Unit 1

Financial Management

Time Value of Cash Flows

 Cash flows occurs at different time
periods

• Purchase of fixed asset-

• In case of firm’s borrowings:

• In case of capital raised through equity

shares:
For comparison need to be adjusted for time
and risk

• The recognition of the time value of money and

risk is extremely vital in financial
decision-making.
• The cash flows can be compared only when they
are adjusted for their timing and risk involved,
through an appropriate rate of return.
 Time preference for money or Time Value of Money
(TVM)
• This represents an individual’s preference for
possession of a given amount of money now,
rather than the same amount at some other period
of time (annual/future etc)
• The reasons for time preference arises due to :
✔ Risks involved
✔ preference for present consumption rather than
the uncertain future
✔ investment opportunities
 Criteria differs based on risks involved
• In case of a firm’s financial
decision-making under certainty, the
criteria of choosing the alternative which
is likely to yield the highest return in the
future is opted, at the market rate of
interest(opportunity cost)
• In cases of risk involvement , the
comparison has to be made with a higher
rate of return to compensate for the risk
involved.
 Required Rate of Return :
• Interest rate expresses the time preference
for money.
• RI will always >0 , even if risk =0
• In reality, some risk exists.
• So RRR = Risk free rate + risk premium
• The RRR is the opportunity cost of capital
in comparable risk- because the investor
could invest his money in assets or securities
of equivalent risk.
 TVM
• Reducing the cash flows to
equivalent amount- that is
common time period- Present ,
future or annual .
• Through compounding and
discounting.
 Variables:
• P= present worth of an amount(0th year)
• F= Future worth of an amount (nth year)
• A = Annuity = annual equivalent amount (Yearly)
• (Annuity is a fixed amount (payment or receipt)
each year for a specified number of years)
• G= growth rate
• i = rate of interest
• n = total number of years
• t = the specific year ( t = 1, 2..........nth year)
 The investment pattern/Interest formulas
• 1. Single payment compound amount factor
• F=P(1+i)n = P(F/P, i , n)
• 2. Single payment discount factor
• P=F [ 1/(1+i)n] = F (P/F, i ,n)
• 3. Series payment compound amount factor
• F= A {[(1+i)n - 1)]/ i } = A(F/A, i , n)
• 4. Sinking fund factor
• A= F { i / [(1+i)n - 1)]} =(A/F , i, n)
• 5. Series Present worth factor
• P= A {[(1+i)n - 1)]/[i(1+i)n ]} = A(P/A ,i ,n)
• 6. Capital Recovery factor
• A =P [i(1+i)n ]} /[(1+i)n - 1)]=P(A/P, i ,n)
 Capital Recovery and Loan Amortization

• If we make an investment today for a given

period of time at a specified rate of interest,
we may like to know the annual income.
Capital recovery is the annuity of an
investment made today, for a specified period
of time, at a given rate of interest.
 The investment pattern/Interest
formulas(contd.)
7. Present value of an uneven cash flow:
n
P = ∑ Ci/ (1+i)^t
t=1
Note: C is the cash flow each year

8. Uniform Gradient Series Formula:

A = A1 + G (A/G , i, n)
Lets Apply:
(Q1) Calculate the present value of $`600 (a)
received one year from now and (b) received at the
end of fifteen years at a rate of return of 5%.
(Q2) Determine the Future value of $700 each paid
at the end of each of the next six years at 8 % rate of
interest.
(Q3) Assuming a 8% rate of discount, compute
the present value of $1,100; $900; $1,500 and
$700 received at the end of one through four
years.
 Contd.
Q.4 A person deposits a sum of Rs. 10000 at the
end of the first year and thereafter increases his
deposit by Rs. 500 each year.Assuming an interest
rate of 6% C.A,find the mature amount he will
receive after 10 years?
Q5. A company has issued debentures of
Rs.50 lakh to be repaid after 7 years. How much
should the company invest in a sinking fund
earning 12 per cent in order to be able to repay
debentures?
 Contd.
• Q6. How long will it take to double your money if
it grows at 12 per cent annually?
• Practice Youself:
• Q7. A company has borrowed `200 crore at 8% p.a. from
a financial institution for 7 years. If the principal and
interest is payable in seven year-end equal instalments.
what is the amount of instalment?
• Q8. Your father will get a gratuity of `350,000 after 10
years from now on his retirement. His employer has
offered to pay him `70,000 per year for 10 years. If your
father’s required rate of return is 1 % p.a. should he
accept the offer?
 9. Present Value of Perpetuity

Perpetuity is an annuity that occurs

indefinitely.
• Ex. For instance, in the case of irredeemable
preference shares (i.e., preference shares
without a maturity), the company is expected to
pay preference dividend perpetually.
•P=A/i
• where, A= annuity and i=rate of interest
Q.9 An investor in an irredeemable perpetual
bond expects $500 annually from an investment.

• Find the present value of his perpetuity at an

annual rate of 10% interest pa.

Q10. You invested in the shares of a company

which has promised to pay you annual dividend of
`$1000 perpetually. Find the Present value of the
divident at a 12% p.a rate of return.
 Answers
• Ans 1 a. 571.43 and b. 288.61
• Ans 2. 5135.15
• Ans 3. 3495
• Ans. 4.Rs. 1,58,304.98 or 1.58 lakh
• Ans.5. Rs. 495000
• Ans6.6 years approx.
• Ans 7: Rs.38 crore approx.
• Ans. 8 Rs. 732354.87 . Yes should accept the
offer.
• Ans.9 $5000
• Ans. 10 $8333.33
Q11 A student borrows $100 per year for
3 years. The loan is to be repaid 2 years
later at a 15% rate of interest. Find the
amount that has to be repaid?

Also draw the cash flow diagram.

Q12. A man who has purchased a new car wants to set aside
enough money in a bank to pay the maintenance cost for
the next 5 years.
The estimated maintenance cost at the end of each year is
given as follows: Assuming a return of 5% find how much he
should deposit now to be able to pay for the maintenance
cost each year.(Solve using Gradient series formula)
Q13.Determine the present value of a
cash inflow of Rs. 3000 at the end of each
year for the next 4 years and Rs. 7000 and
th
Rs. 1000 respectively at the end of the 5
th
and the 6 year at a discount rate of 14%?
Q14.A car has a warranty period of 3 years.
Upon expiration of the warranty period, the
annual maintenance cost starts at $150 and
then increases by $25 per year until the car is
th
sold at the end of the 7 year. Find the
present worth of these expenses at an interest
rate of 10%?
Q15. Exactly 10 years from now Sri Chand
will start receiving an amount of Rs. 3000
each year for the next 16 years. How much
will be the present value of the total
amount he is going to receive at 10%?
 10.Present value of a growing
annuity:
• To understand this formula lets solve the
following question.
• Q16. Find the Present value of an annuity of
Rs 1000 growing at 10% p.a. For 5 years , at a
rate of discount of 12% p.a.

• P= [ A/(i-g)] [1 – ((1+g)/(1+i))^n ]
 Solve

• Q 17.A company paid a dividend of $66 last

year. The dividend stream commencing from
one year , is expected to grow at 10% pa for 15
years and then will be redeemed. Find the
present value of this expected series at a rate
of discount of 21%.
 11. Present Value of a growing perpetuity
• Note: Constantly growing perpetuities are
annuities which grow indefinitely.
• P = [ A/(i-g) ] [ 1- ((1+g)/(1+i)) ^n ]
• So, in case of perpetuities , n tends to ∞
• So the P in this case reduces to :

• P = A / (i-g)
Q 18 . Find the PV if the annuity is
a perpetuity in Q17 above.

• Ref: Q-17 A company paid a

dividend of $66 last year. The
dividend stream commencing from
one year , is expected to grow at
10% pa for 15 years and then will
be redeemed. Find the present value
of this expected series at a rate of
discount of 21%.
 12. Value of an Annuity Due:
• So, we define ‘Annuity due’ as the series of
fixed amount(receipts/payments) , starting at
the beginning of the each interest period, for a
specified number of periods.
• We can get its Future or Present values.
 12. Value of an Annuity Due(contd.)
• Future value of an annuity due :
Fd = A {[(1+i)n - 1)]/ i } (1+i)

Present Value of an annuity due:

P= A {[(1+i)n - 1)]/[i(1+i)n ]} (1+i)

 Solve
Q19. (i)Find the future value of the
annuity and the annuity due of Re 100 for
4 years at 6% pa? Which is greater?
• (ii) Find the present value of the annuity
and the annuity due of the same?
Which one is greater?
Q20. Find the present value of the
annuity and the annuity due of Re.1 at
10% for 4 years. ? Which is greater?
 13. Multi-Period Compounding
• If compounding is done once in a year, it is called
annual or nominal rate of interest.
• But if compounding is done more than once in an
year, it is called the ‘Effective rate of interest’.
[EIR]
• EIR = ([ 1 + (i/m)]^m ) -1
Q21.A bank gives an annual rate of interest of
13% on a public deposit.
Compute the EIR if compounding is done (i) half
yearly (ii) quarterly (iii) monthly and (iv) weekly.
Q22. A saving bank pays 1½% interest rate every 3
months.What is the nominal and the effective rate
per year?
Q23. Find the EIR on 10% for 1 year compounded
semi-annually?
Lets extend the concept of multi-period
compounding.
i. Future value of a sum in case of
Multi-period compounding.
• F=P(1+i/m)mxn
ii. The compounded value of an annuity in
case of multi-period compounding
• F= A {[(1+i/m)^mxn - 1)]/ (i/m)}
iii. Present value of a sum in case of
Multi-period compounding
P= A {[(1+i/m)^mxn - 1)]/[(i/m)(1+i/m)mxn
]}
 Solve
Q24.A company pays 15% per interest rate
compounded quarterly on a 3 year public deposit of
Rs. 1000.Find the total amount on maturity after 3
years?
Q25. Compute the compounded value of Rs 1000 at
12% pa, if compounding is done
• i. Annually
• Ii. Semi-annually
• Iii. Quarterly
• Iv. Monthly for 2 years.
 14. Continuous Compounding
• In some cases , compounding may be done
continuously. For example , the banks may pay
interest continuously, called ‘daily compounding’.
• In such cases, the
• Future value is given as >>
x
• F= Pe ,
• where x= i*n
in
• So it is F = Pe
• where i is the rate of interest and 'n' the number
of interest periods. And e is given as 2.7183
 Continuous Compounding(contd.)

• Likewise , the Present value in

case of continuous compounding
will be
in -in
•P = F / e or P =F* e
 Solve

Q26. Find the future value of

Rs 1000 at 12% pa in case of
continuous compounding.
 Solve By Yourself
• Q1. Your father has promised to give you `100,000 in
cash on your 25th birthday. Today is your 16th birthday.
He wants to know two things: (a) If he decides to make
annual payments into a fund after one year, how much
will each have to be if the fund pays 8%?
• (b) If he decides to invest a lump sum in the account
after one year and let it compound annually, how much
will the lump sum be?
• (c) If in (a) the payments are made in the beginning of
the year, how much will be the value of annuity?
Assuming 8% rate of interest.
 Solve By Yourself Contd.
Q2. Find the Future values(F) given the RRR to be 9% p.a.
(i) of $15,000 invested now for 4 years.
(ii) The future value at the end of five years of an investment of
`6,000 now and of an investment of `6,000 one year from now.
(iii) The future value at the end of eight years of an annual deposit of
`18,000 each year.
(iv) The future value at the end of eight years of annual deposit of
`18,000 at the beginning of each year.
(v) The future values at the end of eight years of a deposit of `18,000
at the end of the first four years and withdrawal of `12,000 per
year at the end of year five through seven.
 Solve By Yourself(contd.)

• Q3. A company receives an annual profit of Rs.

20 lakhs per year for 15 years beginning 1 year
from now. In addition it also receives Rs. 1 lakh
in the 6th year and Rs. 1, 50,000 in the 12th year.
What is the equivalent Present worth and
equivalent annual value of these receipts at
i=15% per year.
 Solve By Yourself(contd.)
• Q4. XYZ Bank pays 12 % and compounds
interest quarterly. If `1,000 is deposited
initially, How much shall it grow at the end of 5
years?
• Q5. Sadhulal Bhai is borrowing `50,000 to buy a
low-income group house. If he pays equal
instalments for 25 years and 4 % interest on
outstanding balance, what is the amount of
instalment? What shall be amount of instalment
if quarterly payments are required to be made?
 Contd.
• Ans.11 : $ 459
• Ans.12 :$766
• Ans. 13 : 12591
• Ans. 14 :$ 439
• Ans.15 : 9955
• Ans. 16. 4039
• Ans. 17 456 Ans. 18 Rs. 600
• Ans. 19 i. 437.46 and 463.71
• ii. 346.63 and 367.48
 contd.
• Ans. 20 3.1698 and 3.4859
• Ans.21 (i) 13.42 (ii) 13.65 (iii) 13.80
(iv) 13.86
• Ans.22 : i=6% and r=6.1% Ans.23
10.25%
• Ans. 24 : F=1555
• 25. 1120, 1123.6 , 1125.50 and 1270.49
• 26. 1127.5