simple Interest An investment of PV dollars growing with simple interest rate of r after t years is worth FV dollars: FV = PV (1 + r t). . Example . The Megabucks Corporation is issuing 10-year bonds paying an annual rate of 6.5%. If you buy $10,000 worth of bonds, how much interest will you earn every six months? How much interest will you earn over the life of the bonds? . . . . . . . Dr. Ji Li Time Value of Money
- 5. Simple Interest and Compound Interest Sinking Funds, Annuities, and Bonds Simple Interest More on Finance Compound Interest Notations and Formulas. Simple Interest An investment of PV dollars growing with simple interest rate of r after t years is worth FV dollars: FV = PV (1 + r t). . Example . A stock fund costs $900 in July 2001 and sells for $892 in July 2002. What is the annual percentage loss of this stock? . . . . . . . Dr. Ji Li Time Value of Money
- 6. Simple Interest and Compound Interest Sinking Funds, Annuities, and Bonds Simple Interest More on Finance Compound Interest Notations and Formulas. Simple Interest An investment of PV dollars growing with simple interest rate of r after t years is worth FV dollars: FV = PV (1 + r t). . Example . You hear the following on your local radio station’s business news: The economy last year grew by 1%. This was the second year in a row in which the economy showed a 1% growth. Because the rate of growth was the same two years in a row, this represents a . simple interest growth, right? . . . . . . Dr. Ji Li Time Value of Money
- 7. Simple Interest and Compound Interest Sinking Funds, Annuities, and Bonds Simple Interest More on Finance Compound Interest Notations and Formulas. Example Suppose that $20, 000.00 is invested in a bank account. Assume that there is no other deposits or withdrawals. How much is in the account after 10 years if (a) the bank pays 6% simple interest rate once a year? (b) the bank pays 6% interest rate compounded annually? (c) the bank pays 6% interest rate compounded monthly? . . . . . . Dr. Ji Li Time Value of Money
- 8. Simple Interest and Compound Interest Sinking Funds, Annuities, and Bonds Simple Interest More on Finance Compound Interest Notations and Formulas. Example Suppose that $20, 000.00 is invested in a bank account. Assume that there is no other deposits or withdrawals. How much is in the account after 10 years if (a) the bank pays 6% simple interest rate once a year? (b) the bank pays 6% interest rate compounded annually? (c) the bank pays 6% interest rate compounded monthly? . Answer to (a) . The account earns 20, 000 × 0.06 = $1, 200 interest every year. In 10 years, the account becomes . FV = PV (1 + rt) = 20, 000(1 + 0.06 × 10) = $32, 000. . . . . . . Dr. Ji Li Time Value of Money
- 9. Simple Interest and Compound Interest Sinking Funds, Annuities, and Bonds Simple Interest More on Finance Compound Interest Notations and Formulas. Example Suppose that $20, 000.00 is invested in a bank account. Assume that there is no other deposits or withdrawals. How much is in the account after 10 years if (a) the bank pays 6% simple interest rate once a year? (b) the bank pays 6% interest rate compounded annually? (c) the bank pays 6% interest rate compounded monthly? . Answer to (b) . In 10 years, the account becomes . FV = 20, 000 (1 + 0.06)10 = $35, 816.95. . . . . . . Dr. Ji Li Time Value of Money
- 10. Simple Interest and Compound Interest Sinking Funds, Annuities, and Bonds Simple Interest More on Finance Compound Interest Notations and Formulas. Example Suppose that $20, 000.00 is invested in a bank account. Assume that there is no other deposits or withdrawals. How much is in the account after 10 years if (a) the bank pays 6% simple interest rate once a year? (b) the bank pays 6% interest rate compounded annually? (c) the bank pays 6% interest rate compounded monthly? . Answer to (c) . In 10 years, the account becomes ( )12×10 0.06 FV = 20, 000 1 + = $36, 387.93. . 12 . . . . . . Dr. Ji Li Time Value of Money
- 11. Simple Interest and Compound Interest Sinking Funds, Annuities, and Bonds Simple Interest More on Finance Compound Interest Notations and Formulas. Compound Interest An investment of PV dollars earning interest at an annual rate of r compounded (reinvested) m times per year for a period of t years is worth FV dollars: ( r )mt FV = PV 1+ . m . . . . . . Dr. Ji Li Time Value of Money
- 12. Simple Interest and Compound Interest Sinking Funds, Annuities, and Bonds Simple Interest More on Finance Compound Interest Notations and Formulas. Compound Interest An investment of PV dollars earning interest at an annual rate of r compounded (reinvested) m times per year for a period of t years is worth FV dollars: ( r )mt FV = PV 1+ . m . Example . Determine the amount of money you must invest at 5% per year, compounded monthly, so that you will be a millionaire in 30 years. . . . . . . . Dr. Ji Li Time Value of Money
- 13. Simple Interest and Compound Interest Sinking Funds, Annuities, and Bonds Simple Interest More on Finance Compound Interest Notations and Formulas. Compound Interest An investment of PV dollars earning interest at an annual rate of r compounded (reinvested) m times per year for a period of t years is worth FV dollars: ( r )mt FV = PV 1+ . m . Effective Rate (APY) . You deposit $100,000 in an account earning interest of 5% compounded annually. What is the APY (Annual Percentage Yield) of your account? . . . . . . . Dr. Ji Li Time Value of Money
- 14. Simple Interest and Compound Interest Sinking Funds, Annuities, and Bonds Simple Interest More on Finance Compound Interest Notations and Formulas. Compound Interest An investment of PV dollars earning interest at an annual rate of r compounded (reinvested) m times per year for a period of t years is worth FV dollars: ( r )mt FV = PV 1+ . m . Constant Dollars . You deposit $100,000 in an account earning interest of 5% compounded annually. What is the APY (Annual Percentage Yield) of your account? Suppose also that inﬂation is running 3% when you make the deposit. How much money will you have two years from now? . . . . . . . Dr. Ji Li Time Value of Money
- 15. Simple Interest and Compound Interest Sinking Funds, Annuities, and Bonds Simple Interest More on Finance Compound Interest Notations and Formulas. Compound Interest An investment of PV dollars earning interest at an annual rate of r compounded (reinvested) m times per year for a period of t years is worth FV dollars: ( r )mt FV = PV 1+ . m . Bonds Part I . How much do you have to pay for a 20-year zero coupon bond with maturity value of $100,000 and a yield of 5.65% annually? . . . . . . . Dr. Ji Li Time Value of Money
- 16. Simple Interest and Compound Interest Sinking Funds, Annuities, and Bonds Simple Interest More on Finance Compound Interest Notations and Formulas. Compound Interest An investment of PV dollars earning interest at an annual rate of r compounded (reinvested) m times per year for a period of t years is worth FV dollars: ( r )mt FV = PV 1+ . m . Bonds Part II . How much do you have to pay for a 20-year zero coupon bond with maturity value of $100,000 and a yield of 5.65% annually? Once purchased, bonds can be sold in the secondary market. How much money would you have received if you sold your bond 5 years before maturity to . an investor looking for a return of 5% annually? . . . . . . Dr. Ji Li Time Value of Money
- 17. Simple Interest and Compound Interest Sinking Funds, Annuities, and Bonds Simple Interest More on Finance Compound Interest Notations and Formulas. Compound Interest An investment of PV dollars earning interest at an annual rate of r compounded (reinvested) m times per year for a period of t years is worth FV dollars: ( r )mt FV = PV 1+ . m . Bonds Part III . How much do you have to pay for a 20-year zero coupon bond with maturity value of $100,000 and a yield of 5.65% annually? Once purchased, bonds can be sold in the secondary market. How much money would you have received if you sold your bond 5 years before maturity to an investor looking for a return of 5% annually? What is your annual yield on your 15-year investment? . . . . . . . Dr. Ji Li Time Value of Money
- 18. Simple Interest and Compound Interest Sinking Funds Sinking Funds, Annuities, and Bonds Annuities More on Finance Amortization Schedule Notations and Formulas More Examples. Time Value of Money 1. Simple Interest and Compound Interest Simple Interest Compound Interest 2. Sinking Funds, Annuities, and Bonds Sinking Funds Annuities Amortization Schedule More Examples 3. More on Finance Sinking Funds and Annuities: New Formulas Perpetuities Net Present Value 4. Notations and Formulas Notations Simple Interest and Compound Interest Sinking Funds, Annuities, and Perpetuities Net Present Value . . . . . . Dr. Ji Li Time Value of Money
- 19. Simple Interest and Compound Interest Sinking Funds Sinking Funds, Annuities, and Bonds Annuities More on Finance Amortization Schedule Notations and Formulas More Examples. Example Suppose you make a deposit of $1000 at the end of every month into an account earning 5% interest per year, compounded monthly. What will be the value of the investment at the end of 30 years? . . . . . . Dr. Ji Li Time Value of Money
- 20. Simple Interest and Compound Interest Sinking Funds Sinking Funds, Annuities, and Bonds Annuities More on Finance Amortization Schedule Notations and Formulas More Examples. Sinking Funds A sinking fund is worth FV dollars if you make a payment of PMT at the end of each compounding period into an account earning interest at an annual rate of r compounded (reinvested) m times per year for t years: (1 + r /m)mt − 1 FV = PMT . r /m . . . . . . Dr. Ji Li Time Value of Money
- 21. Simple Interest and Compound Interest Sinking Funds Sinking Funds, Annuities, and Bonds Annuities More on Finance Amortization Schedule Notations and Formulas More Examples. Sinking Funds A sinking fund is worth FV dollars if you make a payment of PMT at the end of each compounding period into an account earning interest at an annual rate of r compounded (reinvested) m times per year for t years: (1 + r /m)mt − 1 FV = PMT . r /m . Example . At the end of each month you deposit $100 into an account earning 3% annual rate compounded monthly. How much is the account worth in the end of one year? . . . . . . . Dr. Ji Li Time Value of Money
- 22. Simple Interest and Compound Interest Sinking Funds Sinking Funds, Annuities, and Bonds Annuities More on Finance Amortization Schedule Notations and Formulas More Examples. Sinking Funds A sinking fund is worth FV dollars if you make a payment of PMT at the end of each compounding period into an account earning interest at an annual rate of r compounded (reinvested) m times per year for t years: (1 + r /m)mt − 1 FV = PMT . r /m . Retirement Account . Your retirement account has $10,000 in it and ears 5% interest per year compounded monthly. Every month for the next 20 years you will deposit $500 into the account. How much money will be there at the end of those 20 years? . . . . . . . Dr. Ji Li Time Value of Money
- 23. Simple Interest and Compound Interest Sinking Funds Sinking Funds, Annuities, and Bonds Annuities More on Finance Amortization Schedule Notations and Formulas More Examples. Sinking Funds A sinking fund is worth FV dollars if you make a payment of PMT at the end of each compounding period into an account earning interest at an annual rate of r compounded (reinvested) m times per year for t years: (1 + r /m)mt − 1 FV = PMT . r /m . Example . If $2,000 is deposited in an account at the end of each year for the next 12 years, how much will be in the account at the time of the ﬁnal deposit if interest is 5% compounded annually? . . . . . . . Dr. Ji Li Time Value of Money
- 24. Simple Interest and Compound Interest Sinking Funds Sinking Funds, Annuities, and Bonds Annuities More on Finance Amortization Schedule Notations and Formulas More Examples. Example Suppose you deposit an amount PV now in an account earning 5% interest per year, compounded monthly. Starting 1 month from now, the bank will send you monthly payments of $5,000. What must PV be so that the account will be drawn down to $0 in exactly 10 years? . . . . . . Dr. Ji Li Time Value of Money
- 25. Simple Interest and Compound Interest Sinking Funds Sinking Funds, Annuities, and Bonds Annuities More on Finance Amortization Schedule Notations and Formulas More Examples. Annuities An annuity is an account earning compound interest from which periodic withdrawals are made. If the starting balance is PV dollars, you receive a payment of PMT at the end of each compounding period, and the account is down to $0 after for t years, then 1 − (1 + r /m)−mt PV = PMT . r /m . . . . . . Dr. Ji Li Time Value of Money
- 26. Simple Interest and Compound Interest Sinking Funds Sinking Funds, Annuities, and Bonds Annuities More on Finance Amortization Schedule Notations and Formulas More Examples. Annuities An annuity is an account earning compound interest from which periodic withdrawals are made. If the starting balance is PV dollars, you receive a payment of PMT at the end of each compounding period, and the account is down to $0 after for t years, then 1 − (1 + r /m)−mt PV = PMT . r /m . Example . At the end of each month you want to withdraw $100 from an account earning 3% annual rate compounded monthly. How much is it worth right now if you want the account to last for 5 years? . . . . . . . Dr. Ji Li Time Value of Money
- 27. Simple Interest and Compound Interest Sinking Funds Sinking Funds, Annuities, and Bonds Annuities More on Finance Amortization Schedule Notations and Formulas More Examples. Annuities An annuity is an account earning compound interest from which periodic withdrawals are made. If the starting balance is PV dollars, you receive a payment of PMT at the end of each compounding period, and the account is down to $0 after for t years, then 1 − (1 + r /m)−mt PV = PMT . r /m . Car Loan Part I . Mira bought a car worth $30,000 with an initial payment of $6,000. How much does she have to pay in the end of each month for the 5-year car loan with an interest rate of 4% compounded monthly? . . . . . . . Dr. Ji Li Time Value of Money
- 28. Simple Interest and Compound Interest Sinking Funds Sinking Funds, Annuities, and Bonds Annuities More on Finance Amortization Schedule Notations and Formulas More Examples. Annuities An annuity is an account earning compound interest from which periodic withdrawals are made. If the starting balance is PV dollars, you receive a payment of PMT at the end of each compounding period, and the account is down to $0 after for t years, then 1 − (1 + r /m)−mt PV = PMT . r /m . Car Loan Part II . Mira bought a car worth $30,000 with an initial payment of $6,000 on a 5-year car loan with an interest rate of 4% compounded monthly. After making monthly payments over 3 years, she decided to end the loan earlier. How much does she have to pay in her last payment? . . . . . . . Dr. Ji Li Time Value of Money
- 29. Simple Interest and Compound Interest Sinking Funds Sinking Funds, Annuities, and Bonds Annuities More on Finance Amortization Schedule Notations and Formulas More Examples. Amortization Schedule Mira bought a car worth $30,000 with an initial payment of $6,000 on a 5-year car loan with an interest rate of 4% compounded monthly. How much interest does she have to pay in the end of the ﬁrst month? How much outstanding principal is left after Mira makes the ﬁrst payment? How much interest does she have to pay in the end of the second month? How much interest does Mira have to pay in total? . . . . . . Dr. Ji Li Time Value of Money
- 30. Simple Interest and Compound Interest Sinking Funds Sinking Funds, Annuities, and Bonds Annuities More on Finance Amortization Schedule Notations and Formulas More Examples. Example Find the monthly payment on the mortgage if you are buying a $300,000 apartment with a down payment of $60,000 for 30 years at 9% interest rate compounded monthly. Find the total amount you will pay in interest. Produce an amortization schedule for the ﬁrst 12 payments. . . . . . . Dr. Ji Li Time Value of Money
- 31. Simple Interest and Compound Interest Sinking Funds Sinking Funds, Annuities, and Bonds Annuities More on Finance Amortization Schedule Notations and Formulas More Examples. Bonds Suppose that a corporation offers a 20-year bond paying coupon interest rate 4.5% with semiannual coupons. If someone pays $5,000 for bonds with a maturity value of $5,000, he will receive a coupon every 6 months for 20 years for the interest. At the end of the 20 years, he will get the $5,000 back. How much is each coupon worth? Think of the bonds as an investment that will pay the owner a certain amount every 6 months for 20 years, at the end of which it will pay $5,000. How much will a bond trader be willing to pay for the bond if he’s looking for a yield (also called “rate of return”) of 7%? Another trader is looking for 6% yield on her investment. How much will she pay for the same bond? . . . . . . Dr. Ji Li Time Value of Money
- 32. Simple Interest and Compound Interest Sinking Funds Sinking Funds, Annuities, and Bonds Annuities More on Finance Amortization Schedule Notations and Formulas More Examples. Multi-Step Example You wish to provide yourself with an income of $5,000 every 6 months, starting 15 and a half years from now and ending 35 years from now. You deposit $25,000 in the account now, and a guaranteed inheritance of $10,000 which you will receive 10 years from now. You know that these sums will not provide the income you want, so you plan to make periodic deposits to the account at the end of every 6 months for 15 years to make up the difference. How much should the periodic deposits be if all interest is computed at 6% compounded semiannually? . . . . . . Dr. Ji Li Time Value of Money
- 33. Simple Interest and Compound Interest Sinking Funds Sinking Funds, Annuities, and Bonds Annuities More on Finance Amortization Schedule Notations and Formulas More Examples. The Effect of Inﬂation (Turner Example 4, Finance.xlsx) An entrepreneur borrows $10,000 under the following terms: Loan interest rate: 12% Term: 6 years Payment schedule: Monthly Determine the cost of the loan in today’s dollars if inﬂation average 5% over the term of the loan. There are two steps involved to solve this problem: Step 1: Find the monthly payment PMT by ignoring the inﬂation rate. Step 2: Find the present value using the PMT found in an annuity situation. . . . . . . Dr. Ji Li Time Value of Money
- 34. Simple Interest and Compound Interest Sinking Funds and Annuities: New Formulas Sinking Funds, Annuities, and Bonds Perpetuities More on Finance Net Present Value Notations and Formulas. Time Value of Money 1. Simple Interest and Compound Interest Simple Interest Compound Interest 2. Sinking Funds, Annuities, and Bonds Sinking Funds Annuities Amortization Schedule More Examples 3. More on Finance Sinking Funds and Annuities: New Formulas Perpetuities Net Present Value 4. Notations and Formulas Notations Simple Interest and Compound Interest Sinking Funds, Annuities, and Perpetuities Net Present Value . . . . . . Dr. Ji Li Time Value of Money
- 35. Simple Interest and Compound Interest Sinking Funds and Annuities: New Formulas Sinking Funds, Annuities, and Bonds Perpetuities More on Finance Net Present Value Notations and Formulas. Example on Annuities Sally has just received a loan to ﬁnance the purchase of a pretty blue convertible. The amount of the loan is $25,000. Sally is required to transfer to the lending institution a ﬁxed amount at the end of each month starting at the end of the ﬁrst month after she receives her loan. The interest rate on the loan is 9% compounded daily, and the term of the loan is three years. What will be Sally’s monthly payment? . . . . . . Dr. Ji Li Time Value of Money
- 36. Simple Interest and Compound Interest Sinking Funds and Annuities: New Formulas Sinking Funds, Annuities, and Bonds Perpetuities More on Finance Net Present Value Notations and Formulas. Sinking Funds and Annuities — New Formulas Suppose that the number of compounding periods per year, m, is different from the number of payments per year, ppy . Then the interest rate per payment period becomes ( )m/ppy r j = 1+ −1 m and the sinking fund and annuity formulas become ( ) (1 + j)ppy·t − 1 Sinking Fund FV = PMT , j ( ) 1 − (1 + j)−ppy ·t Annuity PV = PMT , j In calculator, input the following: N = ppy · t I%=j% . . . . . . Dr. Ji Li Time Value of Money
- 37. Simple Interest and Compound Interest Sinking Funds and Annuities: New Formulas Sinking Funds, Annuities, and Bonds Perpetuities More on Finance Net Present Value Notations and Formulas. Examples on New Formulas ( ) (1 + j)ppy·t − 1 Sinking Fund FV = PMT , j ( ) 1 − (1 + j)−ppy ·t Annuity PV = PMT , j . . . . . . Dr. Ji Li Time Value of Money
- 38. Simple Interest and Compound Interest Sinking Funds and Annuities: New Formulas Sinking Funds, Annuities, and Bonds Perpetuities More on Finance Net Present Value Notations and Formulas. Examples on New Formulas ( ) (1 + j)ppy·t − 1 Sinking Fund FV = PMT , j ( ) 1 − (1 + j)−ppy ·t Annuity PV = PMT , j . Example . Mira bought a car worth $30,000 with an initial payment of $6,000. How much does she have to pay in the end of each month for the 5-year car loan with an interest rate of 4% compounded daily? . . . . . . . Dr. Ji Li Time Value of Money
- 39. Simple Interest and Compound Interest Sinking Funds and Annuities: New Formulas Sinking Funds, Annuities, and Bonds Perpetuities More on Finance Net Present Value Notations and Formulas. Examples on New Formulas ( ) (1 + j)ppy·t − 1 Sinking Fund FV = PMT , j ( ) 1 − (1 + j)−ppy ·t Annuity PV = PMT , j . Example . Find the monthly payment on the mortgage if you are buying a $300,000 apartment with a down payment of $60,000 for 30 years at 9% interest rate compounded daily. . . . . . . . Dr. Ji Li Time Value of Money
- 40. Simple Interest and Compound Interest Sinking Funds and Annuities: New Formulas Sinking Funds, Annuities, and Bonds Perpetuities More on Finance Net Present Value Notations and Formulas. Perpetuities: Example 1 (Turner Example 5) Determine the initial deposit that must be placed in an account that bears 5% interest compounded monthly in order to withdraw $1,000 every month for ever. Now in the annuity formula 1 − (1 + r /m)−mt PV = PMT , r /m by setting t −→ ∞, the perpetuity formula follows 1 PV = PMT . r /m . . . . . . Dr. Ji Li Time Value of Money
- 41. Simple Interest and Compound Interest Sinking Funds and Annuities: New Formulas Sinking Funds, Annuities, and Bonds Perpetuities More on Finance Net Present Value Notations and Formulas. Perpetuities: Example 2 Determine the initial deposit that must be placed in an account that bears 5.49% interest compounded monthly in order to withdraw $10,000 every 6 months forever, starting a month from now. ( )m/ppy ( ) r 1 − (1 + j)−ppy·t j= 1+ −1 PV = PMT m j Setting t −→ ∞, the perpetuity formula is 1 PMT PV = PMT = ( )m/ppy j r 1+ m −1 . . . . . . Dr. Ji Li Time Value of Money
- 42. Simple Interest and Compound Interest Sinking Funds and Annuities: New Formulas Sinking Funds, Annuities, and Bonds Perpetuities More on Finance Net Present Value Notations and Formulas. Perpetuities: Example 3 1 PMT PV = PMT =( )m/ppy j r 1+ m −1 . Example . Determine the initial deposit that must be placed in an account that bears 2.75% interest compounded daily in order to withdraw $8,000 quarterly forever, starting 3 months from now. . . . . . . . Dr. Ji Li Time Value of Money
- 43. Simple Interest and Compound Interest Sinking Funds and Annuities: New Formulas Sinking Funds, Annuities, and Bonds Perpetuities More on Finance Net Present Value Notations and Formulas. Issuing a Bond (Turner Example 6, Finance.xlsx) A company ﬂoats a $10,000,000 bond issue with a 20 year term. The interest rate on the bond is 3%. How much is each bond interest payment (to the bond holders) made semiannually? The company has set up a sinking fund for the accumulation and dispersion of all funds related to the bond issue and wants to make equal quarterly payments to the fund. Note that the fund will be used both to pay the bond interest due each six months and the bond face value of $10,000,000 twenty years hence. Determine the minimum quarterly payment to the fund that would meet the needs of the company if the interest on the fund is 8% compounded quarterly. . . . . . . Dr. Ji Li Time Value of Money
- 44. Simple Interest and Compound Interest Sinking Funds and Annuities: New Formulas Sinking Funds, Annuities, and Bonds Perpetuities More on Finance Net Present Value Notations and Formulas. Net Present Value Assume r is the APR and m is the number of compounding periods per year. The net present value NPV of a sequence of equally-spaced end-of-period payments (negative) and income (positive) with same compounding and payment/income periods, with an initial cash ﬂow v0 and the payment/income sequence v1 , v2 , . . . , vn is v1 v2 vn NPV = v0 + + + ··· + . 1 + r /m (1 + r /m)2 (1 + r /m)n . . . . . . Dr. Ji Li Time Value of Money
- 45. Simple Interest and Compound Interest Sinking Funds and Annuities: New Formulas Sinking Funds, Annuities, and Bonds Perpetuities More on Finance Net Present Value Notations and Formulas. Net Present Value Assume r is the APR and m is the number of compounding periods per year. The net present value NPV of a sequence of equally-spaced end-of-period payments (negative) and income (positive) with same compounding and payment/income periods, with an initial cash ﬂow v0 and the payment/income sequence v1 , v2 , . . . , vn is v1 v2 vn NPV = v0 + + + ··· + . 1 + r /m (1 + r /m)2 (1 + r /m)n . Example 1 . Suppose you are considering an investment in which you pay $10,000 one year from today and receive an annual income of $3,000, $4,200, and $6,800 at the end of the three years that follow, respectively. Assuming an annual interest rate of 10%, what is the net present value of this investment? . . . . . . . Dr. Ji Li Time Value of Money
- 46. Simple Interest and Compound Interest Sinking Funds and Annuities: New Formulas Sinking Funds, Annuities, and Bonds Perpetuities More on Finance Net Present Value Notations and Formulas. Net Present Value Assume r is the APR and m is the number of compounding periods per year. The net present value NPV of a sequence of equally-spaced end-of-period payments (negative) and income (positive) with same compounding and payment/income periods, with an initial cash ﬂow v0 and the payment/income sequence v1 , v2 , . . . , vn is v1 v2 vn NPV = v0 + + + ··· + . 1 + r /m (1 + r /m)2 (1 + r /m)n . Example 1: Continue . Suppose you are considering an investment in which you pay $10,000 one year from today and receive an annual income of $3,000, $4,200, and $6,800 at the end of the three years that follow, respectively. Determine the Internal Rate of Return, or the interest rate per period which would provide a net present value of $0, after all four years. . . . . . . . Dr. Ji Li Time Value of Money
- 47. Simple Interest and Compound Interest Sinking Funds and Annuities: New Formulas Sinking Funds, Annuities, and Bonds Perpetuities More on Finance Net Present Value Notations and Formulas. NPV: Example 2 You run a takeout pizza business and believe you could improve your return by buying a van to add delivery service. You can buy a van today for $15,000 and belive you will use it for ﬁve years and then sell it for $5,000. After expenses, you estimate that your business will make $4,000 annually by adding the delivery service, starting a year from now. You could make 7.5% by investing in a U.S. Treasure bill over ﬁve years, but you decide that the added risks of the pizza delivery business mean you should earn at least twice that rate. So you set the annual interest rate at 15%. Should you buy the van? . . . . . . Dr. Ji Li Time Value of Money
- 48. Simple Interest and Compound Interest Notations Sinking Funds, Annuities, and Bonds Simple Interest and Compound Interest More on Finance Sinking Funds, Annuities, and Perpetuities Notations and Formulas Net Present Value. Time Value of Money 1. Simple Interest and Compound Interest Simple Interest Compound Interest 2. Sinking Funds, Annuities, and Bonds Sinking Funds Annuities Amortization Schedule More Examples 3. More on Finance Sinking Funds and Annuities: New Formulas Perpetuities Net Present Value 4. Notations and Formulas Notations Simple Interest and Compound Interest Sinking Funds, Annuities, and Perpetuities Net Present Value . . . . . . Dr. Ji Li Time Value of Money
- 49. Simple Interest and Compound Interest Notations Sinking Funds, Annuities, and Bonds Simple Interest and Compound Interest More on Finance Sinking Funds, Annuities, and Perpetuities Notations and Formulas Net Present Value. Notations PV — Present Value FV— Future Value r — Nominal Rate (also called APR) t — Number of years m — Number of compounding periods per year n = mt — Total number of compounding periods iper = r /m — Interest rate per compounding period PMT — The amount of payment ppy — Number of payments per year in an ordinary annuity ( )m/ppy r j = 1+ − 1 — Interest rate per payment period for an annuity m NPV — Net Present Value . . . . . . Dr. Ji Li Time Value of Money
- 50. Simple Interest and Compound Interest Notations Sinking Funds, Annuities, and Bonds Simple Interest and Compound Interest More on Finance Sinking Funds, Annuities, and Perpetuities Notations and Formulas Net Present Value. Simple Interest An investment of PV dollars growing with simple interest rate of r after t years is worth FV dollars: FV = PV (1 + r t). . . . . . . Dr. Ji Li Time Value of Money
- 51. Simple Interest and Compound Interest Notations Sinking Funds, Annuities, and Bonds Simple Interest and Compound Interest More on Finance Sinking Funds, Annuities, and Perpetuities Notations and Formulas Net Present Value. Compound Interest An investment of PV dollars earning interest at an annual rate of r compounded (reinvested) m times per year for a period of t years is worth FV dollars: ( r )mt FV = PV 1+ . m . . . . . . Dr. Ji Li Time Value of Money
- 52. Simple Interest and Compound Interest Notations Sinking Funds, Annuities, and Bonds Simple Interest and Compound Interest More on Finance Sinking Funds, Annuities, and Perpetuities Notations and Formulas Net Present Value. Sinking Funds A sinking fund is worth FV dollars if you make a payment of PMT at the end of each compounding period into an account earning interest at an annual rate of r compounded (reinvested) m times per year for t years: (1 + r /m)mt − 1 FV = PMT . r /m . . . . . . Dr. Ji Li Time Value of Money
- 53. Simple Interest and Compound Interest Notations Sinking Funds, Annuities, and Bonds Simple Interest and Compound Interest More on Finance Sinking Funds, Annuities, and Perpetuities Notations and Formulas Net Present Value. Annuities An annuity is an account earning compound interest from which periodic withdrawals are made. If the starting balance is PV dollars, you receive a payment of PMT at the end of each compounding period, and the account is down to $0 after for t years, then 1 − (1 + r /m)−mt PV = PMT . r /m . . . . . . Dr. Ji Li Time Value of Money
- 54. Simple Interest and Compound Interest Notations Sinking Funds, Annuities, and Bonds Simple Interest and Compound Interest More on Finance Sinking Funds, Annuities, and Perpetuities Notations and Formulas Net Present Value. New Formulars for Sinking Funds and Annuities Suppose that the number of compounding periods per year, m, is different from the number of payments per year, ppy . Then the interest rate per payment period becomes ( )m/ppy r j= 1+ −1 m and the sinking fund and annuity formulas become ( ) (1 + j)ppy·t − 1 Sinking Fund FV = PMT , j ( ) 1 − (1 + j)−ppy ·t Annuity PV = PMT , j . . . . . . Dr. Ji Li Time Value of Money
- 55. Simple Interest and Compound Interest Notations Sinking Funds, Annuities, and Bonds Simple Interest and Compound Interest More on Finance Sinking Funds, Annuities, and Perpetuities Notations and Formulas Net Present Value. Perpetuities A perpetuity is basically an annuity that lasts for ever. 1 PV = PMT . r /m If the compounding periods is not the same as payment periods, then we use the following formula 1 PMT PV = PMT =( )m/ppy j r 1+ m −1 . . . . . . Dr. Ji Li Time Value of Money
- 56. Simple Interest and Compound Interest Notations Sinking Funds, Annuities, and Bonds Simple Interest and Compound Interest More on Finance Sinking Funds, Annuities, and Perpetuities Notations and Formulas Net Present Value. Net Present Value Assume r is the APR and m is the number of compounding periods per year. The net present value NPV of a sequence of equally-spaced end-of-period payments (negative) and income (positive) with same compounding and payment/income periods, with an initial cash ﬂow v0 and the payment/income sequence v1 , v2 , . . . , vn is v1 v2 vn NPV = v0 + + + ··· + . 1 + r /m (1 + r /m)2 (1 + r /m)n . . . . . . Dr. Ji Li Time Value of Money

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