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## Time Value of Money

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**Time Value of Money**FIL 404 Prepared by Keldon Bauer**Cash Flow Time Lines**• You win a contest, and you have the option of taking $1.4 million now or $250,000 per year for five years. • Which should you take? • The answer comes through taking into consideration the time value of money.**2**3 4 5 0 1 Cash Flow Time Lines • The first step is visualizing the cash flows by drawing a cash flow time line. • Time lines show when cash flows occur. • Time 0 is now.**2**3 4 5 0 1 8% $250K $250K $250K $250K $250K Cash Flow Time Lines • Outflows are listed as negatives. • Inflows are positive. • State the appropriate “interest rate,” which represents your opportunity costs**Future Value**• Future value is higher than today, because if I had the money I would put it to work, it would earn interest. • The interest could then earn interest. • Compounding: allowing interest to earn interest on itself.**2**3 4 5 0 1 8% Principal -1 Interest 0.08 0.0864 0.0933 0.1008 0.1088 Prev. Interest 0.00 0.0800 0.1664 0.2597 0.3605 Total 1.08 1.1664 1.2597 1.3605 1.4693 Future Value - Example • If you invest $1,000 today at 8% interest per year, how much should you have in five years (in thousands).**Future Value**• For one year, the future value can be defined as:**Future Value**• The second year, the future value can be stated as follows:**Future Value**• Therefore, the general solution to the future value problem is: • The Excel formula is: • =FV(Interest, Term, Payments, [Present Value], [Type])**Future Value**• Interest can be seen as the opportunity growth rate of money.**2**3 4 5 0 1 8% PV=? $500 Present Value • Present value is the value in today’s dollars of a future cash flow. • If we are interested in the present value of $500 delivered in 5 years:**Present Value**• The general solution to this problem follows from the solution to the future value problem:**Present Value - Excel**• The Excel formula is: • =PV(Interest, Term, Payments, [Future Value], [Type])**Present Value**• Since the discount rate is the opportunity cost, the present value represents what I would have to give up now to get the future value specified.**2**3 4 5 0 1 ?% $100 $500 Interest Rates • If we know the amount we need at time n and the amount we can invest at time zero, then we must only solve for the interest rate.**Interest Rates**• Solving for interest rates algebraically:**2**3 4 5 0 1 ?% $100 $500 Interest Rates - Example In Excel: =RATE(Term, Payment, Present Value, [Future Value], [Type], [Guess])**Time Periods**• If the present value, future value and interest rate are known, but the number of time periods is not. Then n can be found algebraically:**Time Periods - Example**• If we use the last example of investing $100, we want $500 in future, and the current market interest is 8%, n can be found: In Excel: =NPER(Interest, Payment, Present Value, [Future Value], [Type])**Annuities**• Definition: A series of equal payments at a fixed interval. • Two types: • Ordinary annuity: Payments occur at the end of each period. (Default in Excel) • Annuity due: Payments occur at the beginning of each period. (Set the type = 1 in Excel) • In Excel, use the same formulas introduced so far, just specify payment and type.**2**3 4 5 0 1 8% $100 $100 $100 $100 $100 FV=? What is the future value? Ordinary Annuity • Example is a regular payment of $100 for five years earning 8% interest.**Ordinary Annuity – Future Value**• The future value of an ordinary annuity can be found as follows:**2**3 4 5 0 1 8% $100 $100 $100 $100 $100 Ordinary Annuity - Example**2**3 4 5 0 1 8% $100 $100 $100 $100 $100 FV=? What is the future value? Annuity Due • Example is a regular payment of $100 for five years earning 8% interest.**Annuity Due – Future Value**• The future value of an annuity due can be found by noticing that the annuity due is the same as an ordinary annuity, with one more compounding period:**8%**$100 $100 $100 $100 $100 2 3 4 5 0 1 Annuity Due - Example**2**3 4 5 0 1 8% $100 $100 $100 $100 $100 PV=? What is the present value? Ordinary Annuity - Present Value • Example is a regular payment of $100 for five years earning 8% interest.**Ordinary Annuity - Present Value**• The present value of an ordinary annuity can be found as follows:**2**3 4 5 0 1 8% $100 $100 $100 $100 $100 Ordinary Annuity - Example**2**3 4 5 0 1 8% $100 $100 $100 $100 $100 PV=? What is the present value? Annuity Due - Present Value • Example is a regular payment of $100 for five years earning 8% interest.**Annuity Due - Present Value**• The future value of an annuity due can be found as follows:**8%**$100 $100 $100 $100 $100 2 3 4 5 0 1 Annuity Due - Example**Annuities - Finding Interest Rate**• Interest rates cannot be solved directly. • Calculators and computers search for the correct answer (there is only one correct answer). • It guesses and then iteratively goes higher or lower.**Perpetuities**• What would you have to pay to be paid $2,000 per year forever (given a market rate of 8%)?**Uneven Cash Flow Streams**• If payments are irregular or come at irregular intervals, we can still find the PV (or FV). • Take the present value (or future value) of individual payments and sum them together.**$ 92.59**2 3 4 5 0 1 8% $171.47 $238.15 $294.01 $100 $200 $300 $400 $500 $340.29 Uneven Cash Flows - Example $1,136.51 = Present Value**Uneven Cash Flows - Excel**• Excel can do this in one argument: • =NPV(Interest, Array of Payments [starting with payment for time 1]). • If you want to include a payment in time zero, add it to the above argument separately.**$432.00**2 3 4 5 0 1 $349.92 8% $251.94 $136.05 $100 $200 $300 $400 $500 $1,136.51 = Present Value Uneven Cash Flow - Example Future Value = $1,669.91**Finding Interest Rate**• As with annuities, interest rates for uneven cash flow streams cannot be solved directly. • Calculators search for the correct answer, called an IRR (there may be more than one correct answer). • It guesses and then iteratively goes higher or lower.**8%**2 3 4 5 0 1 -$100 $146.93 0 1 2 3 4 5 6 7 8 9 10 4% -$100 $148.02 Compounding • The more often one compounds interest, the faster it grows. Annual Semi-Annual