chapter 5 & 6 time value of money. basic principle : a dollar received today is worth more than...

CHAPTER 5 & 6CHAPTER 5 & 6TIME VALUE OF MONEYTIME VALUE OF MONEY

Basic Principle : A dollar received today is worth more than a dollar received in the future.

• This is due to opportunity costs. The opportunity cost of receiving $1 in the future is the interest we could have earned if we had received the $1 sooner.

• This concept is so important in understanding financial management (investment, stock & bond valuation etc..)

Time Value Of Money

2SITI AISHAH BINTI KASSIM (FM2)

Translate $1 today into its equivalent in the future (compounding) – Future Value

SITI AISHAH BINTI KASSIM (FM2)3

If we can measure this opportunity cost, we can:

Today

Future

?

Translate $1 in the future into its equivalent today (discounting)- Present ValueToda

yFuture

?

Compound interest occurs when interest paid on the investment during the first period is added to the principal; then, during the second period, interest is earned on this new sum.

• Compounding is the process of determining the Future Value (FV) of cash flow.

• The compounded amount (Future Value) is equal to the beginning amount plus interest earned.


Future Value (FV)Future Value (FV)

For example : If we place RM 1000 in a savings account paying 5% interest compounded annually. How much will it be worth at the end of each year ?

• Year 1 = RM1000 (1.05) = RM1050.00• Year 2 = RM1050.00 (1.05) = RM1102.50• Year 3 = RM1102.50 (1.05) = RM1157.63• Year 4 = RM1157.63 (1.05) = RM 1215.51• etc……



RM 1000

0 4321n..

5%

5%

5%

5%

5%

Formula of Future Value (FV) :

FVn = PV (1+i)n or FVn = PV (FVIFi,n)

where;FVn = the FV of the investment at the end of n year n = the number of years i = the annual interest rate PV = original amount invested at beginning of the

first year**(1+i) is also known as compounding factor.



For example : If we place RM1,000 in a savings account paying 5% interest compounded annually. How much will our account accrue in 4 years? PV=RM1,000, i =5% & n=4 years.



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

(FVIFi,n)

FV4=1,000 (1+0.05)4 FV4 = PV

(FVIF5%,4)

=1,000 (1.2155) =1,000

(1.2155)

=RM1,215.50

=RM1,215.50

If Adam invests RM10,000 in a bank where it will earn 6% interest compounded annually. How much will it be worth at the end of a) 1

year and b) 5 years?

Compounded for 1 yeara) FV1= $10000 (1+0.06)1 b) FV1 = PV (FVIF6%,1)

= $10000 (1.06)1 = $10000 (1.0600)

= $10,600.00 = $10,600.00

Compounded for 5 yeara) FV5 = $10000 (1+0.06)5 b) FV5 = PV (FVIF6%,5)

= $10000 (1.06)5 = $10000 (1.3382)

= $13,380.00 = $13,382.00


Example:

Non-annual periods : not annual compounding but occurs semiannually, quarterly, monthly…

If compounding semiannually : FV = PV (1 + i/2)n x 2 or FVn = PV (FVIFi/2,nx2)

If compounding quarterly :• FV = PV (1 + i/4)n x 4 or FVn = PV (FVIFi/4,nx4)

If compounding monthly :• FV = PV (1 + i/12)n x 12 or FVn = PV (FVIFi/12,nx12)


Compound Interest With Non-annual Periods

Present value is the current value of futures sum Finding Present Values (PVs) is called discounting

We can calculate PV by using this equation :

• PV = FVn or PV = FVn (PVIFi,n)

(1+i )n

**[ 1/(1+i)n ] is also known as discounting factor.


PRESENT VALUE (PV)PRESENT VALUE (PV)

For example : What is the PV of $800 to be received 10 years from today if our discount rate is

10%.

PV = 800/(1.10)10 = $308.43

or

PV = $800 (PVIF 10%,10yrs) = $800 (0.3855) = $308.40


PRESENT VALUE (PV)PRESENT VALUE (PV)

In every single sum future value and present value problem, there are 4 variables: FV, PV, i, and n

When doing problems, you will be given 3 of these variables and asked to solve for the 4th variable.

Keeping this in mind makes “time value” problems much easier!


Hint for single sum Hint for single sum problems:problems:

For example: What is the PV of an investment that yields $300 to be received in 2 years and $450 to be received in 8 years if the discount rate is 5%?

PV = $300(PVIF5%,2) + $450(PVIF5%,8)

= $300(0.907) + $450(0.677) = 272.10 + 304.65 = $576.75


PV with Multiple, Uneven Cash Flows

An annuity is a series of equal payments for a specified numbers of years.

There are 2 types of annuities*:- ordinary annuity

- annuity due

*in finance, ordinary annuities are used much more frequently than are annuities due


ANNUITY

100

100

100

100

100

0 31 2 4

Ordinary Annuity is an annuity which the payments occur at the end of each period.

a) Present Value of Annuity (PVA) Present Value of Annuity (PVA) can be calculated by using these equations:

PVAn = PMT / (1+i)n or PVAn = PMT (PVIFAi,n)

For example: Find the PV of $500 received at the end of each year of the next 3 years discounted back to the present at 10%?


Ordinary Annuity

a) PVA3 = (500/1.10)1 + (500/1.10)2 + (500/1.10)3

= 454.55 + 413.22 + 375.66 = $ 1243.43

OR

b) PVA3 = 500 (PVIFA10%,3) = 500 (2.487) = $ 1243.50

Solutions:Solutions:


Ordinary Annuity

b) Future Value of Annuity (FVA) Compound Annuity / Future Value of Annuity (FVA) can be calculated by using these equations :

FVAn = PMT (1+ i)n or FVAn = PMT (FVIFAi,n)

For example : We are going to deposit $15,000 at the end of each year for the next 5 years in a bank where it will earn 9% interest. How much will we get at the end of 5 years?


Ordinary Annuity

a) At the end of 5 years, we will get…FVA5 = 15000(1.09)4 + 15000(1.09)3

+15000(1.09)2 + 15000(1.09)1 +15000 = 21173.72 +19425.44 + 17821.50 + 16350 + 15000

= $89,770.66OR

b) By using FVIFA table FVA5 = 15000 (FVIFA9%,5)

= 15000 (5.9847) = $89,770.50


Ordinary AnnuitySolutions:Solutions:

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

a) Future Value of Annuity Due (FVAD)

FVADn = PMT (FVIFAi,n) (1+i)

b) Present Value of Annuity Due (PVAD)

PVADn = PMT (PVIFAi,n) (1+i)

Annuity Due


For example (FVAD): We are going to deposit $1,000 at the beginning of each year for the next 5 years in a bank where it will earn 5% interest. How much will we get at the end of 5 years?

FVADn = PMT (FVIFAi,n) (1+i) = 1000 (5.526) (1 + 0.05) = $5802.30


Annuity Due

For example (PVAD) : Find the PV of $500 received at the beginning of each year of the next 5 years discounted back to the present at 6%?

PVADn = 500 (PVIFA6%,5) (1+0.06)

= 500 (4.212) (1 + 0.06) = $2,232.36


Annuity Due

Amortized loan is a loan that paid off in equal installments.

To determine the installment (payment) we can use this formula :

PMT = Loans PVIFAi,n

Each installment consists partly of interest and partly of repayment of principal. This breakdown is given in the amortization schedule.

Amortized Loan


For example: Daniel wants to accumulate RM75,000 by the end of five (5) years. Assume that the fund will earn an interest at 9.5% compounded annually.

PMT = $75000/ PVIFA9.5%,5

= $10000 / 6.0446 = $ 12,407.73


Amortized Loan

YearBeginning Balance

(1)

Annual Deposit

(2)

Interest Generated

(3)

Accumulated Amount

(1)+(2)+(3)=(4)

1 0 12,407.73 0 12,407.73

2 12,407.73 12,407.73 1,178.74 25,994.20

3 25,994.20 12,407.73 2,469.45 40,871.38

4 40,871.38 12,407.73 3,882.78 57,161.89

5 57,161.89 12,407.73 5,430.38 75,000

Loan Amortization Loan Amortization ScheduleSchedule


Perpetuity is an annuity that continues forever.

The equation representing the present value of annuity:

PV = PP i

where, PV= PV of the perpetuity PP= Constant dollar amount provided by perpetuity

i = interest rate

Perpetuity


26

SITI AISHAH BINTI KASSIM (FM2)

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chapter 5 & 6 time value of money. basic principle : a dollar received today is worth more than...

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