8. Unit # 4 Present Value for Cash Flows

5/26/2018 8. Unit # 4 Present Value for Cash Flows

1/8

Financial Institution

Zaheer Swati 1

Unit 4

PRESENT VALUE FOR CASH FLOWS

Different Financial instruments have very different streams of cash payments to the holder (known as cash

flows), with very different timing

All else being equal, instruments are evaluated against one another based on the amount of each cash flow and

the timingof each cash flow

The concept of present value(or present discounted value) is based on the commonsense notion that a dollar of

cash flow paid to you one year from now is less valuable to you than a dollar paid to you today. This notion is

true because you could invest the dollar in a savings account that earns interest and have more than a dollar in

one year.

There are two type of PV problem

Single-sum problems

Multi-period cash flow problems

4.1 Present Value Applications

There are four basic types of credit instruments which include present value concepts:

1. Simple Loan2. Fixed Payment Loan3. Coupon Bond4. Discount Bond

4.1.1 Simple Loan

Simple Loans require payment of one amount which equals the loan principal plus the interest (simple or

compound)

Example 4.1: How much would you pay today for loan, which will be worth Rs. 16,000 in three years? Assume interest

is 5%?

Solution: 16,000 / (1 + 0.05)3

Answer: Rs. 13,821.40

$100 at 10%

Year: 0 1 2 3 n

$100 $110 $121 $133

PVn= FV ni1

1

PV of Simple Loan


2/8


Zaheer Swati 2

Unit 4

Example 4.2:You are trying to save up Rs. 2,500 to go on a trip 2 years from now. You are going to invest your money

in a high-interest savings account where it will earn 7.1% interest, compounded quarterly. How much money do you

have to invest now to make sure you have enough for your trip?

Solution:

2,500 / (1 + 0.071/4)2*4

Answer: Rs. 2,171.76

Example 4.3:There are four opportunities of investment of Rs. 950. All cash inflows are presented in below mentioned

table. Assume that discount rate is 14 percent. Which investment is best?

Cash flows 1 2 3 4 5 Total

A 100 200 300 400 500 1,500

B 600 ----- 400 200 300 1,500

C ----- ----- 1,200 ---- 300 1,500

D 500 ----- 400 600 ---- 1,500

Solution:

Cash flows 1 2 3 4 5 Total

A 87.72 153.89 202.49 236.83 259.68 940.61

B 526.32 ----- 269.99 118.42 155.81 1,070.54

C ----- ----- 809.97 ---- 155.81 965.78

D 438.60 ----- 269.99 355.25 ---- 1,063.84

4.1.2 Fixed Payment Loan

Fixed-Payment Loans are loans where the loan principal and interest are repaid in several payments, often

monthly, in equal dollar amounts over the loan term.

Installment Loans, such as auto loans and home mortgages are frequently of the fixed-payment type

Present value of annuity has been used for these types of problems


3/8


Zaheer Swati 3

Unit 4

Example 4.4:You are making car payments of Rs. 315/month for the next 3 years. You know that your car loan has an

interest rate of 12.4% per annum, discounted monthly, what was the initial price of the car?

Solution:

315 [12/124.0

)12/124.01(

11

12*3

]


Example4.5:Mr.Mohammad Ali has received a job offer from a large investment bank as an accountant. His base

salary will be Rs. 35,000 constant to date of retirement. He will receive his first annual salary payment one year from the

day he begins to work. In addition, he will get an immediate Rs. 10,000 bonus for joining the company. Mr. Ali is

expected to work for 25 years. What is the present value of the offer if the discount rate is 12 percent?

Solution:

35,000 [12.0

)12.0.01(

11

25

]

PVA25 = 274,509.87

Bonus = 10,000

Answer: Rs. 284,509.87

Example4.6:You have just joined the investment banking firm. They have offered you two different salary arrangements.

You can have Rs. 50,000 per year for the next 3 years or Rs. 35,000 per year for the next 3 years, along with a Rs. 50,000

bonus today. If the market interest rate is 16%, which salary arrangement do you prefer?

Solution:

50,000 [16.0

)16.01(

11

3

] = Rs. 112,294.48

35,000 [16.0

3)16.01(

11

] =78,606.13 + 50,000 = 128,606.14

Answer: Option B

PVAn= CCF [i

in)1(

11

]

General Formula

Constant Rate


4/8


Zaheer Swati 4

Unit 4

Example4.7:You need to decide how much money to invest each month in order to have Rs. 5,000 in 6 years time. You

will be investing at an interest rate of 7.2%, compounded monthly. What should your monthly payments be?

Solution: CCF = 5,000

[12/072.0

)12/072.01(

11

12*6

]

Answer: Rs. 85.73

Example 4.8:Suppose you borrow Rs. 22,000 at 12 percent compound annual interest to be repaid over the next 6 years.

Equal installment payments are required at the end of each year. What will be installment amount and prepare loan

amortization table (rounding to Rupee)?

Solution:

CCF = PVA

[i

in)1(

11

]

CCF = 22,000

[12.0

)12.01(

11

6

]

CCF = 22,0004.111

CCF = 5,351

Loan Amortization Schedule

1 2 3 4 5

Year End Installment Payments Interest Principal Repayment Outstanding

12 % on 5 2-3 5-4

0 --- --- --- 22,000

1 5,351 22,000 * 0.12 = 2,640 2,711 19,289

2 5,351 19,289 * 0.12 = 2,315 3,036 16,253

3 5,351 16,253 * 0.12 = 1,950 3,401 12,852

4 5,351 12,852 * 0.12 = 1,542 3,809 9,043

5 5,351 9,043 * 0.12 = 1,085 4,266 4,777

6 5,351 4,777 * 0.12 = 573 4,777 0


5/8


Zaheer Swati 5

Unit 4

4.2 Effective Annual Rates

Effective Annual Rates (EAR) are use to compare securities and investments with different compounding cycle

or life

Example 4.9 :National Bank offers a stated annual interest rate of 4.1 percent, compounded quarterly, while Allied Bank

offers a stated annual interest rate of 4.05 percent, compounded monthly. In which bank should you deposit your

money?

Solution:

National Bank EAR = (1 +4

041.0)

4 1 = 4.16%

Allied Bank EAR = (1 +12

0405.0)

12 1 = 4.13%

Answer: National Bank

Example 4.10:What is the PV of $100 received in:

a. Year 10 at a discount rate of 1 percent?

PV = $100/1.0110

Answer: $90.53

b. Year 10 at a discount rate of 13 percent?

PV = $100/1.1310

Answer: $29.46

c. Year 15 at a discount rate of 25 percent?

PV = $100/1.2515

Answer: $ 3.52

Example 4.11:For each of the following, compute the future value:

Present Value Years Interest Rate Future Value

$1,000 4 10% $ 1,464.1

$2,500 6 12.25% $ 5,001.01

EAR = (1 + mAPR

)

m

- 1


6/8


Zaheer Swati 6

Unit 4

Example 4.12:A factory costs $800,000. You believe that it will produce a cash flow of $170,000 a year for 10 years.

If the opportunity cost of capital is 14 percent, what is the PV of the factory?

PVA10= 170,000 [ 14.0)14.01(

11

10

]

Answer: $ 886,739.66

Example 4.13:A machine costs $380,000 and it is expected to produce the following cash flows.

Year 1 2 3 4 5 6 7 8 9 10

CF ($000S) 50 57 75 80 85 92 92 80 68 50

If the cost of capital is 12 percent, should we buy or not?

109876

5432

1.12

$50,000

1.12

$68,000

1.12

$80,000

1.12

$92,000

1.12

$92,000

1.12

$85,000

1.12

$80,000

1.12

$75,000

1.12

$57,000

1.12

$50,000PV

PV= 44,642.8571 + 45,440.0510 + 53,383.5186 + 50,841.4463 + 48,231.2827 + 46,610.0632 + 41,616.1278 +

32,310.6582 + 24,521.4817 + 16,098.6618

Answer: $ 403,687.15

Example 4.14:How much must you deposit today in a bank account?

(a)If you wish to have Rs. 10,000 at the end of 3 months, if the bank pays 5.0% APR?

Solution:

10,000 / (1 + 0.05/12)3


(b)If you wish to have Rs. 50,000 at the end of 24months, if the bank pays 8.0% APR?

Solution:

50,000 / (1 + 0.08/12)24

Answer: Rs. 42,629.82


7/8


Zaheer Swati 7

Unit 4

(c)If you wish to have Rs. 6,000 at the end of 12 months, if the bank pays 9.0% APR?

Solution:

6,000 / (1 + 0.09/12)12


(d)If you wish to have: Rs. 10,000 at the end of 1 months, if the bank pays 5.0% APR?

Solution:

10,000 / (1 + 0.05/12)


(e)If you wish to have Rs. 6,000 at the end of 6 months, if the bank pays 9.0% APR?

Solution:

6,000 / (1 + 0.09/12) 6


(f)If you wish to have Rs. 12,000 at the end of 12 months, if the bank pays 6.0% APR?

Solution:

12,000 / (1 + 0.06/12)12

Answer: Rs. 11,302.86

Example 4.15:Which project, A or B, could be sold for the most money today? Project A will generate cash flows of Rs.

400, Rs. 500, and Rs. 600 in the third, fourth, and sixth years, respectively, of the project's life. Project B will generate

cash flows of Rs. 700 in year two and Rs. 800 in year nine, Investments similar to "A" return 10% on average, while

investments similar to "B" return 11%?

Solution Project A: 310.01

400

+ 410.01

500

+

610.01

600

300.5259 + 341.5067 + 338.6844

Answer: Rs. 980.72

Solution Project B: 211.01

700

+ 911.01

800

568.1357 + 312.7398

Answer: Rs. 880.88

Project A is worth more since it has a higher total PV


8/8


Zaheer Swati 8

Unit 4

Example 4.16:As winner of a breakfast cereal competition, you can choose one of the following prizes:

a. $100,000 now

b. $180,000 at the end of 5 years

c. $11,400 a year forever

d. $19,000 for each of 10 years

If the interest rate is 12 percent, which is the most valuable prize?

Solution:

a. PV = = $100,000

b. PV = $180,000/1.125 = $102,136.83

c. PV = $11,400/0.12 = $95,000

d. PVA10= 19,000 [ 12.0 )12.01(1

110

] = $ 107,354.24

Prize (d) is the most valuable because it has the highest present value

Example 4.17: The Rs. 100 on the time points of 0, 1, 2, 3 are equivalent to Rs. 453 in the time point 4 under a 5%

interest rate?

PVAn= 100 [ 05.0)05.01(

11

3

](1+0.05) = Rs. 454.60

PVn= 453 405,01

1

=Rs. 372.68

-

0 1 2

4

A = 100, -

+

FV= 453, -

5% =

3

A = 100, - A = 100, - A = 100, -

Documents

8. Unit # 4 Present Value for Cash Flows