besley ch. 61 time value of money. besley ch. 62 cash flow time lines cf time lines are a graphical...

Besley Ch. 6 1

Time Value of Money

Besley Ch. 6 2

Cash Flow Time Lines

CF Time Lines are a graphical representation of cash flows associated with a particular financial option.

Time:0 1 2 3 4

One Period

5%Interest Rate (per period)

CF: -100 ?

+ indicates Cash Inflow- indicates Cash Outflow

NOTE: Each tick mark denotes the end of one period.

Besley Ch. 6 3

Cash Flow Time Lines

Outflow: A payment or disbursement of cash, such as for investment, or expenses.

Inflow: A receipt of cash, can be in the form of dividends, principal, annuity payments, etc.

Besley Ch. 6 4

Future Value (FV)

Future Value (FV): The ending value of a cash flow (or series of cash flows) over a given period of time, when compounded for a specified interest rate.

Compounding: The process of calculating the amount of interest earned on interest.

0 1 2 35%

-100 ?Present Value (PV) FV

Besley Ch. 6 5

FV Calculations

Given:PV: $100

i: 5%

n: 1

INT: (PV x i)

Solution:

FVn = PV+INT

= PV + (PV x i)

= PV(1+i)

0 1 2 35%

-100 ?

Solution:FVn = 100+INT

= 100 + (100 x 5%)= 100(1+ 0.05)= 105

Besley Ch. 6 6

FV Calculations

FV1 = PV(1+i)

FV2 = FV1(1+i) = [PV(1+i)](1+i)

FV3 = FV2(1+i) = {[PV(1+i)](1+i)}(1+i)

FVn = PV(1+i)n

0 1 2 35%

-100 ?INT1 INT2 INT35.00 5.25 5.51 =15.76

Value at end of Period: 105.00 110.25 115.76

Besley Ch. 6 7

FV Calculations

Three ways to calculate Time Value of Money (TVM) solutions:Numerical Solution:Calculate solution with formula

Tabular Solutions:Use Interest Factor tables to calculate

Financial Calculator Solutions:Use calculator

Besley Ch. 6 8

Numerical Solution

Future Value Interest Factor for i and n (FVIFi,n) is the factor by which the principal grows over a specified time period (n) and rate (i).

FVIFi,n = (1+i)n

Given: Solution:PV: $1 FVn = PV(1+i)n = PV(FVIFi.n)i: 5% FV5 = 1(1+.05)5

n: 5 FV5 = 1.2763

Besley Ch. 6 9

Tabular Solution

Period (n) 4% 5% 6%1 1.0400 1.0500 1.06002 1.0816 1.1025 1.12363 1.1249 1.1576 1.19104 1.1699 1.2155 1.2625

5 1.2167 1.2763 1.33826 1.2653 1.3401 1.4185

FVIFi,n = (1 + i)n

Given:i: 5%n: 5

FVn = PV(1+i)n = PV(FVIFi.n)

Besley Ch. 6 10

Financial Calculator

Points to remember when using your Financial Calculator: Check your settings:

END / BGN P/Y

Clear TVM memory Five Variables (N, I/Y, PV, PMT, FV) - with any 4 the 5th can be calculated

Given:N: 5

I/Y: 5%

PV: $1

PMT:0

FV: ?

Input:

Output:N I/Y PV PMT FV5 5 -1 0

1.2763

Besley Ch. 6 11

Present Value (PV)

Present Value (PV): The current value of a future cash flow (or series of cash flows), when discounted for a specified period of time an rate.

Discounting: The process of calculating the present value of a future cash flow or series of cash flows.

0 1 2 35%

? 105PV FV

Besley Ch. 6 12

PV Calculations

Given:FV: 105

i: 5%

n: 1

Solution:

FVn = PV(1+i)n Solve for PV

PVn = FVn / (1+i)n = FVn[1/(1+i)n]

= FVn(PVIFi,n)

0 1 2 35%

? 105

Solution:PVn = 105/(1+0.05)1

= 100

Besley Ch. 6 13

PV Calculations

FV2 = FV3/(1+i)

FV1 = FV2/(1+i) = [FV3/(1+i)]/(1+i)

PV = FV1/(1+i) = {[FV1/(1+i)]/(1+i)}/(1+i)

PVn = FVn

0 1 2 35%

?Value at end of Period:

1

(1+i)n

115.76251.05

110.251.051.05

105.00

Given FV

Besley Ch. 6 14

Numerical Solution

Present Value Interest Factor for i and n (PVIFi,n) is the discount factor applied to the FV in order to calculate the present value for a specific time period (n) and rate (i).

PVIFi,n = 1/(1+i)n

Given: Solution:FV: $1 PVn = FV[1/(1+i)n] =

FV(PVIFi.n)i: 5% PV5 = 1 [1/(1+.05)5]n: 5 PV5 = 0.7835

Besley Ch. 6 15

Tabular Solution

Period (n) 4% 5% 6%1 .9615 .9524 .94342 .9246 .9070 .89003 .8890 .8638 .83964 .8548 .8227 .7921

5 .8219 .7835 .74736 .7903 .7462 .7050

PVIFi,n = (1 + i)n

Given:i: 5%n: 5

PVn = FV[1/(1+i)n] = FV(PVIFi.n)

Besley Ch. 6 16


Given:N: 5

I/Y: 5%

PV: ?

PMT:0

FV: -1

Input:

Output:N I/Y PV PMT FV5 5

.7835

0 -1

Besley Ch. 6 17

Annuities

Annuity: a series of equal payments made at specific intervals for a specified period.

Types of Annuities:– Ordinary (Deferred) Annuity - is an annuity in which

the payments occur at the end of each period.

– Annuity Due - is an annuity in which the payments occur at the beginning of each period.

Besley Ch. 6 18

FV Ordinary Annuities

Example: You decide that starting a year from now you will deposit $1,000 each year in a savings account earning 8% interest per year. How much will you have after 4 years?

FVn=PV(1+i)n

FVAn = PMT(1+i)0 + PMT(1+i)1 + PMT(1+i)2 + . . . + PMT(1+i)n-1

0 1 2 3 4

1,000 1,000 1,000 1,000.001,080.001,166.401,259.714,506.11

8%

Besley Ch. 6 19


FVAn represents the future value of an ordinary

annuity over n periods.

FVAn = PMT(1+i)0 + PMT(1+i)1 + PMT(1+i)2 + . . . + PMT(1+i)n-1

= PMT (1+i)n-t = PMT (1+i)t

= PMT (1+i)n-1 = PMT

n

t=1n

t=1

n

t=1

(1+i)n - 1i

Besley Ch. 6 20


Future Value Interest Factor for an Annuity (FVIFAi,n) is the future value interest factor for an annuity (even series of cash flows) of n periods compounded at i percent.

FVIFAi,n = (1+i)n-t = (1+i)n - 1i

n

t=1

Besley Ch. 6 21

Numerical Solution

Given:PMT: $1,000I: 8%N: 4

0 1 2 3 4

1,000 1,000 1,000 1,000.00

8%

(1+i)n - 1iFVAn = PMT

Solution:FVAn = PMT {[(1+i)n – 1]/i}

= 1,000 {[(1+0.08)4 – 1]/0.08]

= 1,000 {4.5061} = $4,506.11

Besley Ch. 6 22

Tabular Solution

Given:PMT: $1,000I: 8%N: 4

FVAn = PMT(FVIFAi,n)

Period (n) 7% 8% 9% 1 1.0000 1.0000 1.0000 2 2.0700 2.0800 2.0900 3 3.2149 3.2464 3.2781 4 4.4399 4.5061 4.5731 5 5.7507 5.8666 5.9847

PVIFAi,n

Besley Ch. 6 23


Given:N: 4

I/Y: 8%

PV: 0

PMT:1,000

FV: ?

Input:


4,506.11

1,0000

Besley Ch. 6 24

FV Annuity Due

Example: You decide that starting today you will deposit $1,000 each year in a savings account earning 8% interest per year. How much will you have after 4 years?

FVn=PV(1+i)n

0 1 2 3 4

1,000 1,000 1,000 1,0001,166.401,259.711,360.494,866.60

8%

1,080.00

FVA(Due)n =PMT (1+i)t = PMT (1+i)n-t x (1+i)n

t=1

n

t=1

(1+i)n - 1iFVA(Due)n = PMT x (1+i)

Besley Ch. 6 25

FV Annuity Due

FVA(Due)n represents the future value of an

annuity due over n periods.

FVA(Due)n = PMT (1+i)tn

t=1

= PMT x (1+i)(1+i)n - 1i

n

t=1= PMT (1+i)n-t x (1+i)

Besley Ch. 6 26

FV Annuity Due

Future Value Interest Factor for an Annuity Due (FVIFA(Due)i,n) is the future value interest factor for an annuity due of n periods compounded at i percent.

FVIFA(Due)i,n = x (1+i)(1+i)n - 1i

Besley Ch. 6 27

Numerical Solution

Given:PMT: $1,000 - BGNI: 8%N: 4

0 1 2 3 4

1,000 1,000 1,0001,000

8%

Solution:FVA(Due)n = PMT [{((1+i)n – 1)/i}x (1+i)}

= 1,000 [{((1+0.08)4 – 1)/0.08}x (1+0.08)}

= 1,000 {4.8666} = $4,866.60

FVA(Due)i,n = PMT x (1+i)(1+i)n - 1i

Besley Ch. 6 28

Tabular Solution


FVA(Due)n = PMT[(FVIFAi,n)(1+i)]

Period (n) 7% 8% 9% 1 1.0000 1.0000 1.0000 2 2.0700 2.0800 2.0900 3 3.2149 3.2464 3.2781 4 4.4399 4.5061 4.5731 5 5.7507 5.8666 5.9847

PVIFAi,n

Besley Ch. 6 29


Given:N: 4

I/Y: 8%

PV: 0

PMT:1,000 - BGN

FV: ?

Input:


-4,866.60

1,0000BGN

Besley Ch. 6 30

PV Ordinary Annuities

Example: You decide that starting a year from now you will withdraw $1,000 each year for the next 4 years from a savings account which earns 8% interest per year. How much do you need to deposit today?

PVAn = PMT[1/(1+i)1] + PMT[1/(1+i)2] + . . . + PMT[1/(1+i)n]

The present value of an annuity is calculated by adding the PV of the individually discounted/compounded cash flows.

0 1 2 3 4

1,000 1,000 1,000 1,000

8%

(925.93)(857.34)(793.83)(735.03)

(3,312.13)

Besley Ch. 6 31


PVAn represents the present value of an ordinary

annuity over n periods.

PVAn = PMT[1/(1+i)1] + PMT[1/(1+i)2] + . . . + PMT[1/(1+i)n]

= PMT (1+i)t

= PMT

n

t=1

1

1 - (1+i)n

i

1

Besley Ch. 6 32


Present Value Interest Factor for an Annuity (PVIFAi,n) is the present value interest factor for an annuity (even series of cash flows) of n periods compounded at i percent.

PVIFAi,n = 1 - (1+i)n

i

1

Besley Ch. 6 33

Numerical Solution

Given:PMT: $1,000I: 8%N: 4

0 1 2 3 4

1,000 1,000 1,000 1,000

8%

Solution:PVAn = PMT {1-[1/(1+i)n]/i}

= 1,000 {1-[1/(1+0.08)4]/0,08}

= 1,000 {3.3121} = $3,312.13

PVAn = PMT 1 - (1+i)n

i

1

Besley Ch. 6 34

Tabular Solution

Given:PMT: $1,000I: 8%N: 4

PVAn = PMT(PVIFAi,n)PVIFAi,n

Periods 7% 8% 9%

1 0.9346 0.9259 0.9174

2 1.8080 1.7833 1.7591

3 2.6243 2.5771 2.5313

4 3.3872 3.3121 3.2397

5 4.1002 3.9927 3.8897

Besley Ch. 6 35


Given:N: 4

I/Y: 8%

PV: ?

PMT:1,000

FV: 0

Input:


-3,3121.13

1,000 0

Besley Ch. 6 36

PV Annuity Due

Example: You decide that starting today you will withdraw $1,000 each year for the next four years from a savings account earning 8% interest per year. How much do you need today?

0 1 2 3 4

1,000 1,000 1,000

8%

(925.93)(857.34)(793.83)

(1,000.00)

(3,577.10)

Besley Ch. 6 37

PV Annuity Due

PVA(Due)n represents the future value of an

annuity due over n periods.

PVA(Due)n = PMT n-1

t=0 (1+i)t

1

= PMT x (1+i)n

t=1 (1+i)t

1

= PMT x (1+i)1 - (1+i)n

i

1

Besley Ch. 6 38

PV Annuity Due

Present Value Interest Factor for an Annuity Due (PVIFA(Due)i,n) is the present value interest factor for an annuity due of n periods compounded at i percent.

PVIFA(Due)i,n= PMT x (1+i)1 - (1+i)n

i

1

Besley Ch. 6 39

Numerical Solution


0 1 2 3 4

1,000 1,000 1,0001,000

8%

Solution:PVA(Due)n = PMT [{(1-1/(1+i)n)/i}x (1+i)]

= 1,000 [{(1-1/(1+0.08)4)/0.08}x (1+0.08)]

= 1,000 {3.5771} = $3,577.10

PVIFA(Due)i,n= PMT x (1+i)1 - (1+i)n

i

1

Besley Ch. 6 40

Tabular Solution


PVA(Due)n = PMT[(PVIFAi,n)(1+i)]PVIFAi,n

Periods 7% 8% 9%

1 0.9346 0.9259 0.9174

2 1.8080 1.7833 1.7591

3 2.6243 2.5771 2.5313

4 3.3872 3.3121 3.2397

5 4.1002 3.9927 3.8897

Besley Ch. 6 41


Given:N: 4

I/Y: 8%

PV: ?

PMT:1,000 - BGN

FV: 0

Input:


-3,577.10

1,000 0BGN

Besley Ch. 6 42

Solving for Interest Rates with Annuities

PVAn=PMT(PVIFAi,n)

-3,239.72 = 1,000(PVIFAi,n)

-3.2397 = PVIFAi,n

Numerical Solution:Trial & Error Solve for PVIFA

0 1 2 3 4

1,000 1,000 1,000 1,000

?%

-3,239.72

PVIFAi,n = 1 - (1+i)n

i

1

Besley Ch. 6 43


Tabular Solution:

-3.2397 = PVIFAi,n

0 1 2 3 4

1,000 1,000 1,000 1,000

?%

-3,239.72

PVIFAi,n

Periods 7% 8% 9%

1 0.9346 0.9259 0.9174

2 1.8080 1.7833 1.7591

3 2.6243 2.5771 2.5313

4 3.3872 3.3121 3.2397

5 4.1002 3.9927 3.8897

Besley Ch. 6 44


Financial Calculator:N: 4

I/Y: ?

PV: -3,239.72

PMT:1,000

FV: 0

0 1 2 3 4

1,000 1,000 1,000 1,000

?%

-3,239.72

Input:

Output:N I/Y PV PMT FV4

9

-3, 239.72 1,000 0

Besley Ch. 6 45

Perpetuities

Perpetuity: A perpetual annuity, an annuity which continues forever.

Consol A perpetual bond issued by the British government where the proceeds were used to consolidate past debts.

PVP = PMT / i

Besley Ch. 6 46

Perpetuities

PVA5%,100 = $19,848

PVA5% of $1000

$20,000

$15,372

$12,462

$7,722

$952$4,329

$0

$5,000

$10,000

$15,000

$20,000

$25,000

1 2 3 4 5 6 7 8 9 10 15 20 30 Perp

Besley Ch. 6 47

Uneven Cash Flow Streams0 1 2 3 4

250 750 750

8%

(231.48)(643.00)(595.37)

(2,204.88)

1,000

(735.03)PVn = FV[1/(1+i)n] = FV(PVIFi.n)

Besley Ch. 6 48

Semiannual and Other Compounding Periods

Simple Interest Rate: The interest rate used to compute the interest rate per period; the quoted interest rate is always in annual terms.

Effective Annual Rate (EAR): The actual interest rate being earned during a year when compounded interest is considered.

Besley Ch. 6 49


Types of Compounding:– Annual Compounding– Semiannual Compounding (Bonds)– Quarterly (Stock Dividends)– Daily (Bank Accounts/Credit Cards)

EAR Formula

EAR = 1+ -1isimplem

m

Besley Ch. 6 50


Annual Percentage Rate (APR): the periodic rate multiplied by the number of period per year.

Besley Ch. 6 51

Fractional Time Periods

Use current formulas and convert time (n) into a fraction.

Besley Ch. 6 52

Amortized Loans

Amortized Loan: a loan that is repaid in equal payments (an annuity) over the life of the loan.

Amortization Schedule: A financial schedule illustrating each payment in the loan, and further breaking that down between principal and interest.

besley ch. 61 time value of money. besley ch. 62 cash flow time lines cf time lines are a graphical...

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