4. Unit # 3 Time VAlue of Money and Interest

Financial Institution

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Unit 3

TIME VALUE OF MONEY & INTEREST

3.1 Time Value of Money A dollar in hand today is worth more than a dollar in hand tomorrow. Why is that?

I could buy something today and thus get the use today of what I buy

I could invest today and gain the return from that investment

I could avoid the loss of value due to inflation in costs

I could lend the money today and gain the interest on that loan

3.2 The Interest Interest rates are among the most closely watched variables in the economy

Its movements are reported almost daily by the media

An interest rate is simply the price of money

Compensation to lender for foregoing other useful investment

Equilibrium price at which demand and supply of fund meet

The intersection between demand and supply represents the equilibrium cost of borrowing funds, k*

This (k*) is called the pure or real rate of interest. This is the interest rate resulting purely as a result of the

interaction between the supply and demand for funds

It’s important to note that interest rates are determined by many other factors besides the real rate of interest

3.3 Why is there interest on a loan? There needs to be a return, given the value today vs. tomorrow

The loss of value from the other potential uses must be recognized

There are risks that the loan may not be repaid.

Interest

Rate (kd)

DEMAND

SUPPLY

k*

Q* Quantity of Funds


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Unit 3

3.4 The Relevant Variables for TVM o The initial amount lent, called the principal amount (P)

o The time period of the loan (n)

o The interest rate (i)

o Frequency of compounding or discounting

3.5 Type of Interest There are two types of interest, simple and compound

Simple interest is applied to the initial amount, called the principal, for a given time period for interest

If the period of the loan is greater than the time period for interest, the simple interest will be repeated, at the same

amount, and accumulate during successive time periods for interest until the end of the time period of loan

Compound interest is applied to the initial sum, plus any previous accumulated interest that has not been paid, for

each successive time period for interest

The rationale for compound interest is that the interest is in fact money that should be in hand at the end of the time

period for interest, i.e., at the time it is due. Therefore, if that interest is not received, it is, in effect, also lent and

therefore should also bear interest

Formula

Example 3.1: What is the future amount that will be available in four years if Rs. 8,000 is invested at 12% per year

simple interest now?

Solution

SIn = P + (P * i * n)

SIn = P + (P * i * n)

SI4 = 8,000 + (8,000 * 0.12 * 4)

SI5 = Rs. 11,840

FVn = PV (1 + i) n Formula for

Compound Interest


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Unit 3

Example 3.2: What is the future amount that will be available in four years if Rs. 8,000 is invested at 12% per year

compound interest now?

Solution

Example 3.3: Calculate simple interest and compound interest assuming that principal amount is Rs. 10,000; interest rate

is 9% for three years. What is the amount difference between compound and simple interest?

Year

Simple Interest Calculation

Compound Interest Calculation

Simple Interest

Calculation

Simple

Interest

Accumulated

Year-end

Balance

Compound

Interest

Calculation

Compound

Interest

Accumulated

Year-end

Balance

Yr 1 Rs. 10,000 * 9% Rs. 900 Rs. 10,900 Rs. 10,000 * 9% Rs. 900 Rs. 10,900

Yr 2 10,000 * 9% 900 11,800 10,900 * 9% 981.00 Rs. 11,881.00

Yr 3 10,000 * 9% 900 12,700 11,880.10 * 9% 1069.29 Rs. 12,950.29

Total Rs. 2,700 Rs. 2,950.29

Difference = 2,950.29 - 2,700 = Rs. 250.29

FVn = PV (1 + i) n

FV4 = PV (1 + 0.12) 4

FV4 =

Continuous

Compound

Simple


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Unit 3

3.6 Different Compounding Cycles or Intra Year Compounding o The value of lump sum amount (one time cash flow) at some future time evaluated at a given interest rate assuming

that compounding takes place more than one time in a year (Intra Year)

o Interest rate reduced while periods of time increase by frequency of compounding (m) i.e. i/m and n*m

Example 3.4: You have Rs. 9,000 to deposit. ABC Bank offers 12 percent per year compounded monthly, while King

Bank offers 12 percent but will only compound annually. How much will your investment be worth in 10 years at each

bank?

Solution: ABC Bank 9,000 (1 + 0.12/12) 10 * 12 Answer: Rs. 29,703.48

King Bank 9,000 (1 + 0.12) 10 Answer: Rs. 27,952.63

Variance Rs. 1,750.85

Example 3.5: If interest is compounded quarterly, how much will you have in a bank account?

(a) If you deposit today Rs. 8,000 at the end of 3 months, if the bank pays 5.0% APR?

Solution: 8,000 (1 + 0.05 / 4)

Answer: Rs. 8,100

(b) If you deposit today Rs. 10,000 at the end of 6 months, if the bank pays 9.0% APR?

Solution: 10,000 (1 + 0.09 / 4) 2

Answer: Rs. 10,455.06

(c) If you deposit today Rs. 80,000 at the end of 12 months, if the bank pays 8.0% APR?

Solution: 80,000 (1 + 0.08 / 4) 1 * 4

Answer: Rs. 86,594.57

(d) If you deposit today Rs. 5,000 at the end of 24 months, if the bank pays 5.0% APR?

Solution: 5,000 (1 + 0.05 / 4) 2 * 4

Answer: Rs. 5,522.43

FVn = PV (1 + i / m) n * m General Formula


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Unit 3

3.7 Future value for stream of constant cash flows (Annuity) If constant cash flows occur at the end of each period/year is called ordinary annuity for instance payment of car

loan, mortgage loan and student loan are examples of ordinary annuity

If payments or receipts are made at the beginning of each year/period, the annuity is an annuity due, rental payment

for apartment and life insurance payments are typical examples of this annuity

Constant Cash flow Stream for Ordinary Annuity

Example 3.6: You decide to work for next 20 years before an early-retirement. For your post-retirement days, you plan

to make a monthly deposit of Rs. 1,000 into a retirement account that pays 12% p.a. compounded monthly. You will

make the first deposit one month from today. What will be your account balance at the end of 20 years?

Solution:

1,000 [ 12/12.01)12/12.01( 12*20 ]

Answer: Rs. 989,255.37

Constant Cash flow Stream for Ordinary Annuity

FVADue = CCF [ ii n 1)1( ] (1+i)

Annuity Due

0 1 2 3 4 5 Rs. 1,000 Rs. 1,000 Rs. 1,000 Rs. 1,000 Rs. 1,000

0 1 2 3 4 5 Rs. 1,000 Rs. 1,000 Rs. 1,000 Rs. 1,000 Rs. 1,000

FVAn = CCF [ ii n 1)1( ]

Ordinary Annuity


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Unit 3

Example 3.7: Suppose you deposit Rs. 1,000 in an account at the start of each of the four years. If the account earns 12

percent annually, how much will be in the account at the end of fourth year?

Solution: 1,000 [ 12.01)12.01( 4 ] (1+0.12)


Example 3.8: If you put Rs. 100 in the market at the end of every year for 20 years at 10%, how much would you end up

with? What if you put the Rs. 100 in at the beginning of every year?

Solution (a): 100 [ 10.01)10.01( 20 ]


Solution (b): 100 [ 10.01)10.01( 20 ] (1+0.10)


Example 4.9: You deposit Rs. 17,000 each year for 10 years at 7%. Then you earn 9% after that. If you leave the money

invested for another 5 years how much will you have in the 15th year?

Solution:

FVAn = CCF [ ii n 1)1(

]

FVA10 = 17,000 [ 07.01)07.01( 10

]

FVA10 = Rs. 234,879.62

FVn = PV (1 + i) n

FV5 = 234,879.62 * (1 + 0.09) 5

FVA5 = Rs. 361,391.41

Example 3.10: Mr. Kazmi has decided to start saving for his post-retirement period. Beginning his 21st birthday has one

year from now. Kazmi plans to invest Rs. 2,000 each birthday into bank, investment earning a 9 percent compound

annually. He will continue his saving plan for ten years and then stop making payments. But his saving in bank will

continue to 6% for 35 more years. What is the present value of these cash flows?

Solution:

2,000 [ 09.01)09.01( 10 ] = Rs. 30,385.86

FV35 = 30,385.86 (1 + 0.06)35

FV35 = 233,548.36

PV45 = 233,548.36/ (1+ 0.10)45


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4. Unit # 3 Time VAlue of Money and Interest