Part Two The Financial Management of Values

Part Two

The Financial Management of Values

Learning Objectives

• Be able to compute the future value and present value

• Be able to compute the return on an investment• Be able to use a financial calculator and a

spreadsheet to solve time value of money problems

• Descibe the conception of value at risk• Understand the risk identification and risk

measurement

The Role of Time Value in Finance

• Most financial decisions involve costs & benefits that are

spread out over time.

• Time value of money allows comparison of cash flows from

different periods.

Question?

Would it be better for a company to invest $100,000 in a product that would return a total of $200,000 in

one year, or one that would return $500,000 after two years?

Answer!

It depends on the interest rate!

The Role of Time Value in Finance

• Most financial decisions involve costs & benefits that are

spread out over time.

• Time value of money allows comparison of cash flows from

different periods.

Basic Concepts

• Future Value: compounding or growth over time

• Present Value: discounting to today’s value

• Single cash flows & series of cash flows can be

considered

• Time lines are used to illustrate these relationships

Computational Aids

• Use the Equations

• Use the Financial Tables

• Use Financial Calculators

• Use Spreadsheets

Computational Aids

Computational Aids

Computational Aids

Computational Aids

Simple Interest

• Year 1: 5% of $100 = $5 + $100 = $105

• Year 2: 5% of $100 = $5 + $105 = $110

• Year 3: 5% of $100 = $5 + $110 = $115

• Year 4: 5% of $100 = $5 + $115 = $120

• Year 5: 5% of $100 = $5 + $120 = $125

With simple interest, you don’t earn interest on interest.

Compound Interest

• Year 1: 5% of $100.00 = $5.00 + $100.00 = $105.00

• Year 2: 5% of $105.00 = $5.25 + $105.00 = $110.25

• Year 3: 5% of $110.25 = $5 .51+ $110.25 = $115.76

• Year 4: 5% of $115.76 = $5.79 + $115.76 = $121.55

• Year 5: 5% of $121.55 = $6.08 + $121.55 = $127.63

With compound interest, a depositor earns interest on interest!

Time Value Terms

• PV0 = present value or beginning amount

• k = interest rate

• FVn = future value at end of “n” periods

• n = number of compounding periods

• A = an annuity (series of equal payments or

receipts)

Four Basic Models

• FVn = PV0(1+k)n = PV(F/P,k,n)

• PV0 = FVn[1/(1+k)n] = FV(P/P,k,n)

• FVAn = A (1+k)n - 1 = A(F/A,k,n) k

• PVA0 = A 1 - [1/(1+k)n] = A(P/A,k,n)

kFV: future valuePV: present valueIF: interest factorA: annuity

Future Value Example

You deposit $2,000 today at 6% interest.

How much will you have in 5 years?

$2,000 x (1.06)5 = $2,000 x (F/P,6%,5) $2,000 x 1.3382 = $2,676.40

Algebraically and Using FVIF Tables

Future Value Example

You deposit $2,000 today at 6% interest.

How much will you have in 5 years?

Using Excel

PV 2,000$ k 6.00%n 5FV? $2,676

Excel Function

=FV (interest, periods, pmt, PV)

=FV (.06, 5, , 2000)

Future Value Example A Graphic View of Future Value

Compounding More Frequently than Annually

• Compounding more frequently than once a year results in a

higher effective interest rate because you are earning on

interest on interest more frequently.

• As a result, the effective interest rate is greater than the

nominal (annual) interest rate.

• Furthermore, the effective rate of interest will increase the

more frequently interest is compounded.

Compounding More Frequently than Annually

• For example, what would be the difference in future value if

I deposit $100 for 5 years and earn 12% annual interest

compounded (a) annually, (b) semiannually, (c) quarterly,

an (d) monthly?

Annually: 100 x (1 + .12)5 = $176.23

Semiannually: 100 x (1 + .06)10 = $179.09

Quarterly: 100 x (1 + .03)20 = $180.61

Monthly: 100 x (1 + .01)60 = $181.67FVn=PV0×(1+k/m)m×n

Compounding More Frequently than Annually

Annually SemiAnnually Quarterly Monthly

PV 100.00$ 100.00$ 100.00$ 100.00$

k 12.0% 0.06 0.03 0.01

n 5 10 20 60

FV $176.23 $179.08 $180.61 $181.67

On Excel

Continuous Compounding• With continuous compounding the number of compounding

periods per year approaches infinity.

• Through the use of calculus, the equation thus becomes:

FVn (continuous compounding) = PV x (ekxn)

where “e” has a value of 2.7183.

• Continuing with the previous example, find the Future value

of the $100 deposit after 5 years if interest is compounded

continuously.

kn

m

nmn ePV

m

kPVFV

00 )1(

Continuous Compounding• With continuous compounding the number of compounding

periods per year approaches infinity.

• Through the use of calculus, the equation thus becomes:

FVn (continuous compounding) = PV x (ekxn)

where “e” has a value of 2.7183.

FVn = 100 x (2.7183).12x5 = $182.22

Nominal & Effective Rates

• The nominal interest rate is the stated or contractual rate of

interest charged by a lender or promised by a borrower.

• The effective interest rate is the rate actually paid or earned.

• In general, the effective rate > nominal rate whenever

compounding occurs more than once per year

EAR = (1 + k/m) m -11

1)/1(

n nmmkEAR

Nominal & Effective Rates

• For example, what is the effective rate of interest on your

credit card if the nominal rate is 18% per year, compounded

monthly?

EAR = (1 + .18/12) 12 -1

EAR = 19.56%

Present Value• Present value is the current dollar value of a future amount

of money.

• It is based on the idea that a dollar today is worth more than

a dollar tomorrow.

• It is the amount today that must be invested at a given rate

to reach a future amount.

• It is also known as discounting, the reverse of

compounding.

• The discount rate is often also referred to as the opportunity

cost, the discount rate, the required return, and the cost of

capital.

Present Value Example

How much must you deposit today in order to have

$2,000 in 5 years if you can earn 6% interest on

your deposit?

$2,000 x [1/(1.06)5] = $2,000 x (P/F,6%,5) $2,000 x 0.74758 = $1,494.52

Algebraically and Using PVIF Tables

Present Value Example

How much must you deposit today in order to

have $2,000 in 5 years if you can earn 6% interest

on your deposit?

FV 2,000$ k 6.00%n 5PV? $1,495

Excel Function

=PV (interest, periods, pmt, FV)

=PV (.06, 5, , 2000)

Using Excel

Present Value Example A Graphic View of Present Value

Annuities• Annuities are equally-spaced cash flows of equal size.

• Annuities can be either inflows or outflows.

• An ordinary (deferred) annuity has cash flows that occur at the

end of each period.

• An annuity due has cash flows that occur at the beginning of

each period.

• The future value of an annuity due will always be greater than

the future value of an otherwise equivalent ordinary annuity

because interest will compound for an additional period.

Annuities

Future Value of an Ordinary Annuity

• Annuity = Equal Annual Series of Cash Flows

• Example: How much will your deposits grow to if you

deposit $100 at the end of each year at 5% interest for three

years.

FVA = 100(F/A,5%,3) = $315.25

Using the FVIFA Tables

0 1 2 3

100 100 100

100X1.05=105

100X(1.05)2=110.25

Future Value of an Ordinary Annuity

• Annuity = Equal Annual Series of Cash Flows

• Example: How much will your deposits grow to if you

deposit $100 at the end of each year at 5% interest for three

years.

Using Excel

PMT 100$ k 5.0%n 3FV? 315.25$

Excel Function

=FV (interest, periods, pmt, PV)

=FV (.05, 3,100, )

Future Value of an Annuity Due

• Annuity = Equal Annual Series of Cash Flows

• Example: How much will your deposits grow to if you deposit $100 at the

beginning of each year at 5% interest for three years.

FVA = 100(F/A,5%,3)(1+k) = $330.96

Using the FVIFA Tables

FVA = 100(3.152)(1.05) = $330.96

100 100 100

100*1.05=105

100*(1.05)2=110.25

100*(1.05)3=115.76

100 100 100

Future Value of an Annuity Due

• Annuity = Equal Annual Series of Cash Flows

• Example: How much will your deposits grow to if you

deposit $100 at the beginning of each year at 5% interest for

three years.

Using Excel

Excel Function

=FV (interest, periods, pmt, PV)

=FV (.05, 3,100, )x(1.05)

=315.25*(1.05)

PMT 100.00$ k 5.00%n 3FV $315.25FVA? 331.01$

Present Value of an Ordinary Annuity

• Annuity = Equal Annual Series of Cash Flows

• Example: How much could you borrow if you could afford

annual payments of $2,000 (which includes both principal

and interest) at the end of each year for three years at 10%

interest?

PVA = 2,000(P/A,10%,3) = $4,973.70

Using PVIFA Tables

2000 2000 2000

2000÷1.12000÷(1.1)2

2000÷(1.1)3

Present Value of an Ordinary Annuity

• Annuity = Equal Annual Series of Cash Flows

• Example: How much could you borrow if you could afford

annual payments of $2,000 (which includes both principal

and interest) at the end of each year for three years at 10%

interest?

Using Excel

PMT 2,000$ I 10.0%n 3PV? $4,973.70

Excel Function

=PV (interest, periods, pmt, FV)

=PV (.10, 3, 2000, )

Present Value of a Mixed Stream

• A mixed stream of cash flows reflects no particular pattern

• Find the present value of the following mixed stream

assuming a required return of 9%.

Using Tables

Year Cash Flow PVIF9%,N PV

1 400 0.917 366.80$

2 800 0.842 673.60$

3 500 0.772 386.00$

4 400 0.708 283.20$

5 300 0.650 195.00$

PV 1,904.60$

Present Value of a Mixed Stream

• A mixed stream of cash flows reflects no particular pattern

• Find the present value of the following mixed stream

assuming a required return of 9%.

Using EXCEL

Year Cash Flow

1 400

2 800

3 500

4 400

5 300

NPV $1,904.76

Excel Function

=NPV (interest, cells containing CFs)

=NPV (.09,B3:B7)

Present Value of a Perpetuity

• A perpetuity is a special kind of annuity.

• With a perpetuity, the periodic annuity or cash flow stream

continues forever.

PV = Annuity/k

• For example, how much would I have to deposit today in

order to withdraw $1,000 each year forever if I can earn 8%

on my deposit?

PV = $1,000/.08 = $12,500

…1000 1000 1000

………………

Loan Amortization

6000=Ax(P/A,10%,4)6000=Ax3.170 A=6000÷3.170=1892.74∴

Thanks for Your Attention