Time value of Money

By: Sajad Nazari

Learning Objectives• Understand how the time value of money works and

why it is important in finance• Learn how to calculate the present value and future

value of single and lump sums ;• To know and identify the different types of annuities for

both present value and future value of both an ordinary annuity and an annuity due;

• Calculate the present value and future value of an uneven cash flow stream;

• Explain the difference between nominal, periodic and effective interest rates

• Discuss the basics loan amortization

Time Value of Money

• It indicates the relationship between time and money.

• A peso received today is worth more than a peso to be received tomorrow."

• This is because the peso amount you received today can be invested to earn interest.

• The single most important in all financial concepts.

Uses of TMV

• Valuation of Stocks and Bonds

• Setting up loan payment schedules

• Making Corporate Decisions regarding investments.

Time Lines

• An important tool used in time value analysis;

• It is a graphical representation used to show the timing of cash flows.

Periods 0 1 2 310%

Definitions:• Interest is money paid for the use of money.

Symbol: I• Principal is the amount of money borrowed or

invested. Symbol P• Interest Rate is the rate, or percent. Symbol: r• Maturity Value / Final Value is the increased

amount resulting from the increase process. Symbol: F

• Time a period of time. Days, monthly, quarterly, semi-annually, annually.

Symbol: t

Simple Interest

• Mr. Lopez invest $10,000 at 5% annual rate for 2 years. How much will be the interest?

• I = Prt• P = Principal R= interest Rate t = time• 0 1 2

• I=Prt• = $10,000 x 5% x 2• =$ 1,000

Periods 5%

PV =$10,000 I =?

Derived Formulas:

• Interest (I) = Prt• Principal (P) = I / rt• Principal (P) = F- i• Rate (r) = I / Pt• Time (t) = I / Pr• Future Value (Fv) = P(1+rt)• Present Value (Pv) = Fv

1+rt

Time Value Concepts

• Future Value• Present Value• Future Value of Ordinary Annuity• Future Value of Annuity Due• Present Value of Ordinary Annuity• Present Value of Annuity Due

FUTURE VALUE

• The amount to which a cash flow or series of cash flows will grow over a given period when compounded at a given interest rate. – Brigham, 2011

• It is the amount of money that will grow to at some point in the future. –Cabrera, 2011

• FV = PV(1+i)n• PV = Present Value / Principal• I = interest rate• n = Number of periods / term

Compounding, Simple Interest and Compound Interest

• Compounding is the arithmetic process of determining the final value of cash flow or series of cash flows when compounded interest is applied. – Brigham, 2011

• Simple Interest occurs when interest is not earned on interest. – Brigham, 2011

• Compound Interest occurs when interest is earned on prior periods’ interest. –Brigham, 2011

Simple interest

• Mr. Lopez invest $10,000 at 5% annual rate for 2 years. How much will be the Maturity value?

• 0 1 2

• F = P(1+rt)n• = $10,000(1.1)• = $11,000

Periods

PV = 10,000 FV = ?

Future Value (compounded)

• Mr. Lopez invest $10,000 compounded at 5% annual rate for 2 years. How much will be the Maturity value?

• 0 1 2

• FV= PV(1+i)n • PV = Present Value• i = interest rate• N= number of years / time

PV = $10,000 FV= ?

Future Value (compounded)• 0 1 2

• PV = 10,000• i = .05• N= 2

• FV= PV(1+i)n FV = PV(1+i)n • =$10,000(1+.05) = $10,500(1.05)• = $10,500 = $11,025

• FV = PV(1+i)n

• = $10,000(1.0250)• =$ 11,025 the maturity value after 2 years

PV=$10,000 $10,500 $11,025

Mr. Lopez invest $10,000 compounded at 5% annual rate for 2 years. How much

will be the Maturity value? (table 1)

• FV= PV(1+i)n • PV = $10,000• i = .05• N= 2• FV = $10,000(1.10250)

= $11,025

Calculating FV using Spread Sheet

Procedure:

Procedure:

Procedure:

Procedure:

Procedure:

Present Value• The value today of a future cash flow

or series of cash flows. –Brigham, 2011

• It is the amount of money today that is equivalent to a given amount to be received or paid in the future. –Cabrera, 2011

• it is just a reverse of the future value, in a way that instead of compounding the money forward into the future, we discount it back to the present.

Formula:

• PV = F V

(1+i)n• FV = Future Value• i = interest rate• n = number of years

Single-Period Case

• Suppose you need $50,000.00 to buy laptop next year. You can earn 10% on your money by putting in on the bank. How much do you have to put up today?

• 0 1

• PV= FV

(1+i)n• = $50,000

(1+.10)1• = $45,454.545 the amount needed to

invest today

PV =? FV = $50,000

Single-Period Case

• Suppose you need $50,000.00 to buy laptop next year. You can earn 10% on your money by putting in on the bank. How much do you have to put up today? (table 2)

• PV= FV

(1+i)n• FV = $50,000 i=.10 n=1• PV = $50,000 (.090909)

=$45,454.5

Using Spread Sheets

Multiple Period Case• Angelo would like to buy a new automobile. He

has $600,000, but the car costs $800,000. If he can earn 12%, how much does he need to invest today in order to buy the car in two years? Does he have enough money, assuming the price will still the same?

• 0 1 2

• FV = Future Value• i = Interest Rate• N = Number of years / Time

PV = ? $800,000

Multiple Period Case• PV= FV

(1+i)n• = $800,000

(1+.12) 2• =$637,755.102 the amount

Angelo must invest today.

• = $600,000 - $637,755.102• = $37,755.102 • Angelo is still short of $37,755.102

Angelo would like to buy a new automobile. He has $600,000, but the car costs $800,000. If he can earn 12%, how much does he need to invest today in order to buy the car in two years? Does he have enough money, assuming the price will still the same? (table 2)

• PV= FV

(1+i)n• FV = $800,000 i=.12 n=2

• PV = $800,000 ( 0.79719)

= $637,752

ANNUITIES

• It is a series of equal sized cash flows occurring over equal intervals of time. –Cabrera, 2011

• A series of equal payment at fixed intervals for a specified number of periods – Brigham, 2011

Two types of Annuity

Ordinary Annuity – exists when the cash flows occur at the end of each period.

• 0 1 2 3

Annuity Due – exists when the cash flows occur at the beginning of each period.

0 1 2 3

Periods

Payments -100 -100 -100

Periods

Payments -100 -100 -100

Future Value of Ordinary Annuity

• It is the future value of a series of equal sized cash flows with the first payment taking place at the end of the first compounding period. The last payment will not earn any interest since it is made at the end of the annuity period. –Cabrera, 2011

• FVA = R (1+i)n-1 i

• R = Periodic Payment• i = Interest Rate• n = time / term

Future Value of Ordinary Annuity (FVA)

• Suppose you deposit $2,000 at the end of year 1, another $2,000 at the end of year 2, how much will you have in 5 years, if you deposit ₱2,000 at the end of each year? Assume a 10% interest rate throughout. (table 3)

• FVA = R (1+i)n-1 I

• R = $2,000 i=.10 n=5• FVA = $2,000(6.10510)• = $12,210.2

Using Spread Sheet

FUTURE VALUE OF AN ANNUITY DUE

• The future value of a series of equal sized cash flows with the first payment taking place at the beginning of the annuity period.

– Cabrera, 2011

• FVAD = R [(1+i)n-1] x (1+i)

I• R= Periodic Payments• i= interest rate• N= number of years / periods

FUTURE VALUE OF AN ANNUITY DUE

• Jose deposits $3,500 every beginning of the month at his bank that credits 3% monthly for a year. How much he will have at the end of the term? (table 5)

• FVAD = R [(1+i)n-1] x (1+i)

I• R = $3,500 i=.03 n=12• FVAD = $3,500 (14.61779)

= $51,162.265

Using spread sheet:

Future Value of Annuity Due (FVAD) (uneven cash flows)

• Suppose you deposit today $100 in an account paying 8%. In one year, you will deposit another $200 and ₱300 at the beginning of the third year, how much will you have in three years?

• FVAD = R [(1+i)n-1] x (1+i)

• FVAD = $100 (1.0800)

= $108.00• FVAD = $308 ( 1.0800)

= $332.64• FVAD = $632.64(1.0800)

= $683.25• FVAD = $683.25

Present Value of Ordinary Annuity

• The Present Value of a series of equal sized cash flows with the first payment taking place at the end of the first compounding period. –Cabrera, 2011

• PVA = R 1-/(1+i)n

i• R = Periodic Payment• i = Interest Rate• n = time / term

Present Value of Ordinary Annuity

• How much is the cash equivalent of the IPAD that can be purchased by giving a down payment of $3,000 and $2,500 payable at the end of each period for 5 months at 5%? (table 4)

• PVA = R 1-/(1+i)n

I• R = $2,500 i=.05 n=5• PVA = $2,500 (4.32948)

= $10,823.7• Cash equivalent = PVA + down payment• = $10,823.7 + $3,000• = $13,823.7

Using spread sheet (PVA)

Present Value of Ordinary Annuity (uneven cash flows)

• You are offered an investment that will pay you $200 in one year, $400 the next year, $600 the next year and $800 at the end of the next year. You can earn 12% on very similar investments. What is the most you should pay for this one?

• PVA = R 1-/(1+i)n

i• PVA = $200 ( 0.89286)

= $178.572• PVA = $578.572 (0.89286) = $559.4398• PVA = $1,159.44 (0.89286)

= $1,035.217• PVA = $1,835.217 (0.89286)

= $1,638.592• PVA = $ 1,638.592

Present Value of Annuity Due

• It is the present value of a series of equal sized cash flows with the first payment taking place at the beginning of the annuity period. –Cabrera,2011

• PVAD = R [1- 1/(1+I)n] x (1+i)

i• R = Periodic Payment• i = Interest Rate• n = time / term

Present Value of Annuity Due

• A machine can be bought for $4,000 down payment and 8 equal monthly payments of $1,200 payable every beginning of the month. If money is worth 6% compounded monthly, what is the cash equivalent of the machine? (table 6)

• PVAD = R [1- 1/(1+I)n] x (1+i)

I• R = $1,200 i=.06 n=8 dp = $4,000

• PVAD = $1,200 (6.58238)

= 7,898.856

• Down payment + PVAD• = $4,000+ $7898.856• = $11,898.856

Present Value of Annuity Due

FINDING ANNUITY PAYMENTS

• What is the annuity payment if the present value is $100,000, the interest rate is 10% and there are 8 periods?

• Present value = $18,744.40176

Finding the Interest Rate, r

• Suppose your parents will retire in 18 years. They currently have $250,000, and they think they will need $1,000,000 at retirement. What annual interest rate must they earn to reach their goal, assuming they don't save any additional funds?

• R = I / Pt• Interest rate = 8%

Finding the number of years, n

• Sometimes, we need to know how long it will take to accumulate a certain some of money

• Ex. How long will it take $10,000 to double if it was invested in the bank that paid 6% per year?

• T = I /Pr

Types of Interest Rates

• Nominal Interest Rate (Quoted or stated) – The contracted, or quoted or stated interest rate. It is also called annual percentage rate (APR); the periodic rate times the number of periods per year.

• Effective Annual Rate – The annual rate of interest actually being earned, as opposed to the quoted rate. This is the rate that would produce the same future value under annual compounding as would more frequent compounding at a given nominal rate.

Amortization

• A method of repaying an interest bearing debt by a series of equal payments at expected time interval.

• A = R [1-(1+i)-n]

i• R = Periodic Payment• i = Interest Rate• n = time / term

• Amortized loan – a loan that is to be repaid in equal amounts on a monthly, quarterly, or annual basis.

• Amortization Schedule - a table showing how a loan will be repaid.

Sample Problem:

• A homeowner borrows $100,000 on a mortgage loan. The loan is to be repaid in five equal payments at the end of each of the next 5 years. The lender charges 6% on the balance at the beginning of each year.

• Annuity Payment = $23,739.64

Amortization Schedule

Year BeginningAmount (1)

Payment (2)

Interest(3)

RepaymentOf Principal (4)

EndingBalance (5)

(1x.06) (2-3) (1-4)

1 $100,000.00 $23,379.64 $6,000.00 $17,739.64 $76,620.36

2 76,620.36 23,379.64 4,597.22 18,872.42 53,240.72

3 53,240.72 23,379.64 3,194.44 20,185.20 29,861.08

4 29,861.08 23,379.64 1,791.665 21,587.98 6,481.44

5 6,481.44 23,379.64 388.886 22,990.75 0.00

References:

• Brigham, Eugene F., Houston, Joel F., Financial Management Fundamentals (12th Edition), Manila Philippines, 2011.

• Ma. Elenita Balatbat Cabrera, Management Accounting Concepts and Applications 2011 Edition

• Arce, Aquino, CoPo, Gabuyo, Laddaran, Mananquil, Mathematics of Investment 2010 Edition.