Introduction of Time Value of money and thier related problems with solution

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CHAPTER 2 Time Value of Money

Learning Objectives:

5 - 2 Learning Objectives Understand the importance of the time value of money Understand the difference between simple interest and compound interest Know how to solve for present value, future value, time or rate Understand annuities and perpetuities Know how to construct an amortization table

The Time Value of Money Concept:

CHAPTER 2 – Time Value of Money 5 - 3 The Time Value of Money Concept Cannot directly compare $1 today with $1 to be received at some future date Money received today can be invested to earn a rate of return Thus $1 today is worth more than $1 to be received at some future date The interest rate or discount rate is the variable that equates a present value today with a future value at some later date

Reasons for Time Value of Money:

CHAPTER 2 – Time Value of Money 5 - 4 Reasons for Time Value of Money Reinvestment opportunities Risk Preference for consumption

Reinvestment opportunities:

CHAPTER 2 – Time Value of Money 5 - 5 Reinvestment opportunities The main reason for the time preference for money is to be found in the reinvestment opportunities for funds which are received early. The fund so invested will earn a rate of return, this would not be possible if the funds are received at a later time. The time preference for money is, therefore expressed generally in terms of a rate of return or more popularly as a discount rate.

Risk:

CHAPTER 2 – Time Value of Money 5 - 6 Risk We live under risk or uncertainty. As an individual is not certain about future cash receipts, he or she prefer received cash now.

Preference for consumption:

CHAPTER 2 – Time Value of Money 5 - 7 Preference for consumption Most people have subjective preference for present consumption over future consumption of goods and services because of the urgency of their present wants or because of the risk of not being in a position to enjoy future consumption that may be caused by illness or death or because of inflation. As money is the means by which individuals acquire most goods and services they may prefer to have money now.

Opportunity Cost:

CHAPTER 2 – Time Value of Money 5 - 8 Opportunity Cost Opportunity cost = Alternative use The opportunity cost of money is the interest rate that would be earned by investing it It is the underlying reason for the time value of money Money today can be invested to be some greater amount in the future Conversely, if you are promised a cash flow in the future, it’s present value today is less than what is promised!

Choosing from Investment Alternatives Required Rate of Return or Discount Rate:

CHAPTER 5 – Time Value of Money 5 - 9 Choosing from Investment Alternatives Required Rate of Return or Discount Rate You have three choices: $20,000 received today $31,000 received in 5 years $3,000 per year indefinitely To make a decision, you need to know what interest rate to use This interest rate is known as your required rate of return or discount rate .

Techniques ::

CHAPTER 2 – Time Value of Money 5 - 10 Techniques : Compounding Discounting Compounding the process of calculating future values of cash flows Discounting the process of calculating present value of cashflows

Simple Interest:

CHAPTER 2 – Time Value of Money 5 - 11 Simple Interest Simple interest is interest paid or received on only the initial investment (or principal) At the end of the investment period, the principal plus interest is received 0 1 2 3 … n I 1 I 2 I 3 I n +P

Simple Interest Example:

CHAPTER 2 – Time Value of Money 5 - 12 Simple Interest Example PROBLEM: Invest $1,000 today for a five-year term and receive 8 percent annual simple interest. SOLUTION: Annual interest = $1,000 × .08 = $80 per year.

Simple Interest General Formula:

CHAPTER 2 – Time Value of Money 5 - 13 Simple Interest General Formula [ 5-1] Where: P = principal invested n = number of years k = interest rate

CHAPTER 2 – Time Value of Money 5 - 14 Compound Interest Compounding (Computing Future Values) Simple interest problems are rare; in finance we are most interested in compound interest Compound interest is interest that is earned on the principal amount invested and on any accrued interest

Compound Interest Example:

CHAPTER 2 – Time Value of Money 5 - 15 Compound Interest Example PROBLEM: Invest $1,000 today for a five-year term and receive 8 percent annual compound interest. How much will the accumulated value be at time 5? SOLUTION:

Compound Interest Example of Interest Earned on Interest:

CHAPTER 2 – Time Value of Money 5 - 16 Compound Interest Example of Interest Earned on Interest PROBLEM: Invest $1,000 today for a five-year term and receive 8 percent annual compound interest. The Interest earned on Interest Effect: Interest (year 1) = $1,000 × .08 = $80 Interest (year 2 ) =($1,000 + $80)×.08 = $86.40 Interest (year 3) = ($1,000+$80+$86.40) × .08 = $93.31

Compound Interest General Formula:

CHAPTER 5 – Time Value of Money 5 - 17 Compound Interest General Formula A= Amount at the end of the period P= Principal at the beginning of the period i= rate of interest n= number of years A = P (1+i) n

Compound Interest Simple versus Compound Interest:

CHAPTER 5 – Time Value of Money 5 - 18 Compound Interest Simple versus Compound Interest Compounding of interest magnifies the returns on an investment R eturns are magnified The longer they are compounded The higher the rate they are compounded (See Figure 5-1 to compare simple and compound interest effects over time)

Compound Interest Simple versus Compound Interest:

CHAPTER 5 – Time Value of Money 5 - 20 Compound Interest Discounting (Computing Present Values) [ 5-3]

Computing Present Value:

CHAPTER 5 – Time Value of Money 5 - 21 Computing Present Value The present Value is an amount today that equates to some larger amount in the future Example : We know we want $1,000,000 when we retire 40 years from today. If we can earn a 10% return on our money, how much should we invest today? Calculator Approach: 1,000,000 FV 0 PMT 40 N 10 I/Y CPT PV 22,094.93

Compound Interest Determining Rates of Return or Holding Periods:

CHAPTER 5 – Time Value of Money 5 - 22 Compound Interest Determining Rates of Return or Holding Periods

Calculating the Rate of Return:

CHAPTER 2 – Time Value of Money 5 - 23 Calculating the Rate of Return If we know the present value, the future value and the number of time periods, we can calculate the rate of return we have earned For example, suppose we invested $5,000 six years ago Today, it is worth $10,000. What is the annually compounded rate of returned? Calculator Approach: 10,000 FV 0 PMT 5,000 +/- PV 6 N CPT I/Y 12.25%

Annuities and Perpetuities Annuities:

CHAPTER 2 – Time Value of Money 5 - 24 Thus far, we have dealt only with single payments, either today or in the future An annuity is a stream of payments that continues for a finite period of time If the payment occurs at the end of the period, it is an ordinary annuity If the payment occurs at the start of the time period, it is an annuity due Annuities and Perpetuities Annuities

COMPOUND VALUE CONCEPT:

CHAPTER 2 – Time Value of Money 5 - 25 Future value of a single amount Future value of series of payments Future value in case of annuities Frequency of compounding COMPOUND VALUE CONCEPT

COMPOUND VALUE CONCEPT:

CHAPTER 2 – Time Value of Money 5 - 26 Future value of a single amount The future value of an amount at the end of period 1 (A) will be equal to the prodcut of the original value (P) and the rate of interest plus 1 This can be expressed in the form of equation A= P(1+r) COMPOUND VALUE CONCEPT

COMPOUND VALUE CONCEPT:

CHAPTER 2 – Time Value of Money 5 - 27 Future value of a single amount The future value of an amount at the end of period 1 (A) will be equal to the prodcut of the original value (P) and the rate of interest plus 1 This can be expressed in the form of equation A= P(1+r) n COMPOUND VALUE CONCEPT

Ex.: If P is Rs.100 r is 10% after a 2 years peiod will be:

CHAPTER 2 – Time Value of Money 5 - 28 Ex.: If P is Rs.100 r is 10% after a 2 years peiod will be A= P(1+r) 2 A=100(1+0.10) 2 = Rs.121 Alternatively A=(P * CVIF r , n ) CVIF- Compound Value Interest Factor r= rate of interest n= Number of year

Above example Ex.: If P is Rs.100 r is 10% after a 2 years peiod will be:

CHAPTER 2 – Time Value of Money 5 - 29 Above example Ex.: If P is Rs.100 r is 10% after a 2 years peiod will be A=(P * CVIF r , n ) Refer Compound value of Re.1 table CVIF 10, 2 = 1.210 100*1.210 = Rs. 121

Future Value of a series of payments:

CHAPTER 2 – Time Value of Money 5 - 30 Future Value of a series of payments The future value of series of three annual receipts of Rs.1000, Rs.500 and Rs.800 respectively at the end of the third year at 5% rate of interest will be A=1000(1.05) 2 + 500(1.05) + Rs.800= 2427.50

Future Value in case of Annuities:

CHAPTER 2 – Time Value of Money 5 - 31 Future Value in case of Annuities Uniformity of cash flows represents a case of annuity Ex. The future value of an equal annual investment of Rs.1000 at the end of a 10 year period at 10% rate of interest will be A=(P * CVIFA r , n ) Refer the Compound value of an annuity of Re.1 table

Future Value in case of Annuities:

CHAPTER 2 – Time Value of Money 5 - 32 Future Value in case of Annuities The CVIFA 10, 10 = 15.937 Amount=1000*15.937=15937

Computation of Present Value of Cash Flows:

CHAPTER 2 – Time Value of Money 5 - 33 Computation of Present Value of Cash Flows Present Value of a single amount Present value of a series of Future Values Present Value in case of Annuity

Formula for determining the future value :

CHAPTER 2 – Time Value of Money 5 - 34 Formula for determining the future value FV PV= ------------- (1+r) n

Present value of a single amount :

CHAPTER 2 – Time Value of Money 5 - 35 Present value of a single amount The present Value of Rs.1000 to be received after 5 years at 10% rate of discount FV PV= ------------- (1+r) n Rs.1000/(1.10) 5 = Rs.621 Alternatively

PowerPoint Presentation:

CHAPTER 2 – Time Value of Money 5 - 36 PV= FV * (PVIF i , n) Refer Present Value of Re.1 table PVIF 10 , 5 = 0.621 1000* 0.621= Rs.621

Ex. A1, A2, A3 are Rs.2000,Rs.3000, Rs.4000 respectively, therefore the Present value at the discount rate of 5% will be:

CHAPTER 2 – Time Value of Money 5 - 37 Ex. A1, A2, A3 are Rs.2000,Rs.3000, Rs.4000 respectively, therefore the Present value at the discount rate of 5% will be PV= 2000/1.05 + 3000/(1.05) 2 + 4000 /(1.05) 3 = Rs.8081

Present Value in case of Annuity:

CHAPTER 2 – Time Value of Money 5 - 38 Present Value in case of Annuity Ex. The Present Value of an investment yielding Rs.1000 annualy for Rs.20 years at a discount rate of 8 % will be PV= FV * (PVIFA i , n) Refer Present Value of an annuity of Re.1 table PVIFA 8 , 20 = 9.81815

PowerPoint Presentation:

CHAPTER 2 – Time Value of Money 5 - 39 Rs.1000* 9.81815= Rs.9818.15

Perpetuity:

CHAPTER 2 – Time Value of Money 5 - 40 Perpetuity There are some speical cases of annuity. This first is perpetuity where annuity is infinite. In otherwords , a perpetuity assures the investor/owner of the funds, periodic cash flows over an infinite period of time. It is normally found in case of preference Shares. P A = A/r

Ex. If a company offers an annual dividend of Rs.20 per share and the risk of investment justifies a rate of return of 14% the present value of perpetuity annuity will be:

CHAPTER 2 – Time Value of Money 5 - 41 Ex. If a company offers an annual dividend of Rs.20 per share and the risk of investment justifies a rate of return of 14% the present value of perpetuity annuity will be P A = A/r 20/0.14 = Rs. 142.86

Deferred Annuity::

CHAPTER 2 – Time Value of Money 5 - 42 Deferred Annuity: The other special case of annuity is deferred annuity. It is regular annuity with the exception that the cash begins to flow only after the deferral period. If in a siz -years annuity the deferral peiod is 2 years, cash flows will begin at the end of the third year. In case, the present value of the cash flow during the deferral period will be subtracted from the present value of the cash flow for the entire period.

Ex. If the annual cash flow is Rs.500, the total period of annuity is 6 years and the deferral period is 2 years, the present value at a rate of discount of 10% will be:

CHAPTER 2 – Time Value of Money 5 - 43 Ex. If the annual cash flow is Rs.500, the total period of annuity is 6 years and the deferral period is 2 years, the present value at a rate of discount of 10% will be A * (PVIFA 10%,6+2 yrs - PVIFA 10%, 2 yrs ) 500*5.3349-1.73554) = Rs.1799.70

PROBLEMS:

CHAPTER 2 – Time Value of Money 5 - 44 Ex. 1. If an investor expects a perpetual sum of Rs.500 annualy from his investment, what is the present value of this perpetuity, if his time preference, or discount rate is 10%. P = A/I 500/0.10 = Rs.5000 PROBLEMS

PROBLEMS:

CHAPTER 2 – Time Value of Money 5 - 45 Ex. 1. If an investor expects a perpetual sum of Rs.500 annualy from his investment, what is the present value of this perpetuity, if his time preference, or discount rate is 10%. P = A/I 500/0.10 = Rs.5000 PROBLEMS

PowerPoint Presentation:

Ex.2 If an amount of Rs.80000 is deposited in a fixed deposit for seven years at 10% compound rate of Interest. How much can one withdraw each year to leave exactly zero in the amount at the end of the seventh year? Solution: Total amount at the end of 7 th year is 155920 CVIFA at 10% for 7 years is 9.487 Annual withdrawal amount is 155920/9.487= Rs.16435.12 CHAPTER 5 – Time Value of Money 5 - 46

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Calculate the present value of each flows of Rs.700 per year for ever (in perpetuity) 1) assuming an interest rate of 7 % 2) Assuming an interest rate of 10% (Rs.10000, Rs.7000) CHAPTER 5 – Time Value of Money 5 - 47

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Ex. 4. If the discount/required rate is 10 per cent, compute the present value of the cashflow streams detailed below A) Rs.100 at the end of year 1 B) Rs.100 at the end of year 4 C) Rs.100 at the end of ( i ) year 3 and (ii) year 5 and D) Rs.100 for the next 10 years (for years 1 through 10 ) CHAPTER 5 – Time Value of Money 5 - 48

PowerPoint Presentation:

P=FV * PVIF i , n 100*0.9091 = 90.91 100*0.683 = 68.30 (100*0.7513) + (100*0.6209) =75.13 + 62.09 =137.22 FV=PVIFA I, n 1000*6.1446 = Rs.614.46 CHAPTER 5 – Time Value of Money 5 - 49

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Ex. 5. An executive is about to retire at the age of 60. His employer has offered him two post-retirement options a) Rs.20,00,000 lump sum b) Rs.2,50,000 for 10 years. Assuming 10 % interest which is a better option PV=A*PVIFA 10,10 = 250000*6.1446) = 1536150 Since the lumpsum of Rs.200000 is worth more now, the executive should opt for it 5 - 50

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Ex.6 ABC Ltd has borrowed Rs.30,00,000 from CanBank Home Finance Ltd to finance the purchase of a house for 15 year. The rate of interest on such loans is 24 per cent per annum. Compute the amount of annual payment/installment. Solution: P/PVIFA 16,5 = 3000000/4.0013 = Rs. 749756.32 5 - 51

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