1

The Time Value of Money

Miss Faith Moono Simwami Mo.simwami@gmail.com

1 - 2

•Understand what is meant by "the time value of money.“

•Understand the relationship between present and future value.

•Describe how the interest rate can be used to adjust the value of cash flows – both forward and backward – to a single point in time.

•Calculate both the future and present value of: (a) an amount invested today; (b) a stream of equal cash flows (an annuity); and (c) a stream of mixed cash flows.

•Distinguish between an “ordinary annuity” and an “annuity due.”

•Use interest factor tables and understand how they provide a shortcut to calculating present and future values.

•Build an “amortization schedule” for an installment-style loan.

Why Study Time Value of Money?

Money has a time value because it can earn more money over time (earning power).

Money has a time value because its purchasing power changes over time (inflation).

Time value of money is measured in terms of interest rate.

Interest is the cost of money—a cost to the borrower and an earning to the lender

The Interest Rate

Simple Interest

Compound Interest

Amortizing a Loan

Compounding More

Than Once per Year

1 - 4

Time value of money has to do with trying to determine the value of cash flows expected in the future. It is essential to note that the value of a dollar today is worth more than a dollar received in the future.

This is due to the element of interest. To store the value of money interest is charged to compensate for the opportunity cost or deferred consumption. Hence in money terms a dollar in the future is equal to a dollar today plus interest earned. By including the element of interest it is possible to compound and discount values.

The Time Value of Money

The Interest Rate

5

Obviously, $10,000 today. You already recognize that there is TIME VALUE TO MONEY!!

Which would you prefer – $10,000 today or $10,000 in 5 years?

Why TIME?

Why is TIME such an important element in your decision?

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

Compound Interest Simple Interest

1 - 6

Interest is the amount we pay to borrow money, or the amount we are paid for money that we lend.

Simple interest means that the

amount paid is the same for each time period of the loan, regardless of whether any money has been repaid or not.

Simple interest is Interest paid (earned) on only the original amount, or principal, borrowed (lent). Simple Interest - Interest earned only on the original investment.

Types of Interest

The most common form of interest is compound interest.

It is called compound interest because the interest accumulated each year is added to the principal, and for each subsequent year interest is earned on this total of principal and interest. The interest thus compounds.

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

Simple Interest Example1

Suppose we invest $5000 at 10% per annum simple interest for 4 years. This means that each year we will earn 10% interest, or $500:

So we would be earning $500 every year the money remained invested, or

$2000:

5005000100

10

200045000100

10

We can use that previous calculation to derive a formula for simple interest:

200045000100

10

This is the interest rate per annum, we will give it the symbol r (rate)

r

P

This is the number of years for which we invested. We will give it the symbol t (time)

t

This is the total interest earned. We will give it the symbol I (interest)

I

This is the amount invested. We will give it the symbol P (principal)

If we graph the amount of the investment against the time for which the money has been invested (t) then we can see that the relationship between these variables is linear.

time in years

121086420

Am

ount

($)

11000

10000

9000

8000

7000

6000

5000

This is an important property of simple interest. Since the amount of increase is constant each year, the growth in the investment is linear.

To determine how much we will have in total (FV) we need to add this interest to the amount of the original investment(PV):

100

PrtPVFV

How do we calculate Future Value (FV) for a lump sum of Money using simple interest?

Simple Interest Example2

11

SI = P(R)(T) =$1,000(0.07)(2) = $140

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?

Simple Interest (FV)

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

FV = Principal + Simple Interest = $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.

Year Initial Amount Interest Final Amount

1 5000

2 5500

3 6050

4 6655

Suppose we invest $5000 at 10% per annum compound interest for 4 years. We can determine the growth of the investment as follows:

How do we calculate Compound Interest?

5005000100

10

5505500100

10

6056050100

10

665.506655100

10

5500

6050

6655

7320.50

How do we calculate Compound Interest for a Future Value (FV) for a lump sum of Money?

trPVFV )1(

There is a formula we can use to calculate compound interest without having to work it out year by year as in the previous table. That is:

This is the compound interest rate per annum.

This is the total amount of the investment after t years.

(The Future Value)

This is the number of years for which we invested the money.

This is the amount initially invested-

(The Present Value)

How do we calculate Compound Interest for a Future Value (FV) for a lump sum of Money? (Using Tables)

t)r, (FVIF PVFV

(1+r)t or (FVIF r,t) from the tables. This is also known as a compound factor.

This is the compound interest rate per annum.

This is the total amount of the investment after t years.

(The Future Value)

This is the number of years for which we invested the money.

This is the amount initially invested-

(The Present Value)

time in years

121086420

Am

ount

(S)

10000

8000

6000

4000

2000

0

If we graph the amount of the investment against the time for which the money has been invested (t) then we can see that the relationship between these variables is not linear.

This is an important property of compound interest. The amount of increase is increasing each year, so the growth in the investment is non linear.

time in years

121086420

Am

ount

($)

10000

8000

6000

4000

2000

0

Type of interest

compound interest

simple interest

To compare simple interest with compound interest we can graph the amount of the investment against year for both types of interest on the same graph.

It is clear that after the first year, the compound interest is a better investment, and the difference between amounts under simple and compound interest increases with time.

Difference between compound interest and simple interest after nine years

Difference between compound interest and simple interest after five years

How do we calculate the Present Value (PV) for a lump sum of Money?

rFVPV

t

1

1 =

There is a formula we can use to calculate the Present Value (PV) for a lump sum of Money.

This is the compound interest rate per annum.

This is the amount initially invested-

(The Present Value)

This is the number of years for which we invested the money.

This is the total amount of the investment after t years.

(The Future Value)

How do we calculate Present Value (PV) for a lump sum of Money? (Using Tables)

t)r, (PVIF FVPV

1/(1+r)t or (PVIF r,t) from the tables. This is also known as a Discount factor.

This is the compound interest rate per annum.

This is the amount initially invested-

(The Present Value)

This is the number of years for which we invested the money.

This the total amount of the investment after t years.

(The Future Value)

Future Value by Formula

19

FVt = = $1,000(1.07)2 = 1,144.90

FVt= PV (1+r)t FVt = PV (FVIF r,t,) – Using Table I

Note: The future value will be more than the simple interest FV since interest is reinvested. For one year the values will be equal ,since interest is not reinvested on compound method.

or

You earned an EXTRA $4.90 in Year 2 with compound over simple interest.

Compound Interest. Example2.

Assume that you deposit $1,000 at a compound interest rate of 7% for 2 years. Find the future value?

FV2

0 1 2

$1,000

7%

FVIFr,t 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

Using Future Value Tables

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

= $1,000 (1.145)= $1,145 [Due to Rounding]

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.

Compound Interest. Example 3.

Calculation based on general formula:

FVt = PV (1 + r)t =

FV5 = $10,000 (1 + 0.10)5 =$16,105.10

Calculation based on Table I:

FV5 = $10,000 (FVIF10%, 5)

FV5 = $10,000 (1.611) = $16,110

[Due to Rounding]

Double Your Money!!!

21

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

We will use the The “Rule-of-72”

Approx. Years to Double = 72 / r% = 72 / 12% = 6 Years [Actual Time is 6.12 Years]

The actual evaluation

FV = PV (1+r)t

FV = PV (1+r)t = 10000 = 5000(1+0.12)t

10000 = 5000(1.12)t

10000/5000 = (5000/5000)(1.12)t

2 = (1.12)t

lg 2 = t (lg 1.12) = lg 2/(lg 1.12) = t

0.30103/0.049218 = t

t= 6.116258 = 6.12 Years

General Present Value Formula

Present Value Example 1:

22

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 discount rate of 7% compounded annually.

PV0= FVt / (1 + r)t

PV0 = FV2 / (1 + r)2 = $1,000 / (1.07)2

= FV2 / (1 + r)2 = $873.44

Period 6% 7% 8%

1 0.943 0.935 0.926

2 0.890 0.873 0.857

3 0.840 0.816 0.794

4 0.792 0.763 0.735

5 0.747 0.713 0.681

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

PV2 = $1,000 (.873) = $873 [Due to Rounding]

PV0 = FVt (PVIFr,t)

Table II

or

or

Present Value Example 2:

Julie Miller wants to know how large of a deposit to make so that the money

will grow to $10,000 in 5 years at a discount rate of 10%.

PV0 = FVt / (1 + r)t PV0 = $10,000 / (1 + 0.10)5 = $6,209.21

Calculation based on Table I:

PV0 = $10,000 (PVIF10%, 5)= $10,000 (0.621) = $6,210.00

[Due to Rounding]

or

23

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

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

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

Types of Annuities

Examples of Annuities

Student Loan Payments Car Loan Payments

Insurance Premiums Mortgage Payments Retirement Savings

0 1 2 3

$100 $100 $100

(Annuity Due)

Beginning of

Period 1

Beginningof

Period 2

Today Equal Cash Flow s

Each 1 Period Apart

Beginning of

Period 3

Annuity Due

25

0 1 2 3

$100 $100 $100

(Ordinary Annuity)

End ofPeriod 1

End of

Period 2

Today Equal Cash Flows Each 1 Period Apart

End ofPeriod 3

Ordinary Annuity

The future value of an ordinary annuity can be viewed as occurring at the end of the last cash flow period, whereas the future value of an annuity due can be viewed as occurring at the beginning of the last cash flow period.

Hint on Annuity Valuation

Formula for Ordinary Annuity – Future Value of an annuity (FVA)

annuityan of valuefuture FV

paymentannuity equal the ere Wh

tr,FVIFA = FV

1 -)+1 (

= FV

a

a

aa

or

r

raa

t

Formula for Annuity Due – Future Value of an annuity (FVAD)

1 -)+1 (

= tr,FVIFA ;

r)t)x(1r,(FVIFA a due FVa

r

raaWhere

t

Period 6% 7% 8%

1 1.000 1.000 1.000

2 2.060 2.070 2.080

3 3.184 3.215 3.246

4 4.375 4.440 4.506

5 5.637 5.751 5.867

Valuation Using Table III

FVAt = CF(FVIFAr%,t)

FVA3 = $1,000 (FVIFA7%,3)

= $1,000 (3.215) = $3,215

Example 1: Ordinary Annuity – FVA

Find the Future Value of an annuity for$1000 paid at the end of each year for 3 years assuming interest is compounded annually at 7%?

Find the future value of an annuity due where $1000 is paid at the beginning of each year for 3 years at 7%?

Example 2: Future Value of an annuity Due (FVAD)

FVADt = CF (FVIFA r%,t)(1 + r) FVAD3 = $1,000(FVIFA7%,3)(1.07) FVAD3 = $1,000 (3.215)(1.07) = $3,440

Formula for Ordinary Annuity – Present Value of an annuity (PVA)

Formula for Annuity Due – Present Value of an annuity Due(PVAD)

annuityan of luePresent va PV

paymentannuity equal the Where

t)r,(PVIFA a PVa

) +1 (-1 =PVA

-

a

a

or

r

ra

t

r).(1by multiply then and

annuityordinary an like dueannuity an out Work

) +1 (-1 =t)r,(PVIFA a;

r)t)x(1r,(PVIFA a due PVa

-

r

raWhere

t

Example of an Ordinary Annuity – PVA

27

A company offer annual payments of $1000 at the end of each year for the next three years. What is the present value of this annuity discounted at 7%?.

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

$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

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

Valuation Using Table IV

PVAt = CF (PVIFA i%,t)

PVA3 = $1,000 (PVIFA7%,3)

= $1,000 (2.624) = $2,624

Example of an Annuity Due – PVAD

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

A company offer annual payments of $1000 at the beginning of each year for the next three years. What is the present value of this annuity discounted at 7%?.

Valuation Using Table IV

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

PVADt = CF (PVIFAr%,n)(1 + r) PVAD3 = $1,000 (PVIFA7%,3)(1.07) = $1,000 (2.624)(1.07) = $2,808

Mixed Flows Example

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

“Piece-At-A-Time” – Example:

29

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 first breaking problem into groups of annuity streams and any single cash flow groups. Then discount each group back to t=0.

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

30

“Group-At-A-Time” – Example:

Perpetuities

Perpetuity is a stream of equal cash flows that is expected to continue forever (infinity). Perpetuity has no time limit

PV perpetuity = PMT/r

)1(

PMT........

)1()1()r+(1 =

3

PMT

2

PMT

1

PMT

rrr

PV

General Formula

where PMT is the equal payment.

Example:

Find the present value of $100 perpetuity discounted at 15% is;

PV perpetuity = PMT/r, where r=15%, PMT is $100. PV perpetuity = 100/0.15 = $666.67

Frequency of Compounding General Formula:

32

Julie Miller has $1,000 to invest for 2 Years at an annual interest rate of 12%. Find the future value when the interest rate is compounded:

Annually; Semi - Annually ; Quarterly; Monthly; Daily

t: Number of Years m: Compounding Periods per Year

r: Annual Interest Rate FVt : Future Value at the end of Year t PV0: PV of the Cash Flow today

FVt = PV0(1 + [r/m])m(t)

Impact of Frequency

Annually: FV2 = 1,000(1 + [0.12/1])(1)(2) = 1,254.40

Semi : FV2 = 1,000(1 + [0.12/2])(2)(2) = 1,262.48 Qrtly: FV2 = 1,000(1 + [0.12/4])(4)(2) = 1,266.77

Monthly : FV2 = 1,000(1 + [0.12/12])(12)(2) = 1,269.73 Daily : FV2 = 1,000(1 + [0.12/365])(365)(2) = 1,271.20

Effective Annual Interest Rate

The interest rate offered or stated by lending institutions known as Nominal or Stated Interest rate or APR (Annual percentage Rate).

When interest is compounded more than once per annum (semi, quarterly, week, daily), comparing such rates with those quoted per annum we need to find the Effective annual Rate (EAR).

EAR is the rate of interest actually being earned as opposed to the stated rate.

Example: 18% Compounded Monthly

What It Really Means? Interest rate per month (r) = 18%/12 = 1.5% Number of interest periods per year (m) = 12

In words, Bank will charge 1.5% interest each month on your unpaid balance, if you borrowed money You will earn 1.5% interest each month on your remaining balance, if you deposited money

: 1.5%

18%

18% compounded monthly

or

1.5% per month for 12 months

=

19.56 % compounded annually

Question: Suppose that you invest $1 for 1 year at 18% compounded monthly. How much interest would you earn?

Solution:

r

rF 1212 )015.01(1$)1(1$

= $1.1956

0.1956 or 19.56%

= 1.5%

18%

18% compounded monthly Example.

Formula: Effective Annual Interest Rate (Yield)

1)/1( MMrEARr = nominal interest rate per year EAR = effective annual interest rate M = number of interest periods per year

EAR. The actual rate of interest earned (paid) after adjusting the nominal

rate for factors such as the number of compounding periods per year.

One-year

1.5%

6.14%

1st Qtr 2nd Qtr 3rd Qtr 4th Qtr

1.5% 1.5% 1.5%

EAR = ( 1 + 0.06 / 4 )4 – 1 = 1.0614 - 1 = 0.0614 or 6.14%

Example 2: Basket Wonders (BW) has a $1,000 Cash Deposit(CD) at the bank. The interest rate is 6% compounded quarterly for 1 year. What is the Effective Annual Interest Rate (EAR)?

Amortization

Amortizing a loan is paying off, liquidating a loan in equal installments at fixed intervals over the life of the loan or obligation. The liquidation can be shown through an amortization schedule.

Amortization Schedule is a table that details the payments, principal paid, interest paid, and remaining principal balance for a loan

Steps to Amortizing a Loan

1) Calculate the payment per period.

2) Determine the interest in Period t.

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.

Steps 1: find the annual installment (a) using annuity formula. PVa = a (PVIFA r,t) where r=6% ,t=3years PVa=K1000 1000=a (2.6730) = ‘a = 374.11

Amortizing a Loan: Example 1.

Prepare an amortization schedule for a K1,000 loan to be paid in 3 equal

installments at the end of each of the next 3 years. Interest is charged at a rate

of 6% per annum.

Step 2. Create an Amortization Schedule

Note: Always find the installment by applying the annuity formula

Step 2. Create an Amortization Schedule

This is theK1000 Loan

(PV Annuity)

This is the equal installments (Annuities)

314.1160 - 374.11

685.89314.11-1000

601000100

6

Always find the installment by applying the annuity formula

39

1. Determine Interest Expense –

Interest expenses – The amount reported by a company or individual as an expense for borrowed money. Interest expense relates to the cost of borrowing money. It is the price that a lender charges a borrower for the use of the lender's money

2. Calculate Debt Outstanding – The quantity of outstanding debt may be used in financing the day-to-day activities of the firm.

Usefulness of Amortization

40

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

7. Check with financial calculator (optional)

Steps to Solve Time Value of Money Problems