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Financial Math Revisited. FIL 341 – Chapter 7 Prepared by Keldon Bauer. Investment vs. Consumption.

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Financial Math Revisited

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## Financial Math Revisited

FIL 341 – Chapter 7

Prepared by Keldon Bauer

### Investment vs. Consumption

• Some people prefer to consume now. Some prefer to invest now and consume later. Borrowing and lending allows us to reconcile these opposing desires which may exist within the firm’s shareholders.

income in period 1

100

A n

B n

80

Some investors will prefer A

60

and others B

40

20

20

40

60

80

100

income in period 0

### Investment vs. Consumption

The grasshopper (G) wants to consume now. The ant (A) wants to wait. But each is happy to invest. Each invests \$185,000 and returns \$210,000 at the end of the year. G wants to consume now so G borrows \$200,000 and repays \$210,000 at the end of the year. The existence of capital markets allows G to consume now and still invest with A in the project.

### Investment vs. Consumption

Dollars Next Year

210

194

• The grasshopper (G) wants to consume now. The ant (A) wants to wait. But each is happy to invest. Each invests \$185,000 and returns \$210,000 at the end of the year. G wants to consume now so G borrows \$200,000 and repays \$210,000 at the end of the year. The existence of capital markets allows G to consume now and still invest with A in the project.

A invests \$185 now and consumes \$210 next year

G invests \$185 now, borrows \$200 and consumes now.

Dollars Now

185 200

### Interest Rates

• Interest rates are the market clearing rates at which all those wishing to invest can find another party or parties willing to forgo consumption.

• To learn more about the mechanisms of how that works take FIL 241 (Markets and Institutions).

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:

### Future Value

• For Excel to calculate a future value, use the following formula:

=FV(Interest Rate, Number of Periods, Payments, PV)

• For the example we have been using of \$1,000 being invested at 8% for 5 years we would type:

=FV(8%, 5, 0, -1,000)

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

• For Excel to calculate a present value, use the following formula:

=PV(Interest Rate, Number of Periods, Payments, PV)

• For the example we have been using of \$500 needed in 5 years, and an opportunity cost of funds of 8% we would type:

=PV(8%, 5, 0, 500)

### 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.

2

3

4

5

0

1

8%

-100

-100

-100

-100

-100

FV=?

What is the future value?

### Ordinary Annuity – Future Value

• Example, suppose you make a regular payment of \$100 for five years (at the end of each year) 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

### FV of Ordinary Annuity - Excel

• For Excel to calculate a future value of an ordinary annuity, use the following formula:

=FV(Interest Rate, Number of Periods, Payments)

• For example, suppose you make a regular payment of \$100 for five years earning 8% interest, we would type:

=FV(8%, 5, -100)

2

3

4

5

0

1

8%

\$100

\$100

\$100

\$100

\$100

FV=?

What is the future value?

### Future Value of Annuity Due

• Example, suppose you make a regular payment of \$100 for five years (at the beginning of each year) 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

### FV of Annuity Due - Excel

• For Excel to calculate a present value, use the following formula:

=FV(Interest Rate, Number of Periods, Payments, PV,1)

• For example, suppose you make a regular payment of \$100 for five years (at the beginning of each year) earning 8% interest:

=FV(8%, 5, -100, 0, 1)

2

3

4

5

0

1

8%

-100

-100

-100

-100

-100

PV=?

What is the present value?

### Ordinary Annuity - Present Value

• Example, suppose you make a regular payment of \$100 for five years (at the end of each year) 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

### PV of Ordinary Annuity - Excel

• For Excel to calculate a present value of an ordinary annuity, use the following formula:

=PV(Interest Rate, Number of Periods, Payments)

• For example, suppose you make a regular payment of \$100 for five years earning 8% interest, we would type:

=PV(8%, 5, -100)

2

3

4

5

0

1

8%

\$100

\$100

\$100

\$100

\$100

PV=?

What is the present value?

### Annuity Due - Present Value

• Example, suppose you make a regular payment of \$100 for five years (at the beginning of each year) 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

### PV of Annuity Due - Excel

• For Excel to calculate a present value, use the following formula:

=PV(Interest Rate, Number of Periods, Payments, PV,1)

• For example, suppose you make a regular payment of \$100 for five years (at the beginning of each year) earning 8% interest:

=PV(8%, 5, -100, 0, 1)

### 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.

• The only adjustment is that the interest is usually stated in APR, and must be divided by the number of compounding periods per year.

### Amortized Loans

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

### Amortized Loans

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

### Amortized Loans - Example

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

### Amortized Loans - Example

• For Excel to calculate a present value, use the following formula:

=PMT(Interest Rate, Number of Periods, Payments, PV)

• For example, the monthly payment on a 30 year loan of \$150,000:

=PMT(8% / 12, 30 * 12, -150000)

### 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.

### 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 \$1,100.65 that is paid toward the principal one would realize that at the beginning one had all \$150,000 outstanding.

### Amortization Schedules

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

• This is just a present value calculation. Using Excel, you would type:

=PV(8% / 12, 359, -1100.65)

### Amortization Schedules

• The difference in principal outstanding is the part of the payment that went toward principal. In this instance, 150,000-149,899.35=\$100.65

• The rest of the payment went toward interest. In this instance that would be 1,100.65-100.65=\$1,000.

### Amortization Schedules: Part II

• Ordinary loans have equal payment, but other payment streams are possible.

• Graduated payment mortgages (GPM) were set up to allow the payment to grow over a specific time period before becoming fixed.

• To find a payment for a GPM using Excel, we will have to set up an amortization schedule.

• Then use Tools and Goal Seek to find the payment.

### Amortization Schedules: Part II

• Set up by showing the principal outstanding before applying the payment, then the amount outstanding after the payment.

• The Goal Seek will change the payment until the outstanding amount at the maturity month is exactly zero.