Business Analysis and Valuation

Dr. Aprajita Pandey

BITS Pilani Department of Economics and Finance
Pilani Campus
 BITS Pilani
Pilani Campus

Lecture No. 3
 Time Preference for Money

• Time preference for money is an individual’s preference for

possession of a given amount of money now, rather than the same
amount at some future time.
• Three reasons may be attributed to the individual’s time preference for
money:
• risk
• preference for consumption
• investment opportunities
 Time Preference for Money
r Central To All Financial Valuation Techniques

r Techniques Used By Investors & Firms Alike

 Required Rate of Return

• Time preference for money is generally expressed by an interest rate .

This rate will be positive even in the absence of any risk. It may
therefore be called risk free rate.
• In reality, an investor will be exposed to some degree of risk.
Therefore, he would also require a rate of return, called risk premium.
• The risk free rate compensates for time while risk premium
compensates for risk.
• The required rate of return may also be called as opportunity cost of
capital.
 The Role of Time Value in Finance

• Most financial decisions involve costs & benefits that are spread out
over time.
• Time value of money allows comparison of cash flows from different
periods.
• Question?
• Would it be better for a company to invest $100,000 in a product that
would return a total of $200,000 after one year, or one that would
return $220,000 after two years?
 Time Value Adjustment

• Two most common methods of adjusting cash flows for time value of
money:
• Compounding—the process of calculating future values of cash flows
and
• Discounting—the process of calculating present values of cash flows.
 Future Value

• The general form of equation for calculating the future value of a lump
sum after n periods may, therefore, be written as follows:

Fn  P (1  i ) n
• The term (1 + i)n is the compound value factor (CVF) of a lump sum
of Re 1, and it always has a value greater than 1 for positive i,
indicating that CVF increases as i and n increase.
 Future Value of a Lump Sum

• Suppose that Rs. 1,000 are placed in a bank’s savings account at 5%

compound interest rate. How much shall it grow at the end of three
years
• Ans. 1157.62
The Power Of Compound Interest
40.00
20%

Future Value of One Dollar ($) 30.00

25.00

20.00 15%

15.00

10.00 10%
5.00
5%
1.00 0%
0 2 4 6 8 10 12 14 16 18 20 22 24
Periods
 Example

• If you deposited Rs. 55650 in a bank, which was paying a 15% interest
rate on a ten-year time deposit, how much would the deposit grow at
the end of ten years?
• 55650 * 4.046 = 225159.90
 Future value of an Annuity

• Annuity is a fixed payment (or receipt) each year for a specified

number of years. If you rent a flat and promise to make a series of
payments over an agreed period, you have created an annuity.
 (1  i ) n  1 
Fn  A  
 i 
• The term within brackets is the compound value factor for an
annuity of Re 1, which we shall refer as CVFA.
Fn =A  CVFA n, i
 Example

• Suppose a firm deposits Rs. 5, 000 at the end of each year for four
years at 6% rate of interest. How much would this annuity accumulate
at the end of the fourth year?

• Ans. 21873
 Sinking Fund

• Sinking fund is a fund, which is created out of fixed payments each

period to accumulate to a future sum after a specified period.
• The factor used to calculate the annuity for a given future sum is
called the sinking fund factor (SFF).

 i 
A = Fn  
 (1  i ) n
 1 
 Example

• Suppose we want to accumulate Rs.21,873 at the end of four years

from now. How much should we deposit each year at an interest rate
of 6% so that it grows to Rs.21,873 at the end of fourth year?
• Ans. 5,000
 Present Value of a Single Cash Flow

• The following general formula can be employed to

calculate the present value of a lump sum to be
received after some future periods:
Fn
P  F 
n  (1  i ) n

(1  i ) n

• The term in parentheses is the discount factor or

present value factor (PVF), and it is always less than
1.0 for positive i, indicating that a future amount has a
smaller present value.

PV  Fn  PVFn ,i
 Example
• Suppose an investor wants to determine the present value of Rs.
50,000 to be received after 15 years. The interest rate is 9%.
• Ans. 13,725
Problem
• Determine the present value of the cash inflows of Rs. 3,000 at the end
of each year for the next four years and Rs. 7,000 and Rs. 1 ,000
respectively at the end of years 5 and 6. The appropriate discount rate
is 14%.
Present Value of an Annuity
• The computation of the present value of an annuity
can be written in the following general form:

1 1 
P  A  
 i i 1  i  
n

• The term within parentheses is the present value

factor of an annuity of Re 1, which we would call
PVFA, and it is a sum of single-payment present
value factors.

P = A × PVAFn, i
 Example
• An investor, who has required interest rate as 10% per year, may have
an opportunity to receive an annuity of Rs.1 for four years.
 Problem
• Jai Chand is planning for his retirement. He is 45 years old today and
would like to have Rs. 300,000 when he attains the age of 60. He
intends to deposit a constant amount of money at 12% each year in the
public provident fund in the SBI to achieve his objective. How much
money should Jai Chand invest at the end of each year, for the next 15
years, to obtain Rs. 300,000 at the end of that period?
 Problem
• An executive is about to retire at the age of 60. His employer has
offered him two post retirement options (a) Rs. 20,00,000 lump sum,
(b) Rs. 2,50,000 for 10 years. Assuming 10 per cent interest, which is
a better option.
Capital Recovery and Loan Amortisation
• Capital recovery is the annuity of an investment
made today for a specified period of time at a given
rate of interest. Capital recovery factor helps in the
preparation of a loan amortisation (loan repayment)
schedule.
 1 
A= P 
 PVAFn ,i 

• The reciprocal of the present value annuity factor is

called the capital recovery factor (CRF).

A = P × CRFn,i
• Suppose you have borrowed a 3-year loan of Rs. 10,000 at 9% from
your employer to buy a motorcycle. If your employer requires three
equal end-of-year repayments, then the annual installment will be?
 Loan Amortization Schedule
End of Payment Interest Principal Repayment Outstanding
Year Balance
0 10,000
1 3,951 900 3051 6,949
2 3,951 625 3,326 3,623
3 3,951 326 3,949 0
 Present Value of Perpetuity
• Perpetuity is an annuity that occurs indefinitely. Perpetuities are not
very common in financial decision-making. But one can find few
examples. For instance in the case or irredeemable preference shares
(i.e., preference shares with out a maturity)
• By definition, in a perpetuity, time period, n, is so large
(mathematically n approaches infinity).
• Present value of a perpetuity = Perpetuity/Interest rate
 Present Value of an Uneven Cash Flow
• Investments made by a firm do not frequently yield constant periodic cash
flows (annuity). In most instances, the firm receives a stream of uneven cash
flows.
• The procedure is to calculate the present value of each cash flow and
aggregate all present values.
 Example
• Consider that an Investor has an opportunity of receiving Rs. 1000,
Rs.1500, Rs.800, Rs. 11,00 and Rs.400 respectively at the end of one
through five years. Find out the present value of this stream of uneven
cash flows, if the investor’s required interest rate is 8%.
• Present Value = + + + +
 Present Value of Growing Annuity
• In Financial Decision making, there are a number of situations where
cash flows may grow at a constant rate. For example, in the case of
companies, dividends are expected to grow at a constant rate.

• PV =
• Note n is raise to the power in this formula not multiplied
 Example
• Assume that to finance your post-graduate studies in an evening
college, you undertake a part-time job for 5 years. Your employer fixes
an annual salary of Rs.1,000 at the end with the provision that you
will get annual increment at the rate of 10%. If your required rate of
return is 12%, you can use the following formula to calculate the
present value of your salary:
• Present Value = + + + +
 Problem
• A company paid a dividend of Rs. 60 last year. The dividend stream
commencing from year one is expected to grow at 10% per annum for
15 years and then end. If the discount rate is 21%, what is the present
value of the expected series?
 Present Value of Growing Perpetuities
• Constantly growing perpetuities are annuities growing indefinitely.
• The calculation of the present value of a constantly growing perpetuity
is given by a simple formula as follows:
• PV =
• If the first year annuity is based on the previous year’s annuity and
starts growing at g, then the Equation will be as follows:
• PV =
 Example
• Suppose a company paid a dividend of Rs.10 at the end of last year.
The dividend is expected to grow at 8% each year forever. If the
discount rate is 10%, what is the present value of dividends?
• PV =
 Future Value of an Annuity Due
• The concepts of compound value and present value of an annuity
discussed earlier are based on the assumption that series of cash flows
occur at the end of the period. In practice, cash flows could take place
at the beginning of the period.
 Example
• Suppose you deposit Rs. 1 in a savings account at the beginning of
each year for four years to earn 6% interest? How much will be the
compound value at the end of 4 years?

•F= ++ +
• = 1.262 +1.191 + 1.124 +1.06 = Rs. 4.637
 Present Value of An Annuity Due
• Let us Consider a 4-year annuity of Rs. 1 each year, the interest rate
being 10%. What is the present value of this annuity if each payment
is made at the beginning of the year?
• Present Value = + + +
 Multiperiod Compounding
• In the discussion, we have assumed that cash flows occurred once a
year. In practice, cash flows could occur more than once a year. For
example, banks may pay interest on savings accounts quarterly.
 Example
• Let us Find out the compound value of Rs.1,000, with an interest rate
being 12% per annum if compounded annually, semi-annually,
quarterly, and monthly for 2 years.
 Problem I
• Your Grand Father is 75 years old. He has total savings of Rs.80,000.
He expects that he will live for another 10 years, and will like to spend
his savings by then . He places his savings into a bank account earning
10% annually. He will withdraw an equal amount each year- the first
withdrawal occurring one year from now- in such a way that his
account balance becomes zero at the end of 10 years. How much will
be his annual withdrawal?
 Problem II
• XY Company is thinking of creating a sinking fund to retire its Rs.
800,000 preference share capital that matures on 31 December 20X8.
The company plans to put a fixed amount into the fund at the end of
each year for eight years. The first payment will be made on 31
December 20X1, and the last on 31 December 20X8.
The company expects that the fund will earn 12% a year. What
annual contribution must be made to accumulate Rs. 8,00,000 as of 31
December 20X8? What would be your answer if the annual contribution
is made in the beginning of the year, the first payment being made on 31
December 20X0
 Problem III
• You have borrowed a car loan of Rs.50,000 from your employer. The
loan requires 10% interest and five equal end-of-year payments.
Prepare a loan amortization schedule?
 Solution
Borrowing 50,000
Interest rate 10%
Annuity factor, 10%, 5 year 3.7908
Annual payment: 50,000/3.7908 13190
Year Outstanding Instalment Interest Repayment
0 50,000 Borrowing
0
50,000
0
Interest rate 10%
1 41,810 Annuity factor, 10%, 5 year13,190
Annual payment: 50,000/3.7908
3.7908
13190
5,000 8,190
2 32,801 13,190 4,181 9,009
3 22,892 13,190 3,280 9,910
4 11,991 13,190 2,289 10,901
5 0 13,190 1,199 11,991
 Problem IV
• Ms. Punam is interested in a fixed annual income. She is offered three
possible annuities. If she could earn 8% on her money elsewhere,
which of the following alternatives, if any, would she choose? Why?
• Pay Rs. 80,000 now in order to receive Rs. 14,000 at the end of each
year for the next 10 years.
• Pay 1,50,000 now in order to receive Rs. 14,000 at the end of each
year for the next 20 years.
• Pay Rs. 1,20,000 now in order to receive Rs.14,000 at the end of for
the next 15 years.
Solution
Interest rate 8%
( i ) Amount now or 80,000
10-year annuity 14,000
PVAF, 8%, 10 year 6.7101
Present value of 10-year annuity [14,000 × 6.7101] 93,941
Ms Punam should choose 10-year annuity offering higher PV.

( ii ) Amount now or 150,000

20-year annuity 14,000
PVAF, 8%, 20 year 9.8181
Present value of 20-year annuity [14,000 x 9.8181] 137,454

(iii) Amount now or 120,000

15-year annuity 14,000
PVAF, 8%, 15 year 8.5595
Present value of 15-year annuity [14,000 × 8.5595] 119,833
Both alternatives are almost the same.