time value of money(latest final ppt)

Basic financial calculations

Parvesh Aghi

Future value of a lump sum Fn = P(1 + i)n OR Fn= P (CVF n,i)

FV (RATE,NPER,PMT,PV,TYPE) Future value of an Annuities FVn = A(CVFAi%,n)

FV(RATE,NPER,PMT,PV,TYPE) Sinking Fund

A = FV CVFAi%,n

PMT( RATE, NPER,PV ,FV,TYPE)

FORMULA’S –FUTURE VALUE

Present value of a lump sum P = Fn / (1 + i)n

PV = Fn * PVF i%,n

PV( RATE, NPER,PMT,FV,TYPE) Present Value of an Annuity

P=A (PVFAi%,n) PV( RATE, NPER,PMT,FV,TYPE Capital Recovery or Loan

amortization A= P * 1/PVFAn

PMT( RATE,NPER,PV,FV,TYPE)

FORMULA’S –PRESENT VALUE

Concept of time value of money

Compounding/future value Future value of a single cash flow Future value of a lump sum Future value of Annuity Sinking fund factor Discounting/Present value Present value of a lump sum Present value of Annuity Capital recovery factor

Key Concepts and Skills

Key Concepts and Skills Be able to compute the future value of an

investment made today Be able to compute the present value of cash

to be received at some future date Be able to compute the return on an

investment Be able to compute the number of periods

that equates a present value and a future value given an interest rate

Be able to use a financial a spreadsheet to solve time value of money problems

Time preference for money is an individual’s preference for possession of a given amount of money now , rather than the same amount at some future time

Three reasons may be attributed to the individual’s time preference for money :

1. Risk2. Preference for consumption3. Investment opportunities

THE PREFERENCE FOR MONEY

The time preference for money is generally expressed by an interest rate.

The rate will be positive even in the absence of any risk

Required rate of return= Risk free rate + Risk premium

ROR may also be called as opportunity cost of capital

Required rate of return

Basic Definitions Present Value – earlier money on a time line Future Value – later money on a time line Interest rate – “exchange rate” between

earlier money and later money◦ Discount rate◦ Cost of capital◦ Opportunity cost of capital◦ Required return

FUTURE VALUES

Future Values Suppose you invest Rs1000 for one year

at 5% per year. What is the future value in one year?◦ Interest = 1000(.05) = 50◦ Value in one year = principal + interest = 1000

+ 50 = 1050◦ Future Value (F) = 1000(1 + .05) = 1050

Suppose you leave the money in for another year. How much will you have two years from now?◦ F = 1000(1.05)(1.05) = 1000(1.05)2 = 1102.50

Future Values: General Formula Fn = P(1 + i)n OR Fn= P (CVF n,i)

◦ F = future value◦ P= present value◦ i = period interest rate, expressed as a decimal◦ n = number of periods◦ CVF = compound value factor table (refer to

future value of lump sum table)

◦ FV (RATE,NPER,PMT,PV,TYPE)◦ Rate= Interest rate◦ NPER =is the number of periods◦ PMT =0 ( Annuity)◦ TYPE =0 (timing of cash flows)◦ PV = - Present value

EXCEL SHEET

Effects of Compounding Simple interest Compound interest Consider the previous example

◦ FV with simple interest = 1000 + 50 + 50 = 1100◦ FV with compound interest = 1102.50◦ The extra 2.50 comes from the interest of .05(50)

= 2.50 earned on the first interest payment

Future Values – Example 2 Suppose you invest the Rs1000 from the

previous example for 5 years. How much would you have?◦ FV = 1000(1.05)5 = 1276.28

The effect of compounding is small for a small number of periods, but increases as the number of periods increases. (Simple interest would have a future value of Rs 1250, for a difference of Rs 26.28.)

Future Values – Example 3 Suppose you had a relative deposit Rs 10 at

5.5% interest 200 years ago. How much would the investment be worth today?◦ FV = 10(1.055)200 = 4,47,189.84

What is the effect of compounding?◦ Simple interest = 10 + 200(10)(.055) = 120◦ Compounding added Rs 447069.84 to the value of

the investment

If you deposited Rs 55650 in a bank , which is paying a 15% interest on a ten year period ,how much the deposit grow at the end of ten years.

FV= 55650 (CVF 10,15%) = 55650*4.046=225159.9

FV= 55650 (1.15)10 = 225159..9

Future value-Example 4

Future value of an AnnuitiesFuture value of an Annuities

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 is a fixed payment (or receipt) each year for a specified number of years.

Examples of Annuities

Student Loan Payments Car Loan Payments Insurance Premiums House loan Payments Retirement Savings Recurring deposit

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 anOrdinary Annuity -- FVA

Example of anOrdinary Annuity -- FVA

Rs1,000 Rs 1,000 1,000

0 1 2 3

3,215 = FVA3

7%

1,070

1,145

Cash flows occur at the end of the period

FVA3 = 1(1.07)2 + 1(1.07)1 + 1(1.07)0

= 1.145 + 1.070 + 1.00 = 3.215

Example of anOrdinary Annuity -- FVA

Example of anOrdinary Annuity -- FVA

Rs1 Rs 1 Rs1 1.000

0 1 2 3

3.215 = FVA3

7%

1.070

1.145

Cash flows occur at the end of the period

FVAn = A(1+i)n-1 + A(1+i)n-2 + ... + A(1+i)1 + A(1+i)0

Overview of an Ordinary Annuity -- FVA

Overview of an Ordinary Annuity -- FVA

A A A

0 1 2 n

FVAn

A = Periodic Cash Flow

Cash flows occur at the end of the period

i% . . .

FVn = A(CVFAi%,n) FVA3 = Rs1,000 (CVFA7%,3)

= 1,000 (3.215) = 3,215

Valuation Using Table IIIValuation Using Table III

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

CVFA= future value of annuity table

Suppose that a firm deposits Rs 5000 at the end of each year for four years at 6 % rate of interest . How much would this annuity accumulate at the end of the 4th year.

Without using table FV = 5000 (1+ 0.6)3+ 5000 (1+ 0.6)2+

5000 (1+ 0.6)1+ 5000 = 5955. + 5618+5300+5000 = 21873

Future value of an annuity Example

With Using Table FVn = A(CVFAi%,n) = 5000 (CVFA6%,4) = 5000 X 4.375 = 21873

Future value of an annuity Example (Contd)

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

FV(RATE,NPER,PMT,PV,TYPE)◦ Rate= Interest rate◦ NPER =is the number of periods◦ PMT = - Annuity◦ TYPE =0◦ PV = 0

EXCEL- FVA

Suppose that we want to accumulate Rs 21875 at the end of four years from now. How much should we deposit each year at an interest rate of 6% so that it grows to Rs 21875 at the end of fourth year ?

We know FV = A(CVFAi%,n)

A = FV CVFAi%,n

= 21875/4.375 =5000

Sinking Fund

PMT( RATE, NPER,PV ,FV,TYPE) PV=0 RATE= Interest FV=future value Type =0

EXCEL- SINKING FUND

PRESENT VALUE

With the compounding technique , the amount of present cash can be converted into an amount cash of equivalent value in future.

However it is common practice to translate the future cash flows into their present values

Present value of a future cash flow (inflow or outflow) is the amount of current cash that is of equivalent value to the decision maker.

Present Values How much do I have to invest today to have

some amount in the future?◦Fn = P(1 + i)n

◦Rearrange to solve for P = Fn / (1 + i)n

◦ or PV = Fn * PVF i%,n When we talk about discounting, we mean

finding the present value of some future amount.

When we talk about the “value” of something, we are talking about the present value unless we specifically indicate that we want the future value.

PVF= present value of lump sum

PV( RATE, NPER,PMT,FV,TYPE) PMT=0 TYPE=0

EXCEL – Present value of lump sum

Present Value – One Period Example Suppose you need Rs 100,000 in one year

for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today?

P = Fn / (1 + i)n

PV = 100,000/ (1.07)1 = 93457.95

Present Values – Example 2 You want to begin saving for you daughter’s

college education and you estimate that she will need Rs 150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today?

◦ PV = 150,000 / (1.08)17 = 40,540.34◦ PV = 150,000 *.27027= 40,540.34

Present Values – Example 3 Your parents set up a trust fund for you 10

years ago that is now worth Rs19,671.51. If the fund earned 7% per year, how much did your parents invest?◦ PV = 19,671.51 / (1.07)10 = 10,000

Suppose that an investor wants to find out the present value of Rs 50000/- to be received after 15 years . Her interest rate is 9%

PV = 50,000/ (1.09)15 = 13,750 PV= 50000 * PVF15%,9

PV= 50000 * .275 = 13750

Present Values – Example 4

Present Value – Important Relationship II For a given time period – the higher the

interest rate, the smaller the present value◦ What is the present value of Rs 500 received in 5

years if the interest rate is 10%? 15%? Rate = 10%: PV = 500 / (1.10)5 = 310.46 Rate = 15%; PV = 500 / (1.15)5 = 248.58

Present Value – Important Relationship I For a given interest rate – the longer the

time period, the lower the present value◦ What is the present value of Rs 500 to be

received in 5 years? 10 years? The discount rate is 10%

◦ 5 years: PV = 500 / (1.1)5 = 310.46◦ 10 years: PV = 500 / (1.1)10 = 192.77

The Basic PV Equation - Refresher PV = FV / (1 + i)n

There are four parts to this equation◦ PV, FV, i and n◦ If we know any three, we can solve for the fourth

A investor may have an investment opportunity of receiving an annuity – a constant periodic amount – for a certain specified numbers of years

We have to find out the present value of the annual amount every year and will have to aggregate all the present values to get the total present value of the annuity.

Present Value of an Annuity

For example , an investor who has a required rate of interest as 10 % per year , may have an opportunity to receive an annuity of Re 1 for 4 years .

The present value of Rs 1 received after one year is P= 1/(1+.10)1= 0.909 After two years ,three years, four years P= 1/(1+.10)2= .826 P= 1/(1+.10)3= .751 P= 1/(1+.10)4= .683 Present value of A= .909+.826+.751+.683=Rs 3.169

The computation of PV of an Annuity can be written in the following general form

Pn = A/(1+i)1 + A/(1+i)2

+ ... + A/(1+i)n

P=A (PVFAi%,n)

A person receives an annuity of Rs 5000 for 4 years . If the rate of interest is 10% ,the present value of Rs 5000 annuity is

P= A ( PVFA4, 10% ) P= 5000 X 3.170 = 15850

Example

PV( RATE, NPER,PMT,FV,TYPE) PMT= Annuity TYPE=0 FV=0

EXCEL= Present value of annuity

If we make an investment today for a given period of time at a specified rate of interest , we may like to know the annual income.

The reciprocal of the present value annuity factor is called the capital recovery factor

P= A*PVFAn i

A= P * 1/PVFAn i

Capital Recovery or Loan amortistion

Suppose that you plan to invest Rs 10000 today for a period of four years . If your rate of interest rate is 10% , how much income per year should you receive to recover your investment.

A= 10000/PFA 10%,4

= 10000/3.170 =3155

PMT( RATE,NPER,PV,FV,TYPE) FV =0 TYPE=0

Excel

Suppose you have borrowed a 3 year loan of Rs 10000/- at 9% from your employer to buy a motorcycle

If your employer requires three equal end of year repayments , the annual installment will be

P=A *PVFAN,I%

10000=A *PVFA3,9%

10000= A *2.531 A= 10000/2.531=Rs 3,951

Loan Amortization

Consider that an investor has an opportunity of receiving following amounts at the end every year for five years

Year 1= 1000 year 2= 1500 Year 3 = 800 Year4 = 1100 Year 5 = 400Find the present value of this stream of cash

flows if the investors required rate of return is 8%

Present value of an uneven cash flows

PV = 1000/(1.08)1+1500/(1.08)2 +800 / (1.08)3+1100/(1.08)4+400/(1.08)5

= 1000*.926

+1500*.857+800*.794+1100*.735+400*.681 = Rs 3927.60

Discounted Cash Flow – System A

Year Cash Flow (Rs) Discount Factor (8%) Present Value (Rs)(CF x DF)

1 1000 .926 926

2 1500 0.857 1285.5

3 800 0.794 635.2

4 1100 0.735 808.5

5 400 0.681 272.4

Total 3927.6

NPV(RATE,VALUE1,VALUE2,….)

EXCEL-Present value of an uneven cash flows

Perpetuity is an annuity that occurs indefinitely

Present value of perpetuity= Perpetuity Interest rate

Present value of perpetuity= P = A/i

Present value of perpetuity

Let us assume that an investor expects a perpetual sum of Rs 50000/- annually from his investment. What is the present value of this perpetuity if his interest rate is 10 %

P = 50000/.10= Rs 5,00,000

ANNUITY DUE

Suppose you deposit Re 1 in a saving account at the beginning of each year for 4 years to earn 6% . How much will be the compound value at the end of 4 years ?

FV = 1 (1.06)4+ 1 (1.06)3+ 1 (1.06)2+ 1 (1.06)1

FV = 1.262+1.191+1.124+1.06 FV = 4.637

You can see that the compound value of an annuity due is more than of an annuity because it earns extra interest for one year ie…(1+i)

value of an annuity due = future value of annuity *(1+i)

F= A* CVFA n,i* (1+i)

Future value of an annuity due

Let us consider a 4 year annuity of Re 1 each year , the interest rate being 10 % What is the present value of this annuity if each payment is made at the beginning of the year ?

PV = 1/(1.10)0 +1/ (1.10)1+1/(1.10)2 +1/(1.10 )3

=1+.0909+.826+.751+=3.487

Present value of annuity due = present value of annuity * (1+i)

P= A* PVFAn,i* (1+i)

Present value of an annuity due

NET PRESENT VALUE

Firm’s financial objective is to maximize the shareholder’s wealth.

Wealth is defined as net present value

Net Present Value (NPV)…of a financial decision is the difference between the present value of cash inflows and the present value of cash outflows.

NPV

The difference between the present value of cash inflows and the present value of cash outflows.

NPV is used in capital budgeting to analyze the profitability of an investment or project. 

NPV compares the value of a Rupee today to the value of that same Rupee in the future, taking inflation and returns into account.

If the NPV of a prospective project is positive, it should be accepted.

However, if NPV is negative, the project should probably be rejected because cash flows will also be negative

For example, if a retail clothing business wants to purchase an existing store, it would first estimate the future cash flows that store would generate, and then discount those cash flows into one lump-sum present value amount, say Rs 565,000.

If the owner of the store was willing to sell his business for less than Rs 565,000, the purchasing company would likely accept the offer as it presents a positive NPV investment.

Conversely, if the owner would not sell for less than Rs 565,000, the purchaser would not buy the store, as the investment would present a negative NPV at that time and would, therefore, reduce the overall value of the clothing company

FORMULA : NPV

CF1=Cash Flow in period 1 CFn = Cash flow in period nCF0 = cash flow today

Investment Decision

PROJECT A PROJECT B PROJECT C

MACHINE OUTFLOW

OUTFLOW

4,00,000 3,90,000 4 50,000

CASH INFLOWS

YEAR 1 70,000 170,000 20,000

YEAR 2 85,000 1 20,000 45,000

YEAR 3 80,000 1,10,000 55,000

YEAR 4 95,000 50 ,000 1, 00,000

YEAR 5 80,000 35,000 1,60,000

YEAR 6 90,000 - 2,00,000

TOTAL IN FLOW 5,00,000 4,85,000 5,60,000

NET INFLOW

1,00,000 95,000 1,30,000

Suppose the required rate of return of the investors or the cost of capital of the company is 9% Calculate which projects should undertaken by the company.

NPV = 70,000/(1.10)1 + 85000/(1.10)2+ 80000/(1.10)3+ 95000/(1.10)4+ 80000/(1.10)5+ 90000/(1.10)6– 4,00,000 =

70000*0.909 +85000* 0.826 +80000* 0.751 +95000* 0.683+ 80000*0.621+90000* 0.564 -4,00,000=

63630+70210+60080+64885+49680+50760

359245-400000=-40755

PROJECT A

Discounted Cash Flow – System A

Year Cash Flow (Rs) Discount Factor (10%)

Present Value (Rs)(CF x DF)

0 - 4,00,000 1.00 -4,00,000

1 +70000 0.909 +63630

2 +85000 0.826 +70210

3 +80000 0.751 +60,080

4 +95000 0.683 +64885

5 +80000 0.620 +49600

6 +90000 0.564 +50760

Total NPV =-40755

NPV = 170000/(1.10)1 + 120000/(1.10)2+ 110000/(1.10)3+ 35000/(1.10)4+ 50000/(1.10)5– 390,000 =

170000*0.909 +120000* 0.826 +110000* 0.751 +50000* 0.683+ 35000*0.621 -390,000=

154530+99120+82610+23905+31000

391165-390000=1165

PROJECT B

Discounted Cash Flow – System A

Year Cash Flow (Rs) Discount Factor (10%)

Present Value (Rs)(CF x DF)

0 -390000 1.00 -390000

1 +170000 0.909 +154530

2 +120000 0.826 +99120

3 +110000 0.751 +82610

4 +35000 0..683 +23905

5 +50000 0..620 +31000

Total NPV =1165

NPV = 20000/(1.10)1 + 45000/(1.10)2+ 55000/(1.10)3+ 100000/(1.10)4+ 160000/(1.10)5+200000/(1.10)6– 4,00,000 =

20000*0.909 +45000* 0.826 +55000* 0.751 +100000* 0.683+ 160000*0.621+200000* 0.564 -4,00,000=

18180 +37170 + 41305 +68300+ 99200 +112800

377115 -400000=-23045

PROJECT C

Discounted Cash Flow – System A

Year Cash Flow (Rs) Discount Factor (10%)

Present Value (Rs)(CF x DF)

0 – 4,00,000 1.00 – 4,00,000

1 +20000 0.909 +18180

2 + 45000 0.826 +37170

3 + 55000 0.751 +41305

4 + 100000 0.683 +68300

5 + 160000 0.620 +99200

6 +200000 0.564 +112800

Total NPV =-23045

No! The NPV is negative. This means that the project is reducing shareholder wealth. [Reject as NPV < 0 ]

NPV Acceptance Criterion

NPV(RATE,VALUE1,VALUE2……..VALUEN) RATE= DISCOUNT RATE VALUE= CASH OUT FLOWS

EXCEL

What rate of interest you earn if you deposit Rs 1000 today and receive Rs 1762 at the end of five years ?

P = Fn / (1 + i)n or

PV = Fn * PVF i%,n

1000 = 1762 * PVF i%,5

PVF i%,5 = 1000/1762 PVF i%,5 = .576

I=12%

Present value and rate of return

RATE(NPER,PMT,PV,FV,TYPE,GUESS)

NPER = period=5 PMT= Annuity=0 PV = -1000 FV= 1762 TYPE=0 Guess= leave blank

EXCEL

A B C D E F G

1 0 1 2 3 4 5

2 -1000 0 0 0 0 1762

3 @IRR(b2..g2)

12%

OR USE THIS EXCEL SHEET

Assume you borrow Rs 70000 from a bank to buy a flat in Patna. You are required to mortgage the flat and pay Rs 11396.93 annually for a period of 15 years . What rate of interest would u be paying ?

P=A (PVFAi%,n) 70000=11396.93 PVFAi%,n)

PVFAi%,15=70000/11396.93 PVFAi%,15=6.142= 14%

RATE(NPER,PMT,PV,FV,TYPE,GUESS)

NPER = 15 PMT= -11396.93 PV = 70000 FV= 0 TYPE=0 Guess= leave blank

EXCEL

Suppose your friend borrows from you Rs 1600 today and would return you Rs 700 ,Rs 600 Rs 500 in year 1 through year 3 as principal plus interest . What rate of return would you earn.

IRR(VALUES,GUESS) VALUES= PUT -1600 700 600 500 ON

EXCEL SHEET GUESS= BLANK GO ANY WHERE ON THE EXCEL SHEET AND

GO TO FORMULA/FINANCIALS/IRR AND SELECT THE VALUES SEE THE ANSWER = .0650=6.5%

EXCEL

The interest rate is usually specified on annual basis in loan agreement or securities ( bonds) as known as the nominal interest rates.

If compounding is done more than once a year , the actual annualized rate of interest would be higher than the nominal interest and it is called the effective interest rate (EIR).

Multi-period compounding

EIR =(1+i/m) n*m -1 i = the annual nominal rate of

interest n =no of years m =no of compounding per year M=1 if annual compounding M= 12 if monthly compounding M=52 if weekly compounding

EIR- FORMULA

Fn = P(1 + i)n

You can get annual rate of interest of 13% on a public deposit with a

company . What is the effective of rate interest

If the compounding is done

A) Half yearly

B) quarterly

C) monthly

D) weekly

USE EIR =(1+i/m) n*m -1 EIR= EIR =(1+.13/2) 1*2 -1 = (1+.065) 2 -1 = (1.065) 2 -1=1.1342-

1=.1342=13.42%

Multi-period compounding

Fn = P(1 + i)n

Fn = P(1 + i/m)n*m

m= no of compounding per year

If a company pays 15% compound quarterly on a 3 year public deposits of Rs 1000 then the total amount compounded after 3 years will be.

Fn = P(1 + i/m)n*m

Fn = 1000(1 + .15/4)3*4

Fn = 1000(1.0375)12

Fn = 1555

THANK YOU

Discounted Cash Flow – System A

PURPOSE GIVEN CALCULATE FORMULA

Future value of a lump

sum

P (Present Value)

F (Future Value )

Fn = P(1 + i)n

Fn= P (CVF n,i) FV =(RATE,NPER,PMT,PV,TYPE)