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## Time Value of Money

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**Time Value of Money**• TVM - Compounding $ Today Future $ Discounting**Future Value (FV)**• Definition - FVn = PV(1 + i)n 1 2 0 N FV = ? PV=x**Future Value Calculations**• Suppose you have $10 million and decide to invest it in a security offering an interest rate of 9.2% per annum for six years. At the end of the six years, what is the value of your investment? • What if the (interest) payments were made semi-annually? • Why does semi-annual compounding lead to higher returns?**Future Value of an Annuity (FVA)**• Definition - 0 1 2 N A A A FVA = ?**Ordinary Annuity vs. Annuity Due**Ordinary Annuity 0 1 2 N i% A A A Annuity Due 0 1 2 N i% A A A**Future Value of an Annuity Examples**• Suppose you were to invest $5,000 per year each year for 10 years, at an annual interest rate of 8.5%. After 10 years, how much money would you have? • What if this were an annuity due? • What if you made payments of $2,500 every six-months instead?**Present Value (PV)**• Definition - PV = P0 = FV / (1 + i)n 1 2 0 N FV = x PV= ?**Present Value Calculations**• How much would you pay today for an investment that returns $5 million, seven years from today, with no interim cashflows, assuming the yield on the highest yielding alternative project is 10% per annum? • What if the opportunity cost was 10% compounded semi-annually? • Why does semi-annual compounding lead to lower present values?**Present Value of an Annuity (PVA)**• Definition - 0 1 2 N A A A PVA = ?**Present Value of an Annuity Examples**• How much would you spend for an 8 year, $1,000, annual annuity, assuming the discount rate is 9%? • What if this were an annuity due? • What if you were to receive payments of $500 every six-months instead?**TVM Properties**• Future Values • An increase in the discount rate • An increase in the length of time until the CF is received, given a set interest rate, • Present Values • An increase in the discount rate • An increase in the length of time until the CF is received, given a set interest rate, • Note: For this class, assume nominal interest rates can’t be negative!**Perpetuities**• Definition - 0 1 2 $ $ $ PVperpetuity = ?**Perpetuity Examples**• What is the value of a $100 annual perpetuity if the interest rate is 7%? • What if the interest rate rises to 9%? • Principles of Perpetuities:**Uneven Cash Flow Streams**• Description - • Ex. Given a discount rate of 8%, how much would you be willing to pay today for an investment which provided the following cash flows:**Uneven Cash Flow Streams**• Ex. Given a discount rate of 8%, what is the future value of the following cash flows stream:**Nominal vs. Effective Rates**• Nominal Rate - • Effective Rate - • What’s the difference?**Nom. vs. Eff. Rate Examples**• Ex. #1: A bond pays 7% interest semi-annually, what is the effective yield on the bond? • A credit card charges 1.65% per month (APR=19.8%), what rate of interest are they effectively charging? • What nominal rate would produce an effective rate of 9.25% if the security pays interest quarterly?**Amortization**• Amortized Loan - • Ex. Suppose you borrow $10,000 to start up a small business. The loan offers a contract interest rate of 8.5%, and must be repaid in equal, annual installments over the next 4 years. How much is your annual payment? • What percentage of your payments go toward the repayment of principal in each year?**Amortization Schedules**Year #1, Principal % = Year #2, Principal % = Year #3, Principal % = Year #4, Principal % =**Continuous Compounding**• Definition/Description -**Does Compounding Matter?**• What is the present value of $200 to be received 2 years from today, if the discount rate is 9% compounded continuously? • How much more would the cash flow be worth if the discount rate were 9% compounded annually? • What is the future value, in 10 years, of a $5,000 investment today, if the interest rate is 8.75% compounded continuously? • How much lower would the future value be if the interest rate were 8.75% compounded annually?