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**FIL 240 Prepared by Keldon Bauer**

Time Value of Money FIL 240 Prepared by Keldon Bauer

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Cash Flow Time Lines You win the American College Student Publishers Contest. 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.

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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. 1 2 3 4 5

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**Cash Flow Time Lines Outflows are listed as negatives.**

Inflows are positive. State the appropriate “interest rate,” which represents your opportunity costs 1 2 3 4 5 8% $250K

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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.

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Future Value - Example If you invest 1,000 today at 8% interest per year, how much should you have in five years (in thousands). 1 2 3 4 5 8% Principal Interest Total Prev. Interest -1 0.08 0.00 1.08 0.0864 0.0800 1.1664 0.0933 0.1664 1.2597 0.1008 0.2597 1.3605 0.1088 0.3605 1.4693

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Future Value For one year, the future value can be defined as:

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Future Value The second year, the future value can be stated as follows:

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Future Value Therefore, the general solution to the future value problem is:

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Future Value Interest can be seen as the opportunity growth rate of money.

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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: 1 2 3 4 5 8% $500 PV=?

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Present Value The general solution to this problem follows from the solution to the future value problem:

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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.

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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. 1 2 3 4 5 ?% $500 $100

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Interest Rates Solving for interest rates algebraically:

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**Interest Rates - Example**

1 2 3 4 5 ?% $500 $100

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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:

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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:

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**Annuities Definition: A series of equal payments at a fixed interval.**

Two types: Ordinary annuity: Payments occur at the end of each period. Annuity due: Payments occur at the beginning of each period.

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Ordinary Annuity Example is a regular payment of $100 for five years earning 8% interest. 1 2 3 4 5 $100 8% What is the future value? FV=?

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**Ordinary Annuity – Future Value**

The future value of an ordinary annuity can be found as follows:

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**Ordinary Annuity - Example**

1 2 3 4 5 $100 8%

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Annuity Due Example is a regular payment of $100 for five years earning 8% interest. 1 2 3 4 5 $100 8% What is the future value? FV=?

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**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:

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Annuity Due - Example 1 2 3 4 5 $100 8%

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**Ordinary Annuity - Present Value**

Example is a regular payment of $100 for five years earning 8% interest. 1 2 3 4 5 $100 8% What is the present value? PV=?

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**Ordinary Annuity - Present Value**

The present value of an ordinary annuity can be found as follows:

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**Ordinary Annuity - Example**

1 2 3 4 5 $100 8%

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**Annuity Due - Present Value**

Example is a regular payment of $100 for five years earning 8% interest. 1 2 3 4 5 $100 8% What is the present value? PV=?

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**Annuity Due - Present Value**

The future value of an annuity due can be found as follows:

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Annuity Due - Example 1 2 3 4 5 $100 8%

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**Annuities - Finding Interest Rate**

Interest rates cannot be solved directly. Calculators search for the correct answer (there is only one correct answer). It guesses and then iteratively goes higher or lower.

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Perpetuities What would you have to pay to be paid $2,000 per year forever (given a market rate of 8%)?

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**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.

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**Uneven Cash Flows - Example**

1 2 3 4 5 $100 $200 $300 $400 $500 8% $294.01 $340.29 $171.47 $238.15 $ 92.59 $1, = Present Value

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**Uneven Cash Flow - Example**

1 2 3 4 5 $100 $200 $300 $400 $500 8% $136.05 $251.94 $349.92 $432.00 Future Value = $1,669.91 $1, = Present Value

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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.

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Compounding The more often one compounds interest, the faster it grows. 1 2 3 4 5 8% -$100 $146.93 Annual 1 2 3 4 10 5 6 7 8 9 Semi-Annual 4% -$100 $148.02

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**Compounding Why is there a difference in future value?**

Because interest is earned on itself faster! How would you adjust to compound monthly? How would you make an adjustment in annuities?

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Effective Annual Rate To convert the other compounding periods to an effective annual compounding rate (EAR) use the following formula:

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**Effective Annual Rate - Example**

8% monthly compounding loan is equal to what in effective annual rate?

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**Fractional Time Periods**

If you invest $100 for nine months at an EAR of 8%, how does one calculate the future value? The same way one did before.

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Amortized Loans A loan with equal payments over the life of the loan is called an amortized loan. Loan mathematics are the same as an annuity. Loan amounts are the present value. Periodic loan payments are the payments.

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Amortized Loans The present value of a monthly loan uses the annuity formula adjusted for monthly payments:

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Amortized Loans Payments on a given loan can be found by solving for PMT in the previous equation:

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**Amortized Loans - Example**

What is the payment on a 30 year loan of $150,000?

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**Amortization Schedules**

Amortization schedules show how much of each payment goes toward principal and how much toward interest. The easiest way of calculating one by hand is by calculating the outstanding loan balance month-by-month, and then taking the difference in loan balance from month to month as the principal portion of the payment.

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**Amortization Schedules**

The portion in the amortized loan formula that says n×m can be interpreted as months remaining. So to find the part of the first $1, that is paid toward the principal one would realize that at the beginning one had all $150,000 outstanding.

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**Amortization Schedules**

After the first month, one has 359 payments left. Therefore the loan principal outstanding is:

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**Amortization Schedules**

The difference in principal outstanding is the part of the payment that went toward principal. In this instance, 150, ,899.35=$100.65 The rest of the payment went toward interest. In this instance that would be 1, =$1,000.

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**Amortization Schedules**

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**Different Types of Interest**

Simple Interest (i or isimple) - The rate we have used thus far to calculate interest. Periodic Interest (i or iperiodic) - The interest paid over a certain period.

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**Different Types of Interest**

Effective Annual Rate (EAR): Described earlier as the rate that would be charged to get the same compounded annual rate. Annual Percentage Rate (APR):

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Chapter 4 The Time Value of Money

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