1

Chapter 4

Time Value of MoneyPrepared by the student

Mahmoud y. Al- Saftawi

6

DB

A defined benefit pension plan is a major type of pension plan in which an employer/sponsor promises a specified monthly benefit on retirement that is predetermined by a formula based on the employee's earnings history, tenure of service and age, rather than depending directly on individual investmentreturns. It is 'defined' in the sense that the benefit formula is defined and known in advance. Conversely, for a "defined contribution pension plan", the formula for computing the employer's and employee's contributions is defined and known in advance, but the benefit to be paid out is not known in advance .

DB

The most common type of formula used is based on the employee’s terminal earnings (final salary). Under this formula, benefits are based on a percentage of average earnings during a specified number of years at the end of a worker’s career .

.

In the private sector, defined benefit plans are often funded exclusively by employer contributions. For very small companies with one owner and a handful of younger employees, the business owner generally receives a high percentage of the benefits. In the public sector, defined benefit plans usually require employee contributions.

Over time, these plans may face deficits or surpluses between the money currently in their plans and the total amount of their pension obligations. Contributions may be made by the employee, the employer, or both. In many defined benefit plans the employer bears the investment risk and can benefit from surpluses.

DC

a defined contribution plan is a type of employer's annual contribution is specified. Individual accounts are set up for participants and benefits are based on the amounts credited to these accounts (through employer contributions and, if applicable, employee contributions) plus any investment earnings on the money in the account. Only employer contributions to the account are guaranteed, not the future benefits. In defined contribution plans, future benefits fluctuate on the basis of investment earnings. The most common type of defined contribution plan is a savings and thrift plan. Under this type of plan, the employee contributes a predetermined portion of his or her earnings (usually pretax) to an individual account

A 401(k)

A 401(k) is a type of retirement savings account in the U.S., which takes its name from subsection 401(k) of the Internal Revenue Code (Title 26 of the United States Code). 401(k) are "defined contribution plans" with annual contributions limited, currently, to $17,500. Contributions are tax-deferred ,deducted from paychecks before taxes and then taxed when a withdrawal is made from the 401(k) account. Depending on the employer's program a portion of the employee's contribution may be matched by the employer.

What is Time Value?

We say that money has a time value because that money can be invested with the expectation of earning a positive rate of return

In other words, “a dollar received today is worth more than a dollar to be received tomorrow”

TVM

Time value of money quantifies the value of a dollar through time

Which would you prefer -- $10,000 today or

$10,000 in 5 years?

Obviously, $10,000 today.

You already recognize that there is

TIME VALUE TO MONEY!!

Why TIME?

Why is TIME such an important element in your decision?

TIME allows you the opportunity to postpone consumption and earn INTEREST.

For present need.

For re-investment purpose.

Future uncertainties.

Time Value of Money

The time value of money is the value of money figuring in a given amount of interestearned or inflation accrued over a given amount of time. The ultimate principle suggests that a certain amount of money today has different buying power than the same amount of money in the future. This notion exists both because there is an opportunity to earn interest on the money and because inflation will drive prices up, thus changing the "value" of the money. The time value of money is the central concept in finance theory

TVM

Uses of Time Value of Money

Time Value of Money, or TVM, is a concept that is

used in all aspects of finance including:

Bond valuation.

Stock valuation.

Accept/reject decisions for project management.

retirement planning.

loan payment schedules.

decisions to invest (or not) in new equipment.

And many others

Time Value Topics

Future value

Present value

Rates of return

Amortization

The Terminology of Time Value

Present Value - An amount of money today, or the current value of a future cash flow

Future Value - An amount of money at some future time period

Period - A length of time (often a year, but can be a month, week, day, hour, etc.)

Interest Rate - The compensation paid to a lender (or saver) for the use of funds expressed as a percentage for a period (normally expressed as an annual rate)

Methods of time value of money

Compounding techniques.

Discounting techniques

Types of Interest

Simple Interest

Interest paid (earned) on only the original amount, or principal.

Compound Interest

Interest paid (earned) on any previous interest earned, as well as on the principal borrowed .

when interest is earned on the interest earned in prior periods, we call it compound interest. If interest is earned only on the principal, we call it simpleinterest.

Simple Interest Formula

Formula SI = P0(i)(N)

SI: Simple Interest

P0: principal (t=0)

i: Interest Rate per Period

n: Number of Time Periods

Simple Interest Example

Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year?

SI = P0(i)(n)= $1,000(.07)(2)= $140

Simple Interest (FV)

What is the Future Value (FV) of the deposit?

FV = P0 + SI = $1,000 + $140= $1,140

Future Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate.

Simple Interest (PV)

What is the Present Value (PV) of the previous problem?

The Present Value is simply the $1,000 you originally deposited. That is the value today!

Present Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate.

Why Compound Interest?

0

5000

10000

15000

20000

1st Year 10th

Year

20th

Year

30th

Year

Future Value of a Single $1,000 Deposit

10% SimpleInterest

7% CompoundInterest

10% CompoundInterest

Fu

ture

Va

lue

(U

.S.

Do

lla

rs)

Types of TVM Calculations

There are many types of TVM calculations

The basic types will be covered in this review module and include:

Present value of a lump sum

Future value of a lump sum

Present and future value of cash flow streams

Present and future value of annuities

Types of TVM Calculations

Present value The current worth of a future sum of money or stream of cash flows given a specified rate of return. Future cash flows are discounted at the discount rate, and the higher the discount rate, the lower the present value of the future cash flows. Determining the appropriate discount rate is the key to properly valuing future cash flows, whether they be earnings or obligations.

Present value of an annuity An annuity is a series of equal payments or receipts that occur at evenly spaced intervals. Leases and rental payments are examples. The payments or receipts occur at the end of each period for an ordinary annuity while they occur at the beginning of each period for an annuity due.

Types of TVM Calculations

Present value of a perpetuity is an infinite and constant stream of identical cash flows.

Future value is the value of an asset or cash at a specified date in the future that is equivalent in value to a specified sum today.

Future value of an annuity (FVA) is the future value of a stream of payments (annuity), assuming the payments are invested at a given rate of interest.

Timelines

v A timeline is a diagram used to clarify the timing of the cash flows for an investment

v Each tick represents one time period

PV FV

Today

0 1 2 3 4 5

Time lines show timing of cash flows.

TIME LINES

CF0 CF1 CF3CF2

0 1 2 3i%

Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.

TIME LINES

The intervals from 0 to 1, 1 to 2, and 2 to 3 are time periods such as years

or months. Time 0 is today, and it is the beginning of Period 1; Time 1 is one

period from today, and it is both the end of Period 1 and the beginning of

Period 2; and so on. they could also

be quarters or months or even days.

.

We can use four different procedures to solve time value problems.

Step-by-Step Approach. TIME LINES

Formula Approach.

Financial Calculators.

Spreadsheets

Future Value

A dollar in hand today is worth more than a dollar to be received in the future—if

you had the dollar now you could invest it, earn interest, and end up with more

than one dollar in the future. The process of going forward, from present values

(PVs) to future values (FVs), is calledcompounding.

Future Value of a Lump Sum

You can think of future value as the opposite of present value

Future value determines the amount that a sum of money invested today will grow to in a given period of time

The process of finding a future value is called “compounding” (hint: it gets larger)

Example of FV of a Lump Sum

How much money will you have in 5 years if you invest $100 today at a 10% rate of return?

1. Draw a timeline

$100

0 1 2 3 4 5

i = 10%?

.

2. Write out the formula using symbols:

FV = PV * (1+i)N

3. Substitute the numbers into the formula:

FV = $100 * (1+0.1)5

4. Solve for the future value:

FV = $161.05

FV of an initial $100 after3 years (i = 10%)

.

FV = ?

0 1 2 3

10%

Finding FVs (moving to the righton a time line) is called compounding.

100

After 1 year

FV1= PV(1 + i)

= $100(1.10)

= $110.00

After 2 years

FV2=

= PV(1+i)

= $100(1.10)

= $121.00

After 3 years

FV3=

= PV(1+i)

= $100(1.10)

= $133.10

In general,

FVN = PV(1 +i)N

Growth of $100 at Various Interest Rates and Time Periods

Present Value

Finding present values is called discounting, and as previously noted, it is the reverse

of compounding: If you know the PV, you can compound to find the FV; or if you know

the FV, you can discount to find the PV.

Present Value of a Lump Sum

Present value calculations determine what the value of a cash flow received in the future would be worth today (time 0)

The process of finding a present value is called “discounting” (hint: it gets smaller)

The interest rate used to discount cash flows is generally called the discount rate

Example of PV of a Lump Sum

How much would $100 received five years from now be worth today if the current interest rate is 10%?

1. Draw a timeline

The arrow represents the flow of money and the numbers under the timeline represent the time period.

Note that time period zero is today

0 1 2 3 4 5

$100?i = 10%

.

2. Write out the formula using symbols:

PV = FV / (1+i)N

3. Insert the appropriate numbers:

PV = 100 / (1 +0 .1)5

4. Solve the formula:

PV = $62.09

What’s the PV of $100 due in 3 years if I/YR = 10%?

.

10%

Finding PVs is discounting, and it’s the reverse of compounding.

100

0 1 2 3

PV = ?

48

Present Value of $1 at Various Interest Rates and Time Periods

49

Finding the interest rate (i)

?%

2

0 1 2 3

-1FV = PV(1 + i)N

$2 = $1(1 + i)3

(2)(1/3) = (1 + i)1.2599 = (1 + i)

i = 0.2599 = 25.99%

50

20%

2

0 1 2 ?

-1FV = PV(1 + i)N

Continued on next slide

Finding the Time to DoubleFinding the number of years (N)

51

Finding the Time to Double Finding the number of years (N)

$2 = $1(1 + 0.20)N

2 = ( 1.20 )( 2 ) = ( 1.20 )

1 log 2 = N log 1.20

للطرفين log 1.20بالقسمة على

N =1 log 2 /log 1.20=3.8

We can working with natural logs

N

N

Double Your Money!!!

.

We will use the “Rule-of-72”.

Quick! How long does it take to double $5,000 at a compound rate of 12% per

year (approx.)?

The “Rule-of-72”

Quick! How long does it take to double $5,000 at a compound rate of 12% per

year (approx.)?

Approx. Years to Double = 72 / i%

72 / 12% = 6 Years

[Actual Time is 6.12 Years]

annuities

An annuity is a cash flow stream in which the cash flows are all equal and occur at regular intervals.

such as bonds provide a series of cash inflows over time, and obligations such as auto

loans, student loans, and mortgages call for a series of payments. If the payments

are equal and are made at fixed intervals, then we have an annuity

annuities

If payments occur at the end of each period, then we have an ordinary (or deferred) annuity.Payments on mortgages, car loans, and student loans are generally made at the ends of the periods and thus are ordinary annuities. If the payments are made at the beginning of each period, then we have an annuity due. Rental lease payments, life insurance premiums, and lottery payoffs (if you are lucky enough to win one!) are examples of annuities due. Ordinary annuities are more common in finance, so when we use the term “annuity” in this book, you may assume that the payments occur at the ends of the periods unless we state otherwise.

Types of Annuities

u An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods.

Ordinary Annuity: Payments or receipts occur at the end of each period.

Annuity Due: Payments or receipts occur at the beginning of each period.

Examples of Annuities

Student Loan Payments

Car Loan Payments

Insurance Premiums

Mortgage Payments

Retirement Savings

Parts of an Annuity

0 1 2 3

$100 $100 $100

End of

Period 2

Today Equal Cash Flows

Each 1 Period Apart

End of

Period 3

(Ordinary Annuity)

End of

Period 1

Parts of an Annuity

.

0 1 2 3

$100 $100 $100

(Annuity Due)

Beginning of

Period 1

Beginning of

Period 2

Today Equal Cash Flows

Each 1 Period Apart

Beginning of

Period 3

60

What’s the FV of a 3-year ordinary annuity of $100 at 10%?

100 100100

0 1 2 310%

110121

FV = 331

61

FV ordinary (deferred) annuity

Formula

The future value of an annuity with N periods and an interest rate of r can be found with the following formula:

= PMT(1+i)N -1

i

= $100(1+0.10)3 -1

0.10= $331

FUTURE VALUE OF AN ANNUITY DUE

Because each payment occurs one period earlier with an annuity due, the payments

will all earn interest for one additional period. Therefore, the FV of an annuity due

will be greater than that of a similar ordinary annuity.

FUTURE VALUE OF ANANNUITY DUE FORMULA

FVAD =

PMT * [ (1 + i ) – 1 ] * (1 + i )

ــــــــــــــــــــــــــــ

i

N

Example of anAnnuity Due -- FVAD

.

FVAD3 = $1,000(1.07)3 + $1,000(1.07)2 + $1,000(1.07)1

= $1,225 + $1,145 + $1,070 = $3,440

$1,000 $1,000 $1,000 $1,070

0 1 2 3 4

$3,440 = FVAD3

7%

$1,225

$1,145

Cash flows occur at the beginning of the period

65

What’s the PV of this ordinary annuity?

100 100100

0 1 2 310%

90.91

82.64

75.13

248.69 = PV

66

PV ordinary annuity? Formula

The present value of an annuity with N periods and an interest rate of I can be found with the following formula:

PMT * [ 1 - (1/1+i) ]ـــــــــــــــــــــــــــــ

i

N

Present Value of Annuities Due

Because each payment for an annuity due occurs one period earlier, the payments

will all be discounted for one less period. Therefore, the PV of an annuity due must

be greater than that of a similar ordinary annuity.

PRESENT VALUE OF ANANNUITY DUE FORMULA

PVAD =

PMT * [ 1 - (1/1+i) ] * (1+i)

ـــــــــــــــــــــــــــــ

i

N

Example of anAnnuity Due -- PVAD

……….

PVADn = $1,000/(1.07)0 + $1,000/(1.07)1 + $1,000/(1.07)2 = $2,808.02

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

0 1 2 3 4

$2,808.02 = PVADn

7%

$ 934.58

$ 873.44

Cash flows occur at the beginning of the period

PERPETUITIES

some securities promise to make payments forever. For example, in the mid-1700s the British government issued some bonds that never matured and whose proceeds were used to pay off other British bonds. Since this action consolidated the government’s debt, the new bonds were called “consols. ”The term stuck, and now any bond that promises to pay interest perpetually is called a consol ,or a perpetuity. The interest rate on the consols was 2.5%, so a consol with a face value of $1,000 would pay $25per year in perpetuity

.

perpetuity

A consol, or perpetuity, is simply an annuity whose promised payments extend out forever. Since the payments go on forever, you can’t apply the step-by-step approach. However, it’s easy to find the PV of a perpetuity with the following

formula:

PV of a perpetuity =

PMT ـــــــــــــــــــــــ

I

UNEVEN, OR IRREGULAR,CASHFLOWS

What is the PV of this uneven cash flow stream?

.

0

100

1

300

2

300

310%

-50

4

90.91

247.93

225.39

-34.15

530.08 = PV

PRESENT VALUE OF AN UNEVEN CASH FLOW STREAM FORMULA

Mixed Flows Example

Julie Miller will receive the set of cash flows below. What is the Present Value at a discount rate of 10%.

0 1 2 3 4 5

$600 $600 $400 $400 $100

PV0

10%

How to Solve?

1. Solve a “piece-at-a-time” by discounting each piece back to t=0.

2. Solve a “group-at-a-time” by firstbreaking problem into groups ofannuity streams and any singlecash flow groups. Then discount each group back to t=0.

“Piece-At-A-Time”

0 1 2 3 4 5

$600 $600 $400 $400 $100

10%

$545.45

$495.87

$300.53

$273.21

$ 62.09

$1677.15 = PV0 of the Mixed Flow

“Group-At-A-Time” (#1)

.

0 1 2 3 4 5

$600 $600 $400 $400 $100

10%

$1,041.60

$ 573.57

$ 62.10

$1,677.27 = PV0 of Mixed Flow [Using Tables]

$600(PVIFA10%,2) = $600(1.736) = $1,041.60

$400(PVIFA10%,2)(PVIF10%,2) = $400(1.736)(0.826) = $573.57

$100 (PVIF10%,5) = $100 (0.621) = $62.10

“Group-At-A-Time” (#2)

.

0 1 2 3 4

$400 $400 $400 $400

PV0 equals

$1677.30.

0 1 2

$200 $200

0 1 2 3 4 5

$100

$1,268.00

$347.20

$62.10

Plus

Plus

FUTURE VALUE OF AN UNEVEN CASH FLOW STREAM

The future value of an uneven cash flow stream (sometimes called the terminal, or horizon, value) is found by compounding each payment to the end of the stream and then summing the future values

FUTURE VALUE OF AN UNEVEN CASH FLOW STREAM FORMULA

Steps to Solve Time Value of Money Problems

1. Read problem thoroughly

2. Create a time line

3. Put cash flows and arrows on time line

4. Determine if it is a PV or FV problem

5. Determine if solution involves a single CF, annuity stream (s), or mixed flow

6. Solve the problem

SEMIANNUAL AND OTHER COMPOUNDING PERIODS

In most of our examples thus far, we assumed that interest is compounded once a year, or annually. This is annual compounding. Suppose, however, that you put $1,000 into a bank that pays a 6% annual interest rate but credits interest each 6 months. This is semiannual compounding

.

quarterly compounding.

monthly compounding.

weekly compounding.

daily compounding.

.

bonds pay interest semiannually; most stockspay dividends quarterly; most mortgages, student loans, and auto loans

involve monthly payments; and most money fund accounts pay interest daily. Therefore,

it is essential that you understand how to deal with non annual compounding.

.

N * (M) COMPOUNDING PERIOD PER YEAR

I / (M) COMPOUNDING PERIOD PER YEAR

Types of Interest Rates

When we move beyond annual compounding, we must deal with the following four

types of interest rates:

• Nominal annual rates, given the symbol INOM

• Annual percentage rates, termed APR rates

• Periodic rates, denoted as IPER

• Effective annual rates, given the symbol EAR or EFF% (or Equivalent) Annual Rate

88

Nominal rate (INOM)

Stated in contracts, and quoted by banks and brokers.

Not used in calculations or shown on time lines

Periods per year (M) must be given.

Examples:

8%; Quarterly

8%, Daily interest (365 days)

NOTE

Note that the nominal rate is never shown on a time line, and it is never used as an input in a financial calculator (except when compounding occurs only once a year). If more frequent compounding occurs, you must use periodic rates

90

Periodic rate (IPER )

IPER = INOM /M, where M is number of compounding

periods per year. M = 4 for quarterly, 12 for monthly,

and 360 or 365 for daily compounding.

Used in calculations, shown on time lines.

Examples:

8% quarterly: IPER = 8%/4 = 2%.

8% daily (365): IPER = 8%/365 = 0.021918%.

91

The Impact of Compounding

Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated I% constant?

Why?

92

The Impact of Compounding (Answer)

LARGER!

If compounding is more frequent than once a year--for example, semiannually, quarterly, or daily--interest is earned on interest more often.

93

FV Formula with Different Compounding Periods

General Formula:

FVn = PV0(1 + [i/m])mn

n: Number of Yearsm: Compounding Periods per Year

i: Annual Interest RateFVn,m: FV at the end of Year n

PV0: PV of the Cash Flow today

94

$100 at a 12% nominal rate with

semiannual compounding for 5 years

= $100(1.06)10 = $179.08

INOMFVN = PV 1 +M

M N

0.12FV5S = $100 1 +

2

2x5

95

FV of $100 at a 12% nominal rate for 5 years with different compounding

FV(Ann.) = $100(1.12)5 = $176.23

FV(Semi.) = $100(1.06)10 = $179.08

FV(Quar.) = $100(1.03)20 = $180.61

FV(Mon.) = $100(1.01)60 = $181.67

FV(Daily) = $100(1+(0.12/365))(5x365) = $182.19

FV Formula with Different Compounding Periods

Julie Miller has $1,000 to invest for 2 Years at an annual interest rate of 12%.

Annual FV2 = 1,000(1+ [.12/1])(1)(2)

= 1,254.40

Semi FV2 = 1,000(1+ [.12/2])(2)(2)

= 1,262.48

.

Qrtly FV2 = 1,000(1+ [.12/4])(4)(2)

= 1,266.77

Monthly FV2 = 1,000(1+ [.12/12])(12)(2)

= 1,269.73

Daily FV2 = 1,000(1+[.12/365])(365)(2)

= 1,271.20

98

Effective Annual Rate (EAR = EFF%)

The EAR is the annual rate that causes PV to grow to the same FV as under multi-period compounding.

99

Effective Annual Rate Example

Example: Invest $1 for one year at 12%, semiannual:

FV = PV(1 + INOM/M)M

FV = $1 (1.06)2 = $1.1236.

EFF% = 12.36%, because $1 invested for one year at 12% semiannual compounding would grow to the same value as $1 invested for one year at 12.36% annual compounding.

100

Comparing Rates

An investment with monthly payments is different from one with quarterly payments. Must put on EFF% basis to compare rates of return. Use EFF% only for comparisons.

Banks say “interest paid daily.” Same as compounded daily.

101

EFF% = 1 + − 1INOM

M

M

EFF% for a nominal rate of 12%, compounded semiannually

= 1 + − 10.12

2

2

= (1.06)2 - 1.0= 0.1236 = 12.36%.

102

Can the effective rate ever be equal to the nominal rate?

Yes, but only if annual compounding is used, i.e., if M = 1.

If M > 1, EFF% will always be greater than the nominal rate.

103

When is each rate used?

INOM: Written into contracts, quoted by banks and brokers. Not used in calculations or shownon time lines.

104

IPER: Used in calculations, shown on time lines.

If INOM has annual compounding,then IPER = INOM/1 = INOM.

When is each rate used? (Continued)

105

When is each rate used? (Continued)

EAR (or EFF%): Used to compare returns on investments with different payments per year.

Used for calculations if and only if dealing with annuities where payments don’t match interest compounding periods.

FRACTIONAL TIME PERIODS

For example, suppose you deposited $100in a bank that pays a nominal rate of 10%, compounded daily, based on a 365-day year. How much would you have after 9months? The answer of $107.79 is found as follows:

FRACTIONAL TIME PERIODS

Now suppose that instead you borrow $100 at a nominal rate of 10% per year,

simple interest, which means that interest is not earned on interest. If the loan is out-standing for 274 days (or 9 months), how much interest would you have to pay? The interest owed is equal to the principal multiplied by the interest rate times the number of periods. In this case, the number of periods is equal to a fraction of a year:

N = 274/365 = 0.7506849.

Interest owed = $100(10%)(0.7506849) = $7.51

Amortization

Steps to Amortizing a Loan

1.Calculate the payment per period.

2.Determine the interest in Period t.(Loan Balance at t-1) x (i% / m)

3.Compute principal payment in Period t. (Payment - Interest from Step 2)

4.Determine ending balance in Period t. (Balance - principal payment from Step 3)

5.Start again at Step 2 and repeat.

109

Amortization

For example,suppose a company borrows $100,000, with the loan to be repaid in 5 equal payments at the end of each of the next 5 years. The lender charges 6% on the balance at the beginning of each year. Here’s a picture of the situation:

It is possible to solve the annuity formula

PV= PMT * [ 1 - (1/1+i) ]ـــــــــــــــــــــــــــــ

i

100000 = PMT * [ 1 - (1/1+0.06) ] = $23,739.64ـــــــــــــــــــــــــــــ

0.06

N

112

Amortization tables are widely used--for home mortgages, auto loans, business loans, retirementplans, and more. They are very important!

Financial calculators (and spreadsheets) are great for setting up amortization tables.

113

Non-matching rates and periods

What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semiannually?

114

Time line for non-matching rates and periods

0 1

100

2 35%

4 5 6 6-mos.periods

100 100

115

Non-matching rates and periods

Payments occur annually, butcompounding occurs each 6 months.

So we can’t use normal annuity valuation techniques.

116

1st Method: Compound Each CF

0 1

100

2 35%

4 5 6

100 100.00110.25121.55331.80

FVA3 = $100(1.05)4 + $100(1.05)2 + $100= $331.80

117

2nd Method: Treat as an annuity, use financial calculator

Find the EFF% (EAR) for the quoted rate:

EFF% = 1 + − 1 = 10.25%0.10

2

2

.

PV= PMT

(1+i)N -1

i

= $100(1+0.1025)3 -1

0.1025

= $331.8

119

What’s the PV of this stream?

0

100

15%

2 3

100 100

90.7082.2774.62

247.59

.

100 / 1.05 = 90.7

100 / 1.05 = 82.27

100 / 1.05 = 74.626

4

2

121

Comparing Investments

You are offered a note that pays $1,000 in 15 months (or 456 days) for $850. You have $850 in a bank that pays a 6.76649% nominal rate, with 365 daily compounding, which is a daily rate of 0.018538% and an EAR of 7.0%. You plan to leave the money in the bank if you don’t buy the note. The note is riskless.

Should you buy it?

122

IPER = 0.018538% per day.

1,000

0 365 456 days

-850

Daily time line

… …

123

Three solution methods

1. Greatest future wealth: FV

2. Greatest wealth today: PV

3. Highest rate of return: EFF%

124

1. Greatest Future Wealth

Find FV of $850 left in bank for15 months and compare withnote’s FV = $1,000.

FVBank = $850(1.00018538)456

= $924.97 in bank.

Buy the note: $1,000 > $924.97.

125

Find PV of note, and compare

with its $850 cost:

PV = $1,000/(1.00018538)456

= $918.95

Buy the note: $918.95 > $850

2. Greatest Present Wealth

126

Find the EFF% on note and compare with 7.0% bank pays, which is your opportunity cost of capital:

FVN = PV(1 + I)N

$1,000 = $850(1 + I)456

Now we must solve for I.

3. Rate of Return

.

لطرفي المعادلة850بالقسمة على

1.176 = (1 + I )

( 1.176 ) = 1 + I

1.000355 = 1 + I

I = 0.0355 * 365

= 13.01 %

456

456

128

P/YR =365NOM% =0.035646(365) = 13.01%

EFF% = [ 1 + (0.1301 /365 ) – 1= 13.89 %

Since 13.89% > 7.0% opportunity cost,buy the note.

Using interest conversion

365