mathematics of finance solutions to the examples in this presentation are based on using a texas...

Mathematics of Finance

Solutions to the examples in this presentation are based on using a Texas

Instruments BAII Plus Financial calculator.

$x today$x today?

BUT WHY?

Postponement of today’s opportunities for investments or consumption to the future would result in OPPORTUNITY COST. TVM captures and explains such lost opportunities.

$x today$x today or $x in future?$x in future?

A matter of Preference or Risk?

Time Value of Money(TVM)

TVM capture the Opportunity Cost

Through:Compounding or determining the

Future Values based on present $s, and

Discounting or determining the Present values based on future $s

CompoundingFuture Value of a single amountFuture Value of an annuityFuture Value of uneven cash flows

DiscountingPresent Value of a single amountPresent Value of an annuityPresent Value of uneven cash flows

CompoundingFuture Value of a single amountFuture Value of an annuityFuture Value of uneven cash flows

DiscountingPresent Value of a single amountPresent Value of an annuityPresent Value of uneven cash flows

TVM capture the Opportunity Cost

Compound Interest

Compounding is paying interest both on principle and interest. For a 2-year savings commitment, the

FV1 = PV + (PV x r) = PV (1 + r)

FV2 = PV (1 + r) + PV (1 + i) x r = PV (1 + r) (1 + r) = PV (1 + r)2

FV1 = 100 + (100 x .05) = 100 (1 + .05) = 105

FV2 = 100 (1 + .05) + 100 (1 + .05) x .05 = 100 (1 + .05)2

= 110.25Note: Present Value = Principal

An investment of $100 today in a savings $100 today in a savings account that pays 5account that pays 5% interest, with interest compounded annually, will result in $110.25 at the end of year 2.

Future Value on a Timeline

0 1 2

$100$100

FVFV

5%

$105$105 $110.25$110.25

PVPV

FVFVnn = PVPV (1+r)n FVFV22 = $100$100 (1.05)2 = $110.25$110.25

Future Value, General Formula

Lets Put The Calculator to Work!

Future Value on TI BAII Plus

Turn the calculator on and change the default setting by:

2nd

Enter

1

I/Y

Press

ENTER

Press

These keystrokes will change the frequency of compounding to once

per year

2

5

100

N

I/Y

PV

CPT, FV

Always Press 2nd, then FV

PressEnter

$110.25


How much money would be in your savings account after 6 years if you deposit $5,000$5,000 today and the bank pay an annual compound interest rate of 7%?

Future Value Example

0 1 2 3 4 5 65 6

$5,000$5,000

FVFV66

7%

Calculator keystrokes: 1.07 yx 6 5000 =

Future Value Solution

Calculation based on the formula:FVFVnn = PV (1+r)n

FVFV55 = $5,000 (1+ 0.07)6

= $7,503.65$7,503.65

7

5000

I/Yr

PV

CPT, FV

N6


PressEnter

7,503.65


Present ValuePresent Value

Having FV = PV(1 + r)n then:

This represents the Discounting process or the process of determining the present value of a single future cash flow.

nrFVPV

)1(

If you need to have a $10,000$10,000 down payment on a house 12 years from now, years from now, how much must you save today in an account that pays 7% interest, compounded annually?

$10,000$10,000

Present Value (Graphic)

0 3 6 6 9 127%

PVPV00

Present Value on TI BAII Plus

7

10000

I/Yr

FV

CPT, PV

N12


PressEnter

4,440.12

Calculator keystrokes:

1.07 yx 12 = 1/x 10000 =

Computing “n” or “i” knowing PV and FV

If John lends Linda $4,000 today for a If John lends Linda $4,000 today for a return of $6,154.50 after 5 years, what rate return of $6,154.50 after 5 years, what rate of annual compound interest does he earn?of annual compound interest does he earn?

4000

6154.50

+/-, PV

FV

CPT, I/Y

N5


PressEnter

9.00%


General Formula:

FVn,m = PVPV00[1 + (r/m)] mn

n: Number of Years

m: Number of Compounding per Year

r: Annual Interest Rate

FVn,m: Future Value at Year n

PVPV00: Present value of amounts

Frequency of Compounding

Frequency of Compounding

Example:If your deposit of $3,000 in a savings account, paying monthly compounded interest based on a 9% annual rate, is maintained for six years how much will be in the account at that time?

PV = $3,000r = 9%/12 = 0.75% per monthn = 6 x 12 = 72 months

Solution, based on formula:

FV= PV (1 + r)n

= 3,000(1.0075)72

= 5,137.66

Calculator Keystrokes:

1.0075 yx 72 3000 =

Frequency of Compounding on (TI BAII Plus )

3000

0.75

PV

I/Y

CPT, FV

N72


PressEnter

$5,137.66

Annuities

Annuities Can Be:Ordinary (starting at the end of each period) orDue (starting at the beginning of each period)

Example of Annuities Are: Any kind of installment payment for retiring a loan Insurance Premiums Savings for Retirement

An AnnuityAn Annuity represents a series of equal payments (or receipts) over EQUAL intervals.

A plan to save $4,000 a year at the end of each year for three years would result in how much savings, considering that your savings account pays 7% interest, compounded annually?

FVAFVA33 = $4,000(1.07)2 + $4,000(1.07)1 + $4,000(1.07)0

= $12,610$12,610

Future Value of an Ordinary Annuity -- FVA

0 1 2 3

$4,000 $4,000 $4,000

$12,859.60 = FVA$12,859.60 = FVA3

End of Year

7%

$4,280

$4,579.60

Future Value (TI BAII Plus)

4000

7

PMT

I/Y

CPT, FV

N3


PressEnter

$12,859.60

Jamshid was approved for a business loan, which required $2,500 annual payment at the end of each next 4 years. The loan carried an annual interest rate of 6%. What was the amount of this loan?

PVAPVA33 = $2,500/(1.06)1 + $2,500/(1.06)2 + $2,500/(1.06)3 + $2,500/(1.06)4

= $8,662.76$8,662.76

Present Value of an Ordinary Annuity -- PVA

$2,500 $2,500 $2,500 $2,500

0 1 2 3 3 4Yearend

6%

$8,662.76 = PVA$8,662.76 = PVA33

$2,358.49$2,224.99 $2,099.05$1,980.23


2500

6

PMT

I/Y

CPT, PV

N4


PressEnter

$8,662.76

Your investment advisor recommends a security that provides $3,000, $5,000, and $7,000 respectively at the end of each of the next 3 years. If you require 12% return on this security, how much would you be willing to pay for it?

PV of Unequal Cash Flows

0 1 2 3

$3000 $5000 7,000 $3000 $5000 7,000

PVPV00

12%12%

Unequal Cash Flow Solution

0 1 2 3

$3,000 $5,000 $7,000$3,000 $5,000 $7,00012%

$2,678.57$2,678.57$3,985.97$3,985.97$4,982.46$4,982.46

$11,647.00 $11,647.00 = = PVPV00

Unequal Cash Flow Solution (TI BAII Plus)

Enter

0

3000

5000

Press

7000

Press CF2nd, then CE/C

ENTER

ENTER

ENTER

ENTER

1

1

1

ENTER

ENTER

ENTER

ENTER

NPV

12 CPT

$11,647.00 Frequency of the cash flows

Computing Yield to Maturity

DXL Industries bond is currently selling for $932.50. This bond is having a coupon interest rate of 11%, and will mature in 20 years. Considering that the bond’s face value is $1,000 and pays interest semiannually, what is the yield to maturity (YTM) on this bond?

YTM Solution on TI BAII Plus

932.50 +/-, PV

1000

(.11 1000) 2=

20 2 =

PMT

FV

N

CPT, I/Y

5.945% for 6 months or 11.89% annually

0 1 2 ……….… 40 55 55 55

1000

Always Press

2nd, then FV

Enter Press

Check your command of the Concepts

Click one of the following problems

1

2

3

Problem #1

Morgan deposited $25,000 in a new savings account that is paying 9% annual interest rate compounded monthly. She will not be able to withdraw her deposit within the next 3 years. What will be the size of deposits in her account in 3 years?

Problem 1 - Select one

$32,716.13$32,375.73$556,280.63

HELP!

TI BAII Plus Solution to #1TI BAII Plus Solution to #1

I/Y

N

PV

CPT, FV

32,716.13

25,000

9 12 =

3 12 =


PressEnter

Click for Next Problem

FV = 25000 (1 + .0075)36

Problem #2

You currently receive $10,000 per year on a contract. You expect it to run another 7 years. Someone wants to buy the contract from you. If you can earn 12% on other investments of this quality, how much would you be willing to sell the contract for?

Possible Answers - Problem 2

$40,020.76$45,637.57$100,890.11

HELP!


PMT

PVA=10000/(1.12)1 + 10000/(1.12)2 +…+ 10000/(1.12)7

10000

I/Y

N

PV

7

12

CPT


PressEnter

0 1 2 3 4 … 7

10000 10000 10000 10000 ... 10000

$45,637.57

Click for Next

Problem

Problem #3

Thompson Corp. has issued a bond with a face value of $1,000. The bond carries a coupon interest rate of 6%, pays interest semi-annually, and will mature in 25 years. How much would you pay for this bond if your required return on similar investments is 8%?

Possible Solutions - Problem 3

$843.78$785.18$388.33

HELP!


Enter Press

1000

30

4

50

PMT

FV

I/Y

N

CPT, PV

30 30 30 1000

0 1 2 ……….… 50


n

t

ntb rr

IPV

)1(

1000

)1(

1

PVb

Click for Next

Problem

Excellent!

A job well done!

Click for Next Problem

Calculating the Future Value

When the frequency of compounding is more than once per year you should adjust both the discount rate, and the time.

Determine the future value of single amount.

Click to return

The Worth of a Contract

The worth of any asset is the present value of its future cash flows.

Terms such as “per year”, “annually”, “every year” are indications that the cash flows are annuities.

Click to return

Valuing a Bond

Consider the coupon payments as annuity and the face value of the bond as a single cash flow at maturity.

Remember that you should adjust the time, the discount rate, and the interest payments to reflect the semi-annual compounding.

Click to return

mathematics of finance solutions to the examples in this presentation are based on using a texas...

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