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; the higher the discount rate, the lower the present value of the future cash flows. Determining the appropriate discount rate is the key to valuing future cash flows properly, whether they be earnings or obligations.[2] 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.[3] Present value of a perpetuity is an infinite and constant stream of identical cash flows.[4] Compound interest (or compounding interest) is interest calculated on the initial principal and also on the accumulated interest of previous periods of a deposit or loan. Thought to have originated in 17th-century Italy, compound interest can be thought of as “interest on interest,” and will make a sum grow at a faster rate than simple interest, which is calculated only on the principal amount. The rate at which compound interest accrues depends on the frequency of compounding; the higher the number of compounding periods, the greater the compound interest. Thus, the amount of compound interest accrued on $100 compounded at 10% annually will be lower than that on $100 compounded at 5% semi-annually over the same time period. Basic Time Value of Money Formula and Example Depending on the exact situation in question, the TVM formula may change slightly. For example, in the case of annuity or perpetuity payments, the generalized formula has additional or less factors. But in general, the most fundamental TVM formula takes into account the following variables: FV = Future value of money PV = Present value of money i = interest rate n = number of compounding periods per year t = number of years Based on these variables, the formula for TVM is: FV = PV x (1 + (i / n)) ^ (n x t) For example, assume a sum of $10,000 is invested for one year at 10% interest. The future value of that money is: FV = $10,000 x (1 + (10% / 1) ^ (1 x 1) = $11,000 The formula can also be rearranged to find the value of the future sum in present day dollars. For example, the value of $5,000 one year from today, compounded at 7% interest, is: PV = $5,000 / (1 + (7% / 1) ^ (1 x 1) = $4,673

1. Time Value for Money
Lecture 4

2. Types of Annuities
• An Annuity represents a series of equal payments
(or receipts) occurring over a specified number of
periods.
• Ordinary Annuity: Payments or receipts occur at
the end of each period.
• For example, straight bonds usually pay coupon payments at the end
of every six months until the bond's maturity date.
• Annuity Due: Payments or receipts occur at the
beginning of each period.
• Rent is an example of annuity due. You are usually required to
pay rent when you first move in at the beginning of the month,
and then on the first of each month thereafter.

3. Examples of Annuities
• Student Loan Payments
• Car Loan Payments
• Insurance Premiums
• Mortgage Payments
• Retirement Savings

4. Parts of an Annuity
• (Ordinary Annuity)
• End of Period 1 End of period 2 End of period 3
0 1 2 3
0 100 100 100
Today
Equal cash flow. each 1
period apart

5. Parts of an Annuity
• (Annuity Due)
• Start of Period 1 Start of period 2 Start of period3
0 1 2 3
100 100 100
Today
Equal cashflow each 1
period apart

6. FVAn = R(1+i)n-1 + R(1+i)n-2 + =R[(1+i)n -1]/i
... + R(1+i)1 + R(1+i)0
Overview of an
Ordinary Annuity -- FVA
R R R
0 1 2 n n+1
FVAn
R = Periodic
Cash Flow
Cash flows occur at the end of the period
i% . . .

7. Example Ordinary Annuity =FVA
• If you are receiving $1000 a year for three years. Lets
further assume that you deposit each annual receipt in
saving account earning 7% interest rate. How much money
you have at the end of 3 years?

8. 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 an
Ordinary 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

9. Example
• Explanation
• Money received at the end of year 1 can further be
invested for two more years @ the rate of 7% which will
be $1145.
• Money received at the end of year 2 can further be
invested for one more year @ the rate of 7% which will be
$1070.
• And at the end of year 3, $1000 will be received which
cant invested further because that’s the last year and
maturity period is over.

10. PVAn = R/(1+i)1 + R/(1+i)2
+ ... + R/(1+i)n
Overview of an
Ordinary Annuity -- PVA
R R R
0 1 2 n n+1
PVAn
R = Periodic
Cash Flow
i% . . .
Cash flows occur at the end of the period

11. 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 an
Ordinary 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

12. Hint
• The future value of an ordinary annuity can be
viewed as occurring at the end of the last cash flow.
• The present value of an ordinary annuity can be
viewed as occurring at the beginning of the first cash
flow period.

13. FVADn = R(1+i)n + R(1+i)n-1 +
... + R(1+i)2 + R(1+i)1
= FVAn (1+i)
Overview View of an
Annuity Due -- FVAD
R R R R R
0 1 2 3 n-1 n
FVADn
i% . . .
Cash flows occur at the beginning of the period

14. 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 an
Annuity 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

15. PVAn = R/(1+i)1 + R/(1+i)2
+ ... + R/(1+i)n
Overview of an
Ordinary Annuity -- PVAD
R R R
0 1 2 n n+1
PVAn
R = Periodic
Cash Flow
i% . . .
Cash flows occur at the end of the period

16. PVADn = $1,000/(1.07)0 + $1,000/(1.07)1 +
$1,000/(1.07)2 = $2,808.02
Example of an
Annuity 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

17. HINT
• future value of an annuity due can be viewed as occurring
at the beginning of the last cash flow period.
• the present value of an annuity due can be viewed as
occurring at the end of the first cash flow period.

18. 1. Read problem thoroughly
2. Determine if it is a PV or FV problem
3. Create a time line
4. Put cash flows and arrows on time line
5. Determine if solution involves a single
CF, annuity stream(s), or mixed flow
6. Solve the problem
Steps to Solve Time Value of
Money Problems

19. Julie Miller will receive the set of cash flows
below. What is the Present Value at a
discount rate of 10%?
Mixed Flows Example
0 1 2 3 4 5
$600 $600 $400 $400 $100
PV0
10%

20. 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 group.
Then discount each group back to t=0.
How to Solve?

21. “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

22. Frequency of Compounding
General Formula:
FVn = PV0(1 + [i/m])mn
n: Number of Years
m: Compounding Periods per Year
i: Annual Interest Rate
FVn,m: FV at the end of Year n
PV0: PV of the Cash Flow today

23. Impact of Frequency
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

24. PV of compounding periods
• PV0 = FVn /(1 + [i/m])mn
Qrtly PV4 = 1266.77/(1+ [.12/4])(4)(2)
=1000

25. Effective Annual
Interest Rate
Effective Annual Interest Rate
The actual rate of interest earned (paid) after
adjusting the stated (nominal) for factors such as
the number of compounding periods per year.
• EAR is the rate compounded annually that provides
the same annual interest rate as nominal rate if it is
compounded annually.
1+EAR= =(1 + [ i / m ] )m (n)
• EAR=(1 + [ i / m ] )m(n) - 1

26. Example
Basket Wonders (BW) has a $1,000 at the bank. The
interest rate is 6% compounded quarterly for 1 year.
What is the Effective Annual Interest Rate (EAR)?
EAR = ( 1 + 6% / 4 )4(1) - 1
= 1.0614 - 1 = .0614 or 6.14%!