time value of money(latest final ppt)
TRANSCRIPT
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)