1. The Time Value of Money
Miss Faith Moono Simwami

2. After studying Chapter 4, you should be able to:
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.
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3. 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

4. The Time Value of Money The Interest Rate Simple Interest
Amortizing a Loan
Simple Interest
The Interest Rate
Compound Interest
Compounding More
Than Once per Year
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.
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5. The Interest Rate
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.

6. Types of Interest Compound Interest Simple Interest
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.
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).
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7. 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:

8. We can use that previous calculation to derive a formula for simple interest:
This is the interest rate per annum, we will give it the symbol r (rate)
This is the number of years for which we invested. We will give it the symbol t (time) r t P I
This is the total interest earned. We will give it the symbol I (interest)
This is the amount invested. We will give it the symbol P (principal)

10. How do we calculate Future Value (FV) for a lump sum of Money using simple interest?
To determine how much we will have in total (FV) we need to add this interest to the amount of the original investment(PV):
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.
This is an important property of simple interest. Since the amount of increase is constant each year, the growth in the investment is linear.

11. Simple Interest Example2
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 = P(R)(T) =$1,000(0.07)(2) = $140
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.

12. How do we calculate Compound Interest?
Suppose we invest $5000 at 10% per annum compound interest for 4 years. We can determine the growth of the investment as follows:
5500
6050
6655

13. How do we calculate Compound Interest for a Future Value (FV) for a lump sum of Money?
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 total amount of the investment after t years.
(The Future Value)
This is the compound interest rate per annum.
This is the number of years for which we invested the money.
This is the amount initially invested-
(The Present Value)

14. How do we calculate Compound Interest for a Future Value (FV) for a lump sum of Money? (Using Tables)
(1+r)t or (FVIF r,t) from the tables. This is also known as a compound factor.
This is the total amount of the investment after t years.
(The Future Value)
This is the compound interest rate per annum.
This is the number of years for which we invested the money.
This is the amount initially invested-
(The Present Value)

15. 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.

16. 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.
Difference between compound interest and simple interest after nine years
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 five years

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

18. How do we calculate Present Value (PV) for a lump sum of Money
How do we calculate Present Value (PV) for a lump sum of Money? (Using Tables)
1/(1+r)t or (PVIF r,t) from the tables. This is also known as a Discount factor.
This is the amount initially invested-
(The Present Value)
This is the compound interest rate per annum.
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)

19. Future Value by Formula
Compound Interest. Example2.
Assume that you deposit $1,000 at a compound interest rate of 7% for 2 years. Find the future value? 7%
$1,000
FV2
Future Value by Formula
FVt= PV (1+r)t or
FVt = PV (FVIF r,t,) – Using Table I
FVt = = $1,000(1.07)2 = 1,144.90
You earned an EXTRA $4.90 in Year 2
with compound over simple interest.
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.

20. Valuation Using Table I
Using Future Value Tables
FVIFr,t is found on Table I at the end of the book.
Valuation Using Table I
FV2 = $1,000 (FVIF7%,2)
= $1,000 (1.145)= $1,145 [Due to Rounding]
Compound Interest. Example 3.
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.
Calculation based on Table I:
FV5 = $10,000 (FVIF10%, 5)
FV5 = $10,000 (1.611) = $16,110
[Due to Rounding]
Calculation based on general formula:
FVt = PV (1 + r)t =
FV5 = $10,000 ( )5 =$16,105.10

21. We will use the The “Rule-of-72”
Double Your Money!!!
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 = = 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
/ = t
t= = 6.12 Years

22. Present Value Example 1:
Table II
General Present Value Formula
PV0= FVt / (1 + r)t
PV0 = FVt (PVIFr,t) or
Present Value Example 1:
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 = FV2 / (1 + r)2 = $1,000 / (1.07)2
= FV2 / (1 + r)2 = $873.44
PV2 = $1,000 (PVIF7%,2)
PV2 = $1,000 (.873) = $873 [Due to Rounding] 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%.
Calculation based on Table I:
PV0 = $10,000 (PVIF10%, 5)= $10,000 (0.621) = $6,210.00
[Due to Rounding]
PV0 = FVt / (1 + r)t
PV0 = $10,000 / ( )5 = $6,209.21 or

23. Types of Annuities
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.
Hint on Annuity Valuation
Examples of Annuities
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.
Student Loan Payments
Car Loan Payments
Insurance Premiums
Mortgage Payments
Retirement Savings

24. Formula for Annuity Due – Future Value of an annuity (FVAD)
Formula for Ordinary Annuity – Future Value of an annuity (FVA)
Formula for Annuity Due – Future Value of an annuity (FVAD)

25. Example 1: Ordinary Annuity – FVA
Example 2: Future Value of an annuity Due (FVAD)
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%?
Valuation Using Table III
FVADt = CF (FVIFA r%,t)(1 + r)
FVAD3 = $1,000(FVIFA7%,3)(1.07)
FVAD3 = $1,000 (3.215)(1.07)
= $3,440
FVAt = CF(FVIFAr%,t)
FVA3 = $1,000 (FVIFA7%,3)
= $1,000 (3.215) = $3,215

26. Formula for Annuity Due – Present Value of an annuity Due(PVAD)
Formula for Ordinary Annuity – Present Value of an annuity (PVA)
Formula for Annuity Due – Present Value of an annuity Due(PVAD)

27. Example of an Ordinary Annuity – PVA
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%?.
Valuation Using Table IV
PVAt = CF (PVIFA i%,t)
PVA3 = $1,000 (PVIFA7%,3)
= $1,000 (2.624) = $2,624

28. Example of an Annuity Due – PVAD
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
PVADt = CF (PVIFAr%,n)(1 + r)
PVAD3 = $1,000 (PVIFA7%,3)(1.07)
= $1,000 (2.624)(1.07) = $2,808
Mixed Flows Example
Julie Miller will receive the set of cash flows below. What is the Present Value at a discount rate of 10%.

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.
“Piece-At-A-Time” – Example:
10%
$ $600 $400 $400 $100
$545.45
$495.87
$300.53
$273.21 $
$ = PV0 of the Mixed Flow

30. “Group-At-A-Time” – Example:
$ $ $ $400
$1,268.00
PV0 equals $
Plus
$ $200
$347.20
Plus
$100
$62.10

31. PV perpetuity = PMT/r Perpetuities
Perpetuity is a stream of equal cash flows that is expected to continue forever (infinity). Perpetuity has no time limit
General Formula
PV perpetuity = PMT/r
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

32. Frequency of Compounding General Formula:
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
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
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

33. 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

34. 18% compounded monthly Example.
Formula: Effective Annual Interest Rate (Yield)
r = 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.

35. 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)?
EAR = ( / 4 )4 – 1 = = or 6.14%
1st Qtr
2nd Qtr
3rd Qtr
4th Qtr
One-year
1.5%
1.5%
1.5%
1.5%
6.14%

36. Amortization Calculate the payment per period.
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
Calculate the payment per period.
Determine the interest in Period t.
Compute principal payment in Period t.(Payment - Interest from Step 2)
Determine ending balance in Period t. (Balance - principal payment from Step 3)
Start again at Step 2 and repeat.

37. Note: Always find the installment by applying the annuity formula
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.
Steps 1: find the annual installment (a) using annuity formula. PVa = a (PVIFA r,t) where r=6% ,t=3years PVa=K =a (2.6730) = ‘a =
Step 2. Create an Amortization Schedule
Year
Beginning Amount (K)
Installment (K)
Interest Paid (K)
Principle Paid (K)
Balance (K) 1
374.11 2 3
Total
1,122.33
Note: Always find the installment by applying the annuity formula

38. Step 2. Create an Amortization Schedule
This is theK1000 Loan
(PV Annuity)
This is the equal installments (Annuities)
Year
Beginning Amount (K)
Installment (K)
Interest Paid (K)
Principle Paid (K)
Balance (K) 1
374.11
60.00
314.11
685.89 2
41.15
332.96
352.93 3
21.18
Total
1,122.33
122.33
1,000.00
Always find the installment by applying the annuity formula

39. Usefulness of Amortization
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.

40. Steps to Solve Time Value of Money Problems
Read problem thoroughly
Create a time line
Put cash flows and arrows on time line
Determine if it is a PV or FV problem
Determine if solution involves a single CF, annuity stream(s), or mixed flow
Solve the problem
Check with financial calculator (optional)