financial mgt chapter#3 time value of money

7/30/2019 Financial Mgt Chapter#3 Time Value of Money

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Time Value of Money



The Time Value of Money

Which would you prefer -- $1000 today or $1000 in 5 years?

Obviously, $1000 today.

Money received sooner rather than later

allows one to use the funds for investmentor consumption purposes. This conceptis referred to as the TIME VALUE OF MONEY!!



The Time Value of Money

A Dollar today is worth more than a dollar tomorrow, because if you invest it todayyou can earn interest on it.

This concept is known as a TIME VALUEOF MONEY.



TIME allows you the opportunity to postponeconsumption and earn INTEREST.

Why TIME?

Why is TIME such an important element inyour decision?



What is Interest?

Money paid (earned) for the use of moneyis called INTEREST.

Interest is Expense for borrower andRevenue for lender.

Interest is the rental charge for

funds/money, just as rentalcharges are made for the use ofbuildings and equipment.



Types of Interest

For example, Jawad gave 500 AFs to Asad for 3 year at a rate of 9%.

Here, Principal or original= 500 AFs Rate of interest = 9%

Simple Interest

Interest paid (earned) on only the originalamount, or principal, borrowed (lent).



Simple Interest Formula

Formula SI = PV(i)(n)

SI: Simple InterestPV: Deposit today (Present Value)

i: Interest Rate per Period

n: Number of Time Periods



What is n?

As already noted, the number of timeperiods in a time value problem isrepresented by n.

n may be a number of years

n may be a number of months

n may be a number of quartersn may be a number of any defined time

periods



What is i?

The interest rate in a time value problem isrepresented by i

i must be expressed as the interest rateper period.

For example if n is a number of years, imust be the interest rate per year.

If n is a number of months, i must be theinterest rate per month.



PV = $1,000, i = 7% or .07, n= 2

SI = ?

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

= $140

Simple Interest Example

Assume that you deposit $1,000 in an account earning7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year?



Problems of Accumulated interest

Question 1:

Assume that you deposit $5,000 in an accountearning 10% simple interest. What is the

accumulated interest at the end of the 5 th

, 8 th

and 10 th year?

Question 2:

Assume that you deposit $15,000, 18,000 and

12,000 in an account earning 12% simpleinterest. What is the accumulated interest at the end of the 7 th year?



What is Future Value?

Future Value is the value at some futuretime of a present amount of money, or aseries of payments, evaluated at a

given interest rate.Or

A sum of money at a future time is termed

as a Future Value.FV2 = $5000 means $5000 after 2 periods

from now



PV = $1000 i = 7% n = 2 years

FV 2 = ?

SI = PV(i)(n)= 1000(.07)(2)=140

FV 2 = P V + SI = $1,000 + $140

= $1,140

Simple Interest (FV)

Assume that you deposit $1,000 in an account earning 7% simple interestfor 2 years. What is the Future Value (FV) of the deposit?



Future Value formula for Simple Interest

FVn = PV + SI

FVn = PV+ PV (i) (n)

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

FVn means future value at the end of period n from today.



Future value Problems of Simple Interest

Q1. Find Future values of $1000 Depositat the end of year 3rd , 5th , 8th , and 10th at simple interest rate of 8 Percent.

Q2. Find future values of $1500, $10,000,$12,000 and $15000 at the end of year 3rd at simple interest rate of 5 Percent.



What is Present Value?

Present Value is the current value of afuture amount of money, or a series of payments, evaluated at a given interestrate.

Or

A sum of money today is called a presentvalue.



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

Simple Interest (PV)

Today you deposit $1000 in a bank offer the interest rateof 10%. What is the Present Value (PV) of this deposit?



Present Value Formula for SimpleInterest

PV=FV/(1+(i)(n))



Present value Problem of Simple Interest

Assume that you need $1,000 in 2 years.Let’s examine the process to determine

how much you need to deposit today at a

simple interest rate of 7%.

FV2 = $1000

i = 7%

n = 2

PV = ?



Solution of Present value problem

Using present value simple interestformula

PV = FVn/(1+(i)(n))

Put the values

PV = 1000/(1+(0.07)(2))

= $ 877.19



Present value Problems of SimpleInterest

Question number 1:

What will be the present value of Afs 10,000, Afs12,000 and Afs 15,000 received in 5 years when

the simple interest rate is 10%. Question number 2:

What will be the present value of Afs 10,000

received in 3 years when the simple interest rateis 10%, 12%, 8% and 5%.



Class Activity

Question number 1:

Assume that you deposit $5,000 in an account earning9% simple interest for 8 years. What is the accumulated interest at the end of the 8 th year?

Question number 2: Find future value of $3500 at the end of year 6th at

simple interest rate of 12% Percent.

Question number 3:

What will be the present value of Afs 15,000 received in4 years when the simple interest rate is 14%.



Class Activity

Question number 1:

Find future value of $25,000 at the end of year 9th at simple interest rate of 15%Percent.

Question number 2:

What will be the present value of Afs50,000 received in 8 years when thesimple interest rate is 10%.



Formula for simple interest rate

i = [FVn/PV – 1]/n



What is Simple Interest Rate?

If you invest $8,000 in a mutual fundtoday, and it grows to be $17,000 after 8years, what is the Simple interest rate that

grow $8,000 to $17,000 in 8 years.

PV = $8000

FV8 = $17000

n = 8

i = ?



Solution:

Formula for Simple Interest Rate

i = [FVn/PV – 1]/n

Put the values i = [17000/8000 – 1]/8

= [2.125 – 1]/8

=(1.125)/8

= 0.14

=14% (approximately)



Numerical related to Simple interest rate

Question number 1:

If you deposit $14,000 in a Bank today, and itgrows to be $25,000 after 10 years, what is the

Simple interest rate that grow $14,000 to$25,000 in 10 years.

Question number 2:

If you deposit $4,000 in a Bank today, and it

grows to be $9,000 after 6 years, what is theSimple interest rate that grow $4,000 to $9,000in 6 years.



Formula for number of periods

n = [FVn/PV – 1]/i



Number of periods in Simple interest

How long will it take for $10,000 to grow to$20,000 at a Simple interest rate of 15%per year?

PV = $10,000

FVn = $20,000

i = 15% or 0.15

n = ?



Solution:

Formula for Number of periods in simpleinterest

n = [FVn/PV – 1]/In = [20000/10000 -1]/0.15

= [2 – 1]/0.15

= 1/0.15 = 6.7 years (approximately)



Numerical related to Simple interest rate

Question number 1:


Question number 2:




Class Activity

Question number 1:

If you deposit $11,000 in a Bank today,and it grows to be $21,000 after 12 years,

what is the Simple interest rate that grow$11,000 to $21,000 in 12 years

Question number 2:




Class Activity

Question number 1:

If you deposit $20,000 in a Bank today,and it grows to be $28,000 after 6 years,

what is the Simple interest rate that grow$20,000 to $28,000 in 6 years

Question number 2:




Compound Interest

Interest paid (earned) on any previousinterest earned, as well as on theprincipal borrowed (lent).



Assume that you deposit $1,000 at acompound interest rate of 7% for 2 years.

Future ValueSingle Deposit (Graphic)

0 1 2

$1,000

FV2

7%



FV1 = PV (1+i)1 = $1,000 (1.07)= $1,070

Compound InterestYou earned $70 interest on your $1,000 depositover the first year.

This is the same amount of interest you wouldearn under simple interest.

Future ValueSingle Deposit (Formula)



FV1 = PV (1+i)1 = $1,000 (1.07)= $1,070

FV2 = FV1 (1+i)1 = PV (1+i)(1+i) = $1,000(1.07)(1.07)= PV (1+i)2 = $1,000(1.07)2

= $1,144.90 You earned an EXTRA $4.90 in Year 2 with compound

over simple interest. This extra $4.90 is on previous70 interest earned for the 1st year.

Future Value

Single Deposit (Formula)



FV1 = PV(1+i)1

FV2 = PV(1+i)2

General Future Value Formula:

FVn = PV(1+i)n

or FVn = PV(FVIFi,n) -- See Table I

General Future ValueFormula



FVIFi,n is found on Table I

at the end of the book.

Valuation Using Table I

Period 6% 7% 8%

1 1.060 1.070 1.080

2 1.124 1.145 1.166

3 1.191 1.225 1.260

4 1.262 1.311 1.360

5 1.338 1.403 1.469



FV2 = $1,000 (FVIF7%,2)= $1,000 (1.145)

= $1,145 [Due to Rounding]

Using Future Value Tables

Period 6% 7% 8%

1 1.060 1.070 1.080

2 1.124 1.145 1.166

3 1.191 1.225 1.260

4 1.262 1.311 1.360

5 1.338 1.403 1.469



Future Value Single Deposit (Formula)

What is the future value (FV) of an initial $100 after 1st , 2nd and 3rd years,if compound interest rate is 10% per year?

After 1 year:

FV1 = PV ( 1 + i ) = $100 (1.10)= $110.00

After 2 years:

FV2 = PV ( 1 + i )2 = $100 (1.10)2

=$121.00 After 3 years:

FV3 = PV ( 1 + i )3 = $100 (1.10)3

=$133.10

After n years (general case):FVn = PV ( 1 + i )n



Julie Miller wants to know how large her deposit of $10,000

today will become at a compound annual interest rate of 10% for 5 years.

Story Problem Example

0 1 2 3 4 5

$10,000

FV5

10%



Calculation based on Table I:

FV5 = $10,000 (FVIF10%, 5) = $10,000 (1.611)= $16,110 [Due to Rounding ]

Story Problem Solution

Calculation based on general formula: FVn = PV(1+i)n

FV5 = $10,000 (1+ 0.10)5

= $16,105.10



Future value Problem

If you invest $1,000 today at an interestrate of 10 percent, how much will it grow tobe after 5 years?

FVn = P V(1+i)n

FV5

= 1,000(1.10)5

= $1,610.51



Future value Problem

If you invested $2,000 today in anaccount that pays 6% interest, withinterest compounded annually, howmuch will be in the account at the end of two years if there are no withdrawals?

Solution:

FVn = PV (1+i)n FV2 = $2,000 (1.06)2

FV2 = $2,247.20



Assume that you need $1,000 in 2 years. Let’sexamine the process to determine how much youneed to deposit today at a discount rate of 7%compounded annually.

0 1 2

$1,000

7%

PV1PV0

Present ValueSingle Deposit (Graphic)



PV = FV2 / (1+i)2 = $1,000 / (1.07)2

= $873.44

Present ValueSingle Deposit (Formula)

0 1 2

$1,000

7%

PV0



PV = FV1 / (1+i)1

PV = FV2 / (1+i)2

General Present Value Formula:

PV = FVn / (1+i)n

or PV = FVn (PVIFi,n) -- See Table II

General Present ValueFormula

etc.



PVIFi,n is found on Table II

at the end of the book.

Valuation Using Table II

Period 6% 7% 8%

1 .943 .935 .926

2 .890 .873 .857

3 .840 .816 .794

4 .792 .763 .735

5 .747 .713 .681



PV2 = $1,000 (PVIF7%,2)= $1,000 (.873)

= $873 [Due to Rounding]

Using Present Value Tables

Period 6% 7% 8%

1 .943 .935 .926

2 .890 .873 .857

3 .840 .816 .794

4 .792 .763 .735

5 .747 .713 .681



Julie Miller wants to know how large of a deposit tomake so that the money will grow to $10,000 in 5years at a discount rate of 10%.

Story Problem Example

0 1 2 3 4 5

$10,000

PV

10%



Calculation based on general formula:PV = FVn / (1+i)n PV = $10,000 / (1+ 0.10)5

= $6,209.21

Calculation based on Table I:PV = $10,000 (PVIF10%, 5)

= $10,000 (.621)= $6,210.00 [Due to Rounding ]

Story Problem Solution



Class Activity

Question number 1: If you invest $8,000 today at a compound annual interest

rate of 12 percent, how much will it grow to be after 6years?

Question number 2:How long does it take to double $75,00 at a compoundrate of 9% per year (approx.) “use the Rule of 72”?

Question number 3:

Julie Miller wants to know how large of a deposit to makeso that the money will grow to $15,000 in 9 years at adiscount rate of 14%.



Class Activity

Question number 1: If you invest $12,000 today at a compound annual

interest rate of 15 percent, how much will it grow to beafter 20 years?

Question number 2: Assume that you need $50,000 in 25 years. How much

you need to deposit today at a discount rate of 18%compounded annually.

F l f R t f i t t d



Formula for Rate of interest under compound interest rate

i = [FVn/PV]1/n - 1

N i l l t d t C d i t t



Numerical related to Compound interestrate

Numerical number 1: If you invest $11,000 in Bonds today, and it

grows to be $50,000 after 8 years, whatcompounded, annualized rate of return did youearn?

Numerical number 2:

If you invest $5,500 in a Business today, and it

grows to be $14,000 after 9 years, whatcompounded, annualized rate of return did youearn?



Numerical number 3: If you invest $7,000 in a mutual fund today, and

it grows to be $13,000 after 9 years, what is theCompound interest rate that grow $7,000 to$13,000 in 9 years.

PV = $7000

FV9 = $13000

n = 9

i = ?



Class Activity

Question number 1: If you invest $35,000 today at a compound interest rate

of 24 percent, how much will it grow to be after 30years?

Question number 2: Julie Miller wants to know how large of a deposit to make

so that the money will grow to $42,000 in 16 years at adiscount rate of 20%.

Question number 3:

If you invest $12,000 in a mutual fund today, and itgrows to be $33,000 after 13 years, what is theCompound interest rate that grow $12,000 to $33,000 in13 years.



We will use the “Rule-of-72”.

Double Your Money!!!

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

(approx.)?



Approx. Years to Double = 72 / i%

72 / 12% = 6 Years [Actual Time is 6.12 Years]

The “Rule-of-72”

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

(approx.)?

F l f N b f P i d d



Formula for Number of Periods under compound interest rate

n = log[FVn/PV]/log[1 + i]

N b f i d i C d i t t



Number of periods in Compound interestRate

Numerical number 1:

How long will it take for $12,000 to grow to$25,000 at a Compound interest rate of

11% per year?

PV = $12,000

FVn = $25,000

i = 11% or 0.11

n = ?



Numerical number 2:

How long will it take for $8,500 to grow to$19,000 at a Compound interest rate of

16% per year?

Numerical number 3:

How long will it take for $13,000 to grow to$22,000 at a Compound interest rate of 10% per year?



Class Activity

Numerical number 1:

If you invest $15,000 in a Bonds today, and it grows tobe $50,000 after 15 years, what compounded,annualized rate of return did you earn?

Numerical number 2:

How long does it take to double $8,000 at a compoundrate of 8% per year (approx.) “use the Rule of 72”.

Numerical number 3:

How long will it take for $18,000 to grow to $27,000 at aCompound interest rate of 14% per year?



Class Activity

Numerical number 1:

If you invest $19,000 in a Bonds today, and it grows tobe $43,000 after 12 years, what compounded,annualized rate of return did you earn?

Numerical number 2:

How long does it take to double $12,000 at a compoundrate of 18% per year (approx.) “use the Rule of 72”.

Numerical number 3:

How long will it take for $14,000 to grow to $34,000 at aCompound interest rate of 9% per year?



Annuity

An Annuity represents a series of equalpayments (or receipts) occurring over a

specified number of equidistant periods.Types of Annuity

i. Ordinary Annuity

ii. Annuity Due

O di A it P t i t



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

0 1 2 3

$100 $100 $100

(Ordinary Annuity)End of

Period 1

End of

Period 2

Today Equal Cash FlowsEach 1 Period Apart

End of

Period 3

A it D P t i t t



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

0 1 2 3

$100 $100 $100

(Annuity Due)Beginning of

Period 1

Beginning of

Period 2

Today Equal Cash FlowsEach 1 Period Apart

Beginning of

Period 3



Examples of Annuities

Student Loan Payments

Car Loan Payments

Insurance Premiums

Retirement Savings

Example of an



FVA3 = $1,000(1.07)2

+$1,000(1.07)1 + $1,000(1.07)0

= $1,145 + $1,070 + $1,000 = $3,215

Example of anOrdinary Annuity -- FVA

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

0 1 2 3 4

$3,215 = FVA3

7%

$1,070

$1,145

Cash flows occur at the end of the period

Formula for Future value of Ordinary Annuity



Formula for Future value of Ordinary Annuity

FVAn = R[(1+i)n – 1]/i

FVAn: Future value of annuity at the end of period n

R: the periodic Receipts or Payments

i: Rate of interest per period

n: Number of time periods

Or

FVAn = R(FVIFAi%,n)


FVIFAi%,n: Future value interest factor of annuity at

interest rate i% and it the end of period “n”

Future Value Ordinary Annuity



Future Value Ordinary AnnuityProblem

If one saves $1,000 a year at the end of every year for three years in an accountearning 7% interest, compounded annually,

how much will one have at the end of thethird year?

R = $1000

n = 3 years i = 7%

FVA3 = ?

Solution of Future Value Ordinary



Solution of Future Value Ordinary Annuity by using formula

Formula:

FVAn = R [(1+i)n – 1]/i

Put the values

FVA3 = 1000[(1+ 0.07)3 – 1]/0.07

= 1000[1.225043 – 1]/0.07

= 1000[0.225043]/0.07 = 1000(3.2149)

= $3,214.9



Future Value Ordinary Annuity



Future Value Ordinary AnnuityProblem

Assume that Sally owns an investmentthat will pay her $400 at the end of eachyear for 20 years. The current interest rate

is 15%. What is the FV of this annuity?



Class Activity

Assume that Karim owns an investmentthat will pay him $1,500 at the end of eachyear for 16 years. The current interest rate

is 14%. What is the FV of this annuity?

Example of an



FVAD3 = $1,000(1.07)3

+$1,000(1.07)2 + $1,000(1.07)1

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

Example of anAnnuity Due -- FVAD

$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

Formula for Future value of Annuity Due



Formula for Future value of Annuity Due

FVADn = R[(1+i)n – 1][i+i]/i

FVADn: Future value of annuity Due at the end of period n



n: Number of time periods Or

FVAn = R(FVIFAi%,n)(1+i)


FVIFAi%,n: Future value interest factor of annuity at interestrate i% and it the end of period “n”



Future Value Annuity Due Problem

If one saves $1,000 a year at the beginningof every year for three years in an accountearning 7% interest, compounded annually,

how much will one have at the end of thethird year?

R = $1000

n = 3 years

i = 7%

FVAD3 = ?

Solution of Future value Annuity Due



Solution of Future value Annuity Dueby using formula

Formula:

FVADn = R [(1+i)n – 1][1+i]/i

Put the values

FVAD3 = 1000[(1+ 0.07)3 – 1][1+.07]/0.07

= 1000[1.225043 – 1][1.07]/0.07

= 1000[0.225043][1.07]/0.07 = 1000(3.2149)(1.07)

= $3,439.943

Solution of Future value Annuity Due



FVADn= R (FVIFAi%,n)(1+i)

FVAD3= $1,000 (FVIFA7%,3)(1+.07)= $1,000 (3.2149)(1.07) = $ $3,439.943

Period 6% 7% 8%

1 1.000 1.000 1.000

2 2.060 2.070 2.080

3 3.184 3.2149 3.246

4 4.375 4.440 4.506

5 5.637 5.751 5.867

Solution of Future value Annuity Dueby using Table



Hint on Annuity Valuation

The future value of an ordinaryannuity can be viewed as

occurring at the end of the lastcash flow period, whereas thefuture value of an annuity due

can be viewed as occurring atthe beginning of the last cashflow period.

F t l A it D P bl



Future value Annuity Due Problem

Assume that Sally owns an investmentthat will pay her $400 at the Beginning of each year for 20 years. The current

interest rate is 15%. What is the FV of thisannuity?

Example of an



PVA3

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

= $934.58 + $873.44 + $816.30= $2,624.32

Example of anOrdinary Annuity -- PVA

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

0 1 2 3 4

$2,624.32 = PVA3

7%

$934.58$873.44$816.30

Cash flows occur at the end of the period

Formula for Present value of OrdinaryA it



Annuity

PVA = R[(1 - [1/(1+i)n]/i PVA Present value of ordinary



n: Number of time periods

Or

PVA = R(PVIFAi%,n)


PVIFAi%,n: Present value interest factor of annuity at interest rate i%

and it the end of period “n”

Present Value Ordinary Annuity



Present Value Ordinary AnnuityProblem

If one agrees to repay a loan by paying$1,000 a year at the end of every year for three years and the discount rate is 7%,

how much could one borrow today? R = $1000

n = 3 years

i = 7%

PVA = ?

Solution of Present value ordinary



Solution of Present value ordinaryannuity by using formula

PVA = R[(1 - [1/(1+i)n]/i

= 1000[1 – [1/(1 + .07)3]/0.07

= 1000[ 1 – [(1/1.225043)]/0.07

= 1000[ 1 – 0.8163]/0.07

= 1000(2.624)

= $2,624



Present value ordinary annuity



y yProblem

Assume that Sally owns an investmentthat will pay her $100 at the end of eachyear for 20 years. The current interest rate

is 15%. What is the PV of this annuity?

Example of an



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

Example of anAnnuity Due -- PVAD

$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

P t V l A it D P bl



Present Value Annuity Due Problem

If one agrees to repay a loan by paying$1,000 a year at the Beginning of everyyear for three years and the discount rate

is 7%, how much could one borrow today? R = $1000

n = 3 years

i = 7%

PVAD = ?

Solution of Present value Annuity



yDue by using formula

PVA = R[(1 - [1/(1+i)n][1 + i]/i = 1000[1 – [1/(1 + .07)3][1 + 0.07]/0.07

= 1000[ 1 – [(1/1.225043)][1.07]/0.07

= 1000[ 1 – 0.8163][1.07]/0.07

= 1000(2.624)(1.07)

= $2,808

Solution of Present value annuity



PVADn = R (PVIFAi%,n)(1+i)PVAD3 = $1,000 (PVIFA7%,3)(1.07)

= $1,000 (2.624)(1.07) = $2,808

yDue by using Table

Period 6% 7% 8%

1 0.943 0.935 0.926

2 1.833 1.808 1.783

3 2.673 2.624 2.577

4 3.465 3.387 3.312

5 4.212 4.100 3.993



Hint on Annuity Valuation

The present value of an ordinaryannuity can be viewed as occurring

at the beginning of the first cashflow period, whereas the futurevalue of an annuity due can be

viewed as occurring at the end of the first cash flow period.

Present value annuity Due Problem



Present value annuity Due Problem

Assume that Sally owns an investmentthat will pay her $400 at the beginning of each year for 20 years. The current

interest rate is 15%. What is the PV of thisannuity?

Future and Present Value of Ordinary



y Annuity

Question 1: Assume that Asad owns an investment that will

pay him $3,000 at the end of each year for 8

years. The current interest rate is 18%. What isthe FV of this annuity?

Question 2:

Assume that Wais owns an investment that will

pay him $4,500 at the end of each year for 12years. The current interest rate is 20%. What isthe PV of this annuity?

Future and Present Value of Annuity



yDue

Question 1: Assume that Ahmad owns an investment that

will pay him $1,400 at the Beginning of each

year for 16 years. The current interest rate is24%. What is the FV of this annuity?

Question 2:

Assume that Jawad owns an investment that will

pay him $3,300 at the beginning of each year for 9 years. The current interest rate is 16%. Whatis the PV of this annuity?



CLASS ACTIVITY

Future and Present Value of Ordinary



y Annuity

Question 1: Assume that Wali owns an investment that will

pay him $1,800 at the end of each year for 12


Question 2:

Assume that Akram owns an investment that will

pay him $2,700 at the end of each year for 15years. The current interest rate is 18%. What isthe PV of this annuity?

Future and Present Value of Annuity



Due

Question 1: Assume that Ali owns an investment that will pay

him $2,500 at the Beginning of each year for 13


Question 2:

Assume that Fahad owns an investment that will

pay him $4,200 at the beginning of each year for 16 years. The current interest rate is 8%. Whatis the PV of this annuity?

financial mgt chapter#3 time value of money

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