Time Value of Money

Chapter 5

Important Concepts to be discussed

1.Opportunity cost and TVOM2.Simple vs. Compound Interest3.Lump sum payments4.Annuities5.Perpetuities6.Complex Cash Flow Streams7.Compounding Frequency8.EAR and APR

Introduction

• ‘A dollar in hand today is worth more than a dollar to be received in future’.

Example 1Today 30 Days from now

Option 1 $100 $100

Option 2 $100 $105

• In case of Option 1 the seller would go for $100 today.• In case of Option 2 the seller would be indifferent between

receiving $100 today or going for $105, 30 days from now.• $5 thus is seller’s opportunity cost which can be defined as:

‘the cost of selecting an alternative is the benefit foregone from the next best alternative’.

1. Opportunity Cost

• The Economist.com defines opportunity cost as,– “the true cost of something is what we give up to

get it”.• Opportunity cost of an investment is the ROR

available on the best alternative investment of similar risk.

Opportunity Cost (contd.)

• Personal Opportunity Cost– Personal opportunity cost refers to the time,

health or energy. – For example, time spent on studying usually

means lost time for leisure or working.

Opportunity Cost (contd.)

• Financial Opportunity Cost– Financial opportunity cost involve monetary value

of decisions made.– For example, the purchase of an item from your

savings means you can no longer earn interest on these funds.• As we know the true cost of something is what we give

up to get it so by purchasing an item we give up the chance to earn interest on these savings.

Measuring Opportunity Cost using Time Value of Money

• TVOM can be used to measure financial opportunity cost using interest calculations. – For example spending $1000 from a savings

account that was giving 4% a year means an opportunity cost of $40 in lost interest.

$1000 x 0.04 = $40

2. Future Value of Lump Sum Payments

• Future Value refers to the ‘amount of money an investment will grow into over a period of time at some given interest rate’.

• Defined in another way, ‘Future value refers to the cash value of an investment at sometime in future’.

2. Future Value of Lump Sum

• Example 2• Investing for a Single Period (e.g. 1 year)– Suppose we invested $100 in an account that pays

10% interest rate per year. By the end of the year we’ll have 110 in our account.

2. Future Value of Lump Sum

• Calculation• 100 x 10% = 10• Principal i interest pmt.

• At the end of the year we’ll have $ 100 +$ 10

$ 110

2. Future Value of Lump Sum

• Formula• FV = PV + (PV x i)• 110 = 100 + (100 x 10%)

110 = 100 + 10$110 = $110

FV = PV + (PV x i)FV = PV (1 + i)FV = 100 (1 + 0.10)FV = 100 x 1.10FV = $110

2. Future Value of Lump Sum

• Example 3• Investing for more than period (e.g. 2 years)– Going back to our $100 investment what we will have

after 2 years?– During 1st year FV = PV (1+i) (1+i)

• FV = PV (1+ i) FV = 100 (1.10) (1.10)• FV = 100 (1.10) = $110 FV = $121

– During 2nd year or• FV = PV (1+ i) • FV = 110 (1.10) = $121

This 121 is the future value of $100, two years from now at 10% ROI.

FV = PV (1+i)n

FV = PV (1+i)2

FV = 100 (1+0.10)n

FV = 100 (1.10)2

FV = $121

Simple vs. Compound InterestExample 4

Compounding

• The process of leaving your money and any accumulated interest in an investment for more than one period, thereby reinvesting the interest is called ‘compounding’.

• Compounding the interest means earning interest on interest so we call the result ‘compound interest’.

We can generalize this as . . .

FV = PV (1 + i)n

FutureValue

FutureValue

Present ValuePresent Value InterestRate

InterestRate

Numberof Compounding

Periods

Numberof Compounding

Periods

Future Value of a Single Amount

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NumericalsExample 5• Find the following future values:• a. An initial £ 500 compounded for 1 year at 6 percent.• b. An initial £ 500 compounded for 2 years at 6 percent.Example 6• What’s the future value of $100 after 3 years if it earns 10%, annual

compounding?Example 7• For example, you earn $500 from your summer job and want to save for

European trip in the next three years. How much will you have when you go for the trip if you deposit the money in a savings account that earns 10% interest? Using the time line for this problem, complete the equation:

Revision of Last Session• Future Value of a Lump Sum• The future value of a lump sum usually is the easiest time value concept

to understand. • The term “lump sum” refers to a single-sum payment or receipt at one

point in time. • The future value of a single sum is the future amount of an initial

deposit when it is compounded for a given number of periods and at a given interest rate.

• Compounding is the process whereby interest is earned upon interest. • When a deposit is made, interest is earned on the deposit in the first

period; in subsequent periods, interest is earned not only on the original deposit but also on the interest earned in each of the previous compounding periods. Thus, interest is earned on increasing amounts over time.

Simple Interest• Interest is the fee paid to use someone else’s money. • Interest on loans of a year or less is frequently calculated as simple

interest, which is paid only on the amount borrowed or invested and not on past interest.

• The amount borrowed or deposited is called the principal. • The rate of interest is given as a percent per year, expressed as a decimal.

For example, 6% = .06 and 11 1/2 % = .115. • The time during which the money is accruing interest is calculated in

years. (6 months mean 6/12 = 0.5 year, 9 months mean 9/12 = 0.75 year)• Simple interest is the product of the principal, rate, and time.• Simple interest is normally used only for loans with a term of a year or less

and for bonds (A typical bond pays simple interest twice a year ).

Simple Interest (Example 8)

Simple Interest (Example 9)

Compound Interest• With annual simple interest, you earn interest each year on

your original investment. • With annual compound interest, however, you earn interest

both on your original investment and on any previously earned interest.

• To see how this process works, suppose you deposit $1000 at 5% annual interest. The following chart shows how your account would grow with both simple and compound interest: (Example 10)

Compound Interest

• As the chart shows, simple interest is computed each year on the original investment, but compound interest is computed on the entire balance at the end of the preceding year.

• So simple interest always produces $50 per year in interest, whereas compound interest produces $50 interest in the first year and increasingly larger amounts in later years (because you earn interest on your interest).

Question: Simple vs. Compound InterestExample 11

• What is the future value of $100 after 5 years– at 10% compound interest?– At 10% simple interest?

FV = PV (1 + i)n

FutureValue

FutureValue

Present ValuePresent Value InterestRate

InterestRate

Numberof Compounding

Periods

Numberof Compounding

Periods

Future Value of a Single Amount when Interest is Compounded

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Future Value of Lump Sum Amount when Interest is Compounded

(Example 12 & 13)

Present Value of a Lump Sum Amount

• The process of determining the present value of a payment or a stream of payments that is to be received in the future.

• The formula for future value of lump sum payment, FV = PV(1 + i)n, has four variables: FV , PV , i , and n .

• Given the values of any three of these variables, the value of the fourth can be found. In particular, if the future value, i , and n are known, then PV can be found.

• Here, PV is the amount that should be deposited today to produce FV dollars in n periods.

Present Value of a Lump Sum Amount (Example 14)

Interest Rate of a Lump Sum Amount (Example 15)

Summing Up: Lump Sum Payments

• FV = PV (1 + i)n

• PV = FV/ (1 + i)n • Similarly we can also calculate i and n from the

above formula, if any of the 3 variables are given.

Time Line

• Graphical representation to show timing of the cash flows.• Number above tick mark represent end of year values. • For instance 1 means end of year 1 and beginning of year 2. 2

means end of year 2 and beginning of year 3. • Interest rate is placed directly above the time line.• Cash flows are placed directly below the time line.

Annuities

• So far, only lump-sum deposits and payments have been discussed. Many financial situations, however, involve a sequence of payments at regular intervals, such as weekly deposits in a savings account or monthly payments on a car loan.

• Such periodic payments are now the subject of our discussion.

Annuities

• Definition: A stream of equal payments occurring at fixed intervals for a specified time.

• For a payment to be classified as annuity certain conditions must be met:1. Payment amount should be same.2. Time interval between occurrence of any two periods

should be same.3. Payment last for a certain time period.– For example: $1500 (1) deposited at the end of each year

(2) for the next 6 years (3) in an account paying 8% interest compounded annually.

Ordinary Annuities

• Characteristics– In an ordinary annuity payment occurs at the end

of each period.– The first payment starts one period from the

beginning of the timeline. (because for ordinary annuity payments occurs at end of each of period, so for example 1st period ends at 1 on timeline, not 0 which is the beginning of timeline)

– Last payment is at the end of the timeline.

Ordinary Annuities

• Calculating FV Ordinary Annuity• Ordinary Annuities —ones where the payments are made at the end of

each period. • Example 16

FV of Ordinary Annuity• Step by step method (we treat each pmt as a single amount)

• Formula method

FVn = FV of annuity at the end of nth period. PMT = annuity payment deposited or received at the end of each period. i = interest rate per period n = number of periods for which annuity will last.

Example 17 (Future Value of an Ordinary Annuity Stream)

Jill has been faithfully depositing $2,000 at the end of each year over the past 5 years into an account that pays a guaranteed 8% per year. How much money has she have accumulated in the account?

$11733.202

FV of Ordinary Annuity (Example 18)

Solving for PMT in an Ordinary Annuity (when FV is given in question)

Instead of figuring out how much money you will accumulate (i.e. FV), you may like to know how much you need to save each period (i.e. PMT) in order to accumulate a certain amount at the end of n years.

In this case, we know the values of n, i, and FVn in the formula and we need to determine the value of PMT.

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Examples 19 You would like to have $25,000 saved 6 years from now to pay

towards your down payment on a new house. If you are going to make equal annual end-of-year payments to an investment account that pays 7%, how big do these annual payments need to be?

20 How much must you deposit in a savings account each year, earning 8% interest in order to accumulate $5,000 at the end of 10 years?

21 If you can earn 12% on your investments, and you would like to accumulate $100,000 for your child’s education at the end of 18 years, how much must you invest annually to reach your goal?

Verify the answers: 3494.89; 345.15;1793.7340

Solving for ‘n’

Taking data from Example 19: FV =25000, i = 7%, PMT = $3494.895, n = ?

n = 6 years

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Solving for ‘i’

Taking data from Example 19: FV =25000, i = ?, PMT = $3494.895, n = 6

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The Present Value of an Ordinary Annuity

The present value of an ordinary annuity measures the value today of a stream of cash flows occurring in the future.

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• Also verify through formula. 44

Example 20: We compute the PV of each single cash flow and sum them up.

The Present Value of an Ordinary Annuity

Example 21: (a) What is the value today or lump sum equivalent of

receiving $3,000 every year for the next 30 years if the interest rate is 5%

(b) what will be the value of annuity after 30 years?

For the example, FV=199,316.54. PV=46,117.35.

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Formulas for the Present and Future Values of an Ordinary Annuity

Checkpoint 5.1: Check Yourself

The Present Value of an Ordinary Annuity Your grandmother has offered to give you $1,000 per year for the next 10

years. What is the present value of this 10-year, $1,000 annuity discounted back to the present at 5%?

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Checkpoint 5.1

Verify the answer:7721.73;

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Checkpoint 5.2: Check Yourself

What is the present value of an annuity of $10,000 to be received at the end of each year for 10 years given a 10 percent discount rate?

Verify the Answer: 61,445.67

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Annuities Due Annuity due is an annuity in which all the cash flows occur at the beginning of

the period. For example, rent payments on apartments are typically annuity due as rent is paid at the beginning of the month.

Computation of future value of an annuity due requires compounding the cash flows for one additional period, beyond an ordinary annuity.

Computation of present value of an annuity due requires discounting the cash flows for one less period than an ordinary annuity.

Formula Adjustment :FV or PV (annuity due) = (FV or PV (ordinary annuity)x(1+i)

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Example Example 22 In example 21, we calculated the future value of 30-year

ordinary annuity of $3,000 earning 5% to be $199,316.54. What will be the future value if the deposits of $3,000 were made at the beginning of the year i.e. the cash flows were annuity due?FVAD=199,316.54 x 1.05= $ 209282.367

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Checkpoint 5.3 Checkpoint 5.2 where we computed the PV of 10-year ordinary

annuity of $10,000 at a 10% discount rate to be equal to $61,446. What will be the present value if $10,000 is received at the beginning of each year i.e. the cash flows were annuity due?PVAD=61446x1.1= $67590.6

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Class Practice and Homework

• Attempt Question # 5-1, 5-2, 5-3, 5-4, 5-6, 5-9, 5-10, 5-12, 5-13, 5-14, 5-15 (of your Textbook) for homework.

Assignment Questions

• Please start solving Assignment 4, Part I.