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Time Value of Money TVM - Compounding $ Today Future $ Discounting David M. Harrison, Ph.D. Texas Tech University Real Estate Investments Future Value (FV) Definition - FVn = PV(1 + i)n 1 0 2 FV = ? PV=x » David M. Harrison, Ph.D. Texas Tech University N Real Estate Investments 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? David M. Harrison, Ph.D. Texas Tech University Real Estate Investments Future Value of an Annuity (FVA) Definition - (1 i ) n 1 FVAn i 0 » 1 2 N A A A FVA = ? David M. Harrison, Ph.D. Texas Tech University Real Estate Investments Ordinary Annuity vs. Annuity Due Ordinary Annuity 0 1 2 N A A A 1 2 N A A i% Annuity Due 0 i% A David M. Harrison, Ph.D. Texas Tech University Real Estate Investments 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? David M. Harrison, Ph.D. Texas Tech University Real Estate Investments Present Value (PV) Definition - PV = P0 = FV / (1 + i)n 1 0 2 FV = x PV= ? » David M. Harrison, Ph.D. Texas Tech University N Real Estate Investments 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? David M. Harrison, Ph.D. Texas Tech University Real Estate Investments Present Value of an Annuity (PVA) Definition 1 1 (1 i ) n PVA i 0 PVA = ? 1 2 N A A A » David M. Harrison, Ph.D. Texas Tech University Real Estate Investments 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? David M. Harrison, Ph.D. Texas Tech University Real Estate Investments 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! David M. Harrison, Ph.D. Texas Tech University Real Estate Investments Perpetuities Definition - Paym ent PMT PVPerpetuity Interest Rate i 0 1 2 $ $ $ PVperpetuity = ? David M. Harrison, Ph.D. Texas Tech University Real Estate Investments 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: » » David M. Harrison, Ph.D. Texas Tech University Real Estate Investments 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: David M. Harrison, Ph.D. Texas Tech University Year Cashflow 1 2 3 4 5 100 200 250 200 400 Present Value Real Estate Investments Uneven Cash Flow Streams Ex. Given a discount rate of 8%, what is the future value of the following cash flows stream: David M. Harrison, Ph.D. Texas Tech University Year Cashflow 1 2 3 4 5 100 200 250 200 400 Future Value Real Estate Investments Nominal vs. Effective Rates Nominal Rate - Effective Rate - What’s the difference? m NominalRate EAR 1 1 m David M. Harrison, Ph.D. Texas Tech University Real Estate Investments 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? David M. Harrison, Ph.D. Texas Tech University Real Estate Investments 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? David M. Harrison, Ph.D. Texas Tech University Real Estate Investments Amortization Schedules Year 1 2 Beg. Bal. $10,000 PMT INT PRIN 3 4 Year #1, Principal % = Year #2, Principal % = Year #3, Principal % = Year #4, Principal % = David M. Harrison, Ph.D. Texas Tech University Real Estate Investments End. Bal. Continuous Compounding Definition/Description - FVn PV in FVn e in e FVn PV e David M. Harrison, Ph.D. Texas Tech University in Real Estate Investments 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? David M. Harrison, Ph.D. Texas Tech University How much lower would the future value be if the interest rate were 8.75% compounded annually? Real Estate Investments