The Time Value of Money A core concept in financial management

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The Time Value of Money A core concept in financial management. Lesson Objectives . To introduce the time value concept Calculate present and future values of any set of expected future cash flows. Time Value ???. Rs.1000 you received today or Rs.1000 will be received tomorrow.

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The Time Value of Money

A core concept in financial management

Lesson Objectives

To introduce the time value concept

Calculate present and future values of any set of expected future cash flows.

Time Value ???

What do you prefer?

Simple reason is your time preference for money.

Therefore, you may expect to get an extra cash amount as compensation for delaying.

It is called the interest.

This Interest for the Time Preference for Money is called the Time Value of Money

Why we need a premium for future cash flows?

Alternative uses of money.

(Investment opportunities)

Individual’s preference for early consumption (time preference theory )

The risk associated with future cash flows

The interest rate for the time value of money can be regarded as the opportunity cost

Comparison of CF at different Time intervals

• A rupee received today is more valuable than a rupee received tomorrow.
• Thus, cash flows in different time periods cannot be compared as they are.
• There are two ways
• Future value
• Present value
Translate Rs.1 today into its equivalent in the future (COMPOUNDING).

Today

Future

?

Today

Future

?

• Translate Rs.1 in the future into its equivalent today (DISCOUNTING).

?

ASSUMPTIONS

A point in time is denoted by the letter “t”.

Unless otherwise stated, t=0 represents today (the decision point).

Unless otherwise stated, cash flows occur at the end of a time interval.

Cash inflows are treated as positive amounts, while cash outflows are treated as negative amounts.

Compounding frequency is the same as the cash flow frequency.

The Time Line

Beginning of the fourth year

End of the

third year

Today

t=1

t=2

t=3

t=4

t=0

Cash Flows

Single cash flow

Annuity

Multiple/ uneven cash flows

Future Value and Compounding Process

Future value

Is the total of the principle amount and the interest accumulated on the principle for a given period.

Is the sum which an initial amount of principle (or present value (PV)) is expected to grow over a given (n) period at a given interest rate.

Example 1:

Suppose you place Rs.100 in a savings account that earns 6% interest compounded annually.

How much can you get at the end of each period?

Future Value Formula

Let PV = Present Value

FVn= Future Value at time n

r = interest rate (ordiscount rate) per period.

Future Value Factor

r = 15%

r = 10%

r = 5%

r = 0%

More Frequent Compounding

Interest may be compounded more than once a year.

The Nominal Rate (Annual Percentage Rate (APR)) is the periodic rate times the number of periods per year.

The Effective rate (Annual Percentage Yield (APY)) is the “true” annually compounded interest rate.

Effect of Compounding Frequency on Future Value

Find the future value at the end of one year if the present value is Rs.20,000 and the interest rate is 16%. Use the following compounding frequencies:

• Annual Compounding
• Semiannual Compounding
• Quarterly Compounding
• Monthly Compounding
• Daily Compounding
• Continuous Compounding
Annual Compounding - Once a year

The periodic rate is 16%.

APY = APR = 16%

Compounding m-times in a year

Semi - Annual Compounding

Since m = 2, the periodic rate is 8%.

APY > APR

Quarterly Compounding

Since m = 4, the periodic rate is 4%.

Continuous Compounding

With continuous compounding, m becomes very large.

• As m approaches infinity, the value of (1+r/m)mn goes to er n. Thus,
• Then the effective rate = er - 1, where e = 2.71828.
• Thus, effective rate = (2.71828)0.16 - 1 = 0.17351 or 17.351%.
• FV = Rs.20,000 (1.17351)1 = Rs.23,469.39
Present Value

Future value

Thus the Present Value

When we get the present value, the interest rate is referred as the Discount Rate and this process is called as Discounting

1. If you will receive Rs.100 one year from now, what is the PV of that Rs.100 if your opportunity cost is 6%2. If you receive Rs.100 5 years from now, what is the PV of that Rs.100, if your opportunity cost is 6%?3. What is the PV of Rs.1,000 to be received 15 years from now if your opportunity cost is 7%?

Present Value - single sums

Present Value Factor and Time

r = 0%

r = 5%

r = 10%

r = 15%

1. If you sold land for Rs.11,933 that you bought 5 years ago for Rs.5,000, what is your annual rate of return?2. Suppose you placed Rs.100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to Rs.500?

Present Value - single sums

Ex.

Multiple Cash Flows

PV of multiple cash flows = the sum of the present values of the individual cash flows.

FV of multiple cash flows at a common point in time = the sum of the future values of the individual cash flows at that point in time.

How do we find the FV/PV of a cash flow stream when cash flows are different? (Use a 10% interest rate).

Uneven Cash Flows

FV

-10,000 2,000 4,000 6,000 7,000

0

1

2

3

4

Annuities

An annuity is a series of identical cash flows that are expected to occur each period for a specified number of periods.

Thus, CF1 = CF2 = CF3 = Cf4 = ... = CFn

Examples of annuities:

Installment loans (car loans, mortgages).

Coupon payment on corporate bonds.

Types of Annuities

Ordinary Annuity:

An annuity with End-of-Period cash flows, beginning one period from today.

Annuity Due:

An annuity with Beginning-of-Period cash flows.

Deferred Annuity:

An annuity that begins more than one period from today.

Future Value of an Annuity

1,000 1,000 1,000 1,000

0

1

2

3

4

When it has n periods, the equation is

Simplification

When you substitute above variables and simplify the equation, You can arrive at

If you invest Rs.1,000 at the end of each next 3 years, at 8%, how much would you have after 3 years?

Future Value - annuity

Future Value Interest Factor for Annuity (FVIFA) Tables can be used to get the answer

Present Value Of an Annuity

1,000 1,000 1,000 1,000

0

1

2

3

4

When it has n periods, the equation is

Simplification

When you substitute above variables and simplify the equation, You can arrive at

What is the PV of Rs.1,000 CF at the end of each of the next 3 years, if the opportunity cost is 8%?

Present Value Of an Annuity

Present Value Interest Factor for Annuity (PVIFA) Tables can be used to get the answer

Using an interest rate of 8%, we find that:

The Future Value (at 3) is Rs.3,246.40.

The Present Value (at 0) is Rs.2,577.10.

0 1 2 3

Earlier, we examined this “ordinary” annuity:

1000 1000 1000

Same 3-year time line,

Same 3 Rs.1000 cash flows, but

The cash flows occur at the beginning of each year, rather than at the end of each year.

This is an “annuity due.”

0 1 2 3

Annuity Due

1000 1000 1000

If you invest Rs.1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3?

Equation and the Solution is

Future Value - annuity due

What is the PV of Rs.1,000 at the beginning of each of the next 3 years, if your opportunity cost is 8%?

0 1 2 3

Present Value - Annuity due

1000 1000 1000

• Equation and the solution is
Deferred Annuity

The first cash flow in a deferred annuity is expected to occur later than t=1.

The PV of the deferred annuity can be computed as the difference in the PVs of two annuities.

Deferred Annuity

An annuity’s first cash flow is expected to occur 3 years from today. There are 4 cash flows in this annuity, with each cash flow being Rs.500. At an interest rate of 10% per year, find the annuity’s present value.

0

1

2

3

4

5

6

Rs.500

Rs.500

Rs.500

Rs.500

Example

Cash flows from an investment are expected to be Rs.40,000 per year at the end of years 4, 5, 6, 7, and 8. If you require a 20% rate of return, what is the PV of these cash flows?

After graduation, you plan to invest Rs. 400 per month in the stock market. If you earn 12% per year on your stocks, how much will you have accumulated when you retire in 30 years?

If you borrow Rs.100,000 at 7% fixed interest for 30 years in order to buy a house, what will be your monthly house payment?

Upon retirement, your goal is to spend 5 years travelling around the world. To travel in style, it will require Rs.250,000 per year at the beginning of each year.

If you plan to retire in 30 years, what are the equal monthly (end of month) payments necessary to achieve this goal?

The funds in your retirement account will compound at 10% per annum on monthly basis.

Team Assignment

Present Value of Your Bank Loan

Cynthia Smart agrees to repay her loan in 24 monthly installments of Rs.250 each. If the interest rate on the loan is 0.75% per month, what is the present value of her loan payments?

You wish to retire 25 years from today with Rs.2,000,000 in the bank. If the bank pays 10% interest per year, how much should you save each year to reach your goal?

Rob Steinberg borrows Rs.10,000 to be repaid in four equal annual installments, beginning one year from today. What is Rob’s annual payment on this loan if the bank charges him 14% interest per year?

Loan Amortization Schedule

It shows how a loan is paid off over time.

It breaks down each payment into the interest component and the principal component.

We will illustrate this using Rob Steinberg’s 4-year Rs.10,000 loan which calls for annual payments of Rs.3,432.05. Recall that the interest rate on this loan is 14% per year.

APR and APY for an Installment Loan

Suppose you borrow Rs.5,000 from the bank and promise to repay the loan in 12 equal monthly installments of Rs.437.25 each, with the first payment to be made one month from today.

What is the APR?

What is the APY?

Perpetuity

A perpetuity is an annuity with an infinite number of cash flows.

The present value of cash flows occurring in the distant future is very close to zero.

At 10% interest, the PV of Rs.100 cash flow occurring 50 years from today is Rs.0.85!

The PV of Rs.100 cash flow occurring 100 years from today is less than one penny

Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity.

Mathematically,

(PVIFA r, n ) =

1

n

1 -

(1 + r)

r

We said that a perpetuity is an annuity where n = infinity. What happens to this formula when n gets very, very large?

1

n

1 -

(1 + r)

r

1 r

When n gets very large,

this reaches to zero.

So we’re left with PVIFA =

So, the PV of a perpetuity is very simple to find:

PMT

PV=

r

Present Value of a Perpetuity

What should you be willing to pay in order to receive Rs.10,000 annually forever, if you require 8% per year on the investment?

Find the present value of a perpetuity of Rs.270 per year if the interest rate is 12% per year.

The First Commerce Bank offers a Certificate of Deposit (CD) that pays you Rs.5,000 in four years. The CD can be purchased today for Rs.3,477.87. Assuming you hold the CD to maturity, what annual interest rate is the bank paying on this CD?

Ex.

Solving for an Unknown Interest Rate on a Loan

Erin Clapton borrows Rs.10,000 from her bank, and agrees to repay the loan in six equal annual installments of Rs.2,100 each. The first payment will be made one year from today. What interest rate is the bank charging her?

Solving for an Unknown Interest Rate on a Loan

at r = 8%, PVA6 = Rs.9,708.05

at r = 6%, PVA6 = Rs.10,326.38

at r = 7%, PVA6 = Rs.10,009.83

Rs.2,100

Rs.2,100

Rs.2,100

Rs.2,100

Rs.2,100

Rs.2,100

0

1

2

3

4

5

6

Rs.10,000