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Meaning of Time value of money, Concept of Time Line, Calculating Future value, Present value, Discounting, Compounding, Different types of Interest Rates and their calculations, Finding difference between CI and SI, Application of Time value of money

Contents

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What is Time value of money?

The time value of money is one of the most important concepts in finance. Money that the firm has in its possession today is more valuable than future payments because the money it now has can be invested and earn positive returns.

Money received now can be invested to earn additional cash (interest).

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Time Value Terminology

• Future value (FV) is the amount an investment is worth after one or more periods.

• Present value (PV) is the current value of future cash flows of an investment.

• Simple interest refers to interest earned only on the original capital investment amount.

• Compound interest refers to interest earned on both the initial capital investment and on the interest reinvested from prior periods.

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Time Line.

A horizontal line on which time zero appears at the leftmost end and future periods are marked from left to right; can be used to depict investment cash flows is called time

CF0 CF1 CF3CF2

0 1 2 3

i%

Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.

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Time line for uneven CFs: -$500 at t = 0 and $100, $75, and $50 at the end of Years 1 through 3.

100 50 75

0 1 2 3i%

-500

100 100 100

0 1 2 3

i%

-500

Time line for even CFs

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Here, FV = Future ValuePV = Present Valuei = Interest raten = Number of yearm = Number of Periods in a year

Future Value is the amount to which a cash flow or series of cash flows will grow over a given period of time when compounded at a given interest rate.Compounding is the process of determining the value of a cash flow or series of cash flows some time in the future when compounded interest is applied.

ni 1PV FV

mn

m

i 1 PV FV

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FV Problem 01: Single Sum

FV = ?

0 1 2 3i = 10%

-100

What’s the FV of an initial $100 after 3 years if i = 10%?

FV3 = PV(1 + i)3 = $100(1.10)3 = $133.10.

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FV problem 02: Various Compounding Period

What will $1 000 amount to in 5 years’ time if interest is 12% per annum, compounded annually?

762.30 $1

1.7623 $1000

0.12 1 $1000 FV 5

FV problem 03: Various Compounding Period

What will $1 000 amount to in 5 years’ time if interest is 12% per annum, compounded monthly? $1816.70

1.8167 $1000

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12% 1 $1000

m

12% 1$1000 FV

512

tm

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FV of various Compounding Period

Find the amount to which $500 will grow under each of the following conditions:

a. 12% compounded annually for 5 years

b. 12% compounded semiannually for 5 years

c. 12% compounded quarterly for 5 years

d. 12% compounded monthly for 5 years

e. 12% compounded daily for 5 years

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Future Value of Multiple Cash Flows

You deposit $1,000 now, $1,500 in one year, $2,000 in two years and $2,500 in three years in an account paying 10% interest per annum. How much do you have in the account at the end of the third year?

$1 000 (1.10)3 = $1 331$1 500 (1.10)2 = $1 815$2 000 (1.10)1 = $2 200$2 500 1.00 = $2 500Total = $7 846

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Multi-Period Future ValueSuppose Mr. Amin deposited Tk 50000 in a bank. The bank promised to give 10% interest per annum to Mr. Amin. What will be the future value after 4 Years, 6 years, and 8 years if compounded A. Annually, B. Half yearly C. Quarterly and D. Monthly ? Check the error on the result!

Period After 4 Years

After 6 years

After 8 years

Annually 73205 88578.05 107179.44

Half Yearly 73872.77 89792.82 109143.73

Quarterly 74225.28 90436.30 110187.85

Monthly 74349.63 90663.66 110557.36

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Finding The Interest Rate (Rate of Return)

Problem: Abir Hasan, who recently won $ 10,000 in the lottery, wants to buy a car in 5 years. Abir estimates that the car will cost $ 16,105 at that time. What interest rate must he earn to be able to afford the car?

%

).(

).(

).()(

.)(

,$

,$)(

)(,$,$

)(

.

.

10

161051

610511

610511

610511

00010

105161

10001010516

1

200

200

5

1

5

15

5

5

5

r

r

r

r

r

r

r

rPVFV t

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Simple versus Compound Interest

Assume that you win a lottery prize of Tk. 10000. suppose you invest your winnings at 6% for 20 years. How much MORE do you have at the end of the 20 years at the end of 20 years because your “interest earns interest”, that is , is compounded annually than if your investment just earns “simple interest”?

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PV Problem 01: What’s the PV of $100 due in 3 years if i = 10%?

PV = $100

11.10

= $100 0.7513 = $75.13.

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ni+1

FV =PV

mn

mi

+1

FV =PV

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Practice: Multi-Period FVAsia Bank pays 8% and compounds interest quarterly. If you want to get Tk 10000 at the end of five years from now, what you have to deposit now?

Tk 6729.71

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PV Problem 02: Present Value of Multiple Cash Flows

You will get $1 500 in one year, $2000 in two years and $2 500 in three years in an account paying 10% interest per annum. What is the present value of these cash flows?

$2 500 (1.10)-3 = $1 878

$2 000 (1.10)-2 = $1 653

$1 500 (1.10)-1 = $1 364

Total = $4 895

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Example of PV of a Cash Flow StreamJoe made an investment that will pay $100 the first year, $300 the second year, $500 the third year and $1000 the fourth year. If the interest rate is ten percent, what is the present value of this cash flow stream?

0 1 2 3 4

?

$100 $300 $500 $1000

???

i = 10%

Draw a timeline:

PV = [100/(1+.1)1]+[$300/(1+.1)2]+[500/(1+.1)3]+[1000/(1.1)4]

PV = $90.91 + $247.93 + $375.66 + $683.01PV = $1397.51

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Present Value of Various Compounding Periods

Find the present value of $500 due in the future, under each of the following conditions:

a. 12% interest compounded annually, discounted back five years

b. 12% interest rate, semiannually compounding, discounted back five years

c. 12% interest rate, quarterly compounding, discounted back five years

d. 12% interest rate, monthly compounding, discounted back five years

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Problems: FV

Assume that it is now January 1, 2005. On January 1, 2006, you will deposit $1000 into a savings account that pays 8%.

1. If the bank compounds interest annualy, how much will you have in your account on January1, 2009?

2. What would your January 1, 2009, balance be if the bank used quarterly compounding rather than annual compounding?

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Finding the Time to Double: The Rule of 72

The ‘Rule of 72’ states:If you earn r % per year, your money will be doubled in about 72 / r % years.

For example, if you invest at 6%, your money will double in about 12 years.

This rule is only an approximate rule.

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Sample Practice at Home

1. If you invest Tk 500 today in an account that pays 6% interest compounded annually, how much will be in your account after two years?

2. Which amount is worth more at 14%: Tk1000 in hand today or Tk 2000 due in six years?

3. The average cost to attend 1 year of a public university today is $7,414. If your child will begin college 11 years from now and his first college payment will be made then, how much will one year of college cost if college costs rise 4% per year?

4. What is the present value of $100 invested at 12% per annum, compounded semi-annually, for 2 years?

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Sample Practice at Home

4. Suppose you decide that you will need $40,000 in 18 years to pay for your new child's college education. If you can earn 4%, compounded annually, on your money, how much do you need to invest today?

5. Suppose you decide that you will need $40,000 in 18 years to pay for your new child's college education. If you can earn 4%, compounded quarterly, on your money, how much do you need to invest today?

6. If your initial investment is $230 and you expect to receive $350 in 3 years, what is the rate of return on the investment?

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Practice