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## Chapter 4 Time Value of Money (cont.)

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**Present value of multiple cash flows**Nominal interest rate and real interest rate Effective interest rate Chapter 4 Time Value of Money(cont.)**Multiple Cash Flows**• Usually an investment involve multiple/a stream of (negative/positive) cash flows instead of just one payment and one initial investment. One term deposit Several term deposits that end at the same time Several withdrawals out of one deposit**FV of Multiple Cash Flows**• The future value of several cash flows paid (or several cash flows received) at a certain point of time can be calculated by adding up the future values of each of the cash flows. • N specifies how many periods away from now is the FV that we want to calculated. • Ct denotes the actual cash flow that is paid/received at the end of the tth period.**FV of Multiple Cash Flows**Example: If you make one term deposit of $300 now and another 2 of $200 at the end of each of the following two years, and all the deposit expires at the end of the 4th year from now. Interest rate is 8%. How much will your bank account balance be? (draw a time line and assign values to variables in the formula)**FV of Multiple Cash Flows**Example: (cont.)**PV of Multiple Cash Flows**• The present value of several cash flows paid (or several cash flows received) in future can be calculated by adding up the present values of each of the cash flows. • Ct denotes the actual cash flow that is paid/received at the end of the tth period.**PV of Multiple Cash Flows**Example: If you need to make 3 payments at different point of time: one of $250 now, a second payment of $300 at the end of next year (the first year) and a third one of $500 at the end of the year after next (the second year) . Interest rate is 8%. How much money should you have in your bank account now so that you would be able to make all the three payments at the specified time? (draw a time line and assign values to variables in the formula)**PV of Multiple Cash Flows**Example: (cont.)**Multiple Cash Flows**• Using financial calculators: • Calculate the FV/PV of each cash flows independently then sum the results together • Make sure the correct t (i.e. N) is used for each cash flow • When there are several cash flows paid and also several cash flows received, the formula to be used are the same: • Make sure the correct sign is given to each cash flow**Perpetuities & Annuities**Perpetuity: A stream of level cash payments that never ends. Annuity: Equally spaced level stream of cash flows for a limited period of time.**Perpetuities**Assume: • Deposit $100 • Annual interest rate is 8% and it never changes • Interests are withdrawn at the end of every year but never the principal Cash flows: • Pay $100 now • Receive $8 at the end of every year forever**Perpetuities**PV of Perpetuity: the value of all future cash flows from a perpetuity in terms of a one time payment now Formula: for a perpetuity whose cash flows occur at the end of every period starting from now. C = cash payment r = interest rate / discount rate**Perpetuities**Example - Perpetuity In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today if the rate of interest is 10%?**Perpetuities**Example - continued If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today?**Annuities**• Annuity can be viewed as the difference between two perpetuities**Annuities**PV of Annuity: the value of all future cash flows from an annuity in terms of a one time payment now Formula: for an annuity whose cash flows occur at the end of every period starting from now and lasting for t periods. C = cash payment every period r = interest rate t = number of periods cash payment is received**Annuities**PV Annuity Factor (PVAF) - The present value of $1 a year for each of t years. [Table A.3 on page 704 ] • Find the appropriate PVAF according to the right t and r**Annuities**Example - Annuity To purchase a car, you are scheduled to make 3 annual installments of $4,000 per year starting one year from now. Given a rate of annual interest of 10%, what is the price you are paying for the car (i.e. what is the PV)?**Annuities**• Example – Annuity (cont.)**Annuity Due Calculation**• Adjust your financial calculator • Switch from “End” to “Begin • The inputs are the same as an ordinary annuity • Example: start paying the installments right now**Switch From “End” to “Begin”**• HP Press {shift} (i.e. the yellow button) and then press {BEG/END} • TI • Press {2nd}, then {BGN} • Press {2nd}, then {SET} • Press {2nd}, then {QUIT} • To switch back from “Begin” to “End”, just repeat the procedure**Annuity Due Calculation (cont.)**• PV of and annuity due equals the multiple of the PV of the ordinary annuity and (1+r) • Both annuities have the same annual payment and number of periods • Example: start paying the installments right now • Calculate the PV of corresponding ordinary annuity • Multiply by (1+r)**Annuities Applications**• Present Value of payments • Implied interest rate for an annuity • Calculation of periodic payments • Mortgage payment • Annual income from an investment payout • Future Value of annuity**Present Value of payments**• Example: In 1992, a nurse in a Reno casino won the biggest jack pot - $9.3 million. That sum was paid in 20 annual installments of $465,000. What is the PV? r=10% (draw a time line and assign values to variables in the annuity formula)**Home Mortgages**• Example:Suppose you are buying a house that costs $125,000, and you want to put down 20% ($25,000) in cash. Assume that the mortgage loan lasts 30 years, i.e. 360 months. What will be your monthly payment for each option, if the monthly interest rate is 1%? (draw a time line and assign values to variables in the annuity formula)**Future Value of Annuity**Example - Future Value of annual payments You plan to save $4,000 every year for 20 years starting from the end of this year, and then retire. Given a 10% rate of interest, what will be the balance of your retirement account in 20 years?**Inflation**Inflation: Rate at which prices as a whole are increasing. • Consumer price index, CPI Real Interest Rate: Rate at which the purchasing power of the return of an investment increases. • Real value of money Nominal Interest Rate: Rate at which money invested grows. • Nominal value of money • The quoted interest rate**Inflation**• Exact formula • Approximation formula**Inflation**• Let r= real interest rate, i=inflation rate, and R= nominal interest rate.**Inflation**Example If the interest rate on one year government bonds is 5.0% and the inflation rate is 2.2%, what is the real interest rate?**Effective Interest Rates**• Effective Annual Interest Rate - Interest rate that is annualized using compound interest. • Give the actual annual interests • Annual Percentage Rate - Interest rate that is annualized using simple interest. • Only a way to quote interest rates • Imposed by legal requirements**Effective Interest Rates**Example Given APR of 12% and monthly compounding, what is the Effective Annual Rate(EAR)? • First, calculate month interest rate • Then, calculate the annual rate after compounding**Amortizing Loan**• Mortgage Amortization (page 88) • Periodic Payment = Amortization + Periodic Interest • Periodic Interest = interest rate * prior period loan balance Example: pay off 100,000 mortgage loan in 360 months at interest rate of 1% per month**Amortizing Loan**Summary: • Each periodic payment include amortization and interests due. • As the loan approaches maturity, the amortizations paid increase every period. • As the loan approaches maturity, the loan balances and interests due decrease every period. • The last amortization is just enough to payoff the last part of principal.**Problem 25 on page 108 (4/e 24 on page 105)**Annuity Values You want to buy a new car, but you can make an initial payment of only $2,000 and can afford monthly payments of at most $400. • If the APR on auto loans is 12% and you finance the purchase over 48 months, what is the max price you can pay for the car? • How much can you afford if you finance the purchase over 60 months?**Problem 28 on Page 109 (Problem 27 on Page 105)**• Rate on a Loan If you take out an $8,000 car loan that calls for 48 monthly payments of $240 each, what is the APR of the loan? What is the EAR?**Problem 37 on Page 109 (Problem 36 on Page 106)**Amortizing Loan You take out a 30-year $100,000 mortgage loan with an APR of 6% and monthly payments. In 12 years you decide to sell your house and pay off the mortgage. What is principal balance on the loan