this is a lecture on time value of money which explains the topic time value of money in a very easy and simple way... it also explains some examples on the topic... plus definition of rate of return, real rate of return, inflation premium, nominal interest rate,market risk, maturity risk,liquidity risk,and default risk,

1. The time value of money:

2. “Don't waste your time with
explanations: people only hear
what they want to hear.”
Paulo Coelho

3. Time value of money:
 Which would you prefer -- $10,000 today or $10,000 in
5 years?
 Obviously, $10,000 today.
 You already recognize that there is TIME VALUE TO
MONEY!!

4. Time value of money:
 Why TIME?
 Why is TIME such an important element in your
decision?
 TIME allows you the opportunity to postpone
consumption and earn INTEREST.

5.  A dollar in hand today is worth more than a dollar to
be received in the future because, if you had it now,
you could invest it, earn interest, and end up with
more than dollar in future.
 Any amount of money promised in the future is
uncertain and riskier than others.

6.  The time value of money principle is concerned with two
topics: (1) future value, and (2) present value. As shown in
the illustration below, the two are mirror images of one
another. (Year 0 stands for "at the present time" or "right
now" since year 1 would be 1 year from now, etc.)
 In a future value problem, we know the amount of money
that we have to invest today (i.e., the present value). What
we don't know is how much money we will have in the
future (i.e., the future value).
 In a present value problem, we know the amount of money
that we want to have (or expect to have) in the future. What
we don't know is how much money we need to invest today
in order to attain that money in the future.

8. Time line:
 An important tool used in the time value of money
analysis; it is a graphical representation used to show
the timing of cash flows.
 0 1 2 3 4 5 6 7
 Moving money through time – that is, finding the
equivalent value to money at different points in time –
involves translating values from one period to another.

9.  Translating a value to the present is referred to as
discounting.
 Translating a value to the future is referred to as
compounding.
 Translating money from one period involves interest,
which is how the time value of money and risk enter
into the process.

10. Future value (FV):
 The amount to which a cash flow or a series of cash flows
will grow over a period of time when compounded at a
given interest rate.
 It can be calculated as;
FV = PV (1+i)n
 Where,
 FV = future value or ending amount
 PV = present value or beginning amount
 i = interest rate per period
 n = the number of periods
 (1+i)n = future value (interest) factor

11. Compounding:
 The arithmetic process of determining the final value
of a cash flow or series of cash flows when compound
interest is applied.
 It is the process of going from today’s values or present
values (PVs) to future values (FVs), over a period of
time (over the time line).

12. Present value:
 The value today of a future cash flow or a series of cash flows.
The present value of a cash flow due n years in the future is the
amount which, if it were on hand today, would grow to equal the
future amount.
 It can be calculated as;
 Where,
 PV = present value
 FV = future value
 i = interest rate or rate of return
 n = number of periods
 1/(1+i)n = present value (interest) factor

13. Present value:
 From the formula for present value we know that;
 As the number of discount periods, n, becomes larger,
the discount factor becomes smaller and the present
becomes less, and
 As the interest rate per period, i, becomes larger, the
discount factor becomes smaller and the present value
becomes less.
 Therefore, the present value is influenced by both the
interest rate (i.e. the discount rate) and the number of
discount periods.

14. Discounting:
 The process of finding the present value of a cash flow
or a series of cash flows;
 discounting is the reverse of compounding.
 Translating a value back in time – referred to as
discounting – requires determining what a future
amount or cash flow is worth today.
 Discounting is used in valuation because we often
want to determine the value today of some future value
or cash flow (e.g. what a bond is worth today if it
promised interest and principal repayment in the
future).

15. Interest:
 The principal is the amount borrowed.
 Interest is the compensation for the opportunity cost
of funds and the uncertainty of repayment of the
amount borrowed; that is, it represents both the price
of time and the price of risk.
 The price of time is compensation for the opportunity
cost of funds.
 The price of risk is compensation for bearing risk.

17. Simple interest:
 Interest paid (earned) on only the principal amount, or principal borrowed (lent).
 Formula:
SI = (Po)(i)(n)
 Where,
 SI = simple interest
 Po = principal amount (lent or borrowed)
 i = interest rate
 n = number of periods
 Example;
 Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years.
What is the accumulated interest at the end of the 2nd year?
 SI = (Po)(i)(n)
 SI = $1000(0.07)(2)
 SI = $140

18. Simple interest (FV):
 What is the Future Value (FV) of the deposit?
 FV = Po + SI
 FV = $1000 + $140
 FV = $1140
 Future Value is the value at some future time of a
present amount of money, or a series of payments,
evaluated at a given interest rate.

19. Simple Interest (PV):
 What is the Present Value (PV) of the previous
problem?
 The Present Value is simply the $1,000 you originally
deposited. That is the value today!
 Present Value is the current value of a future amount
of money, or a series of payments, evaluated at a given
interest rate.

20. Compound Interest:
 Interest paid (earned) on any previous interest
earned, as well as on the principal borrowed (lent).
 Example:
 Assume that you deposit $1,000 at a compound interest
rate of 7% for 2 years.
 0 1 2
 $1,000
 FV2

21. Future value single deposit:
 FV1 = Po (1+i)1
 FV1 = $1,000 (1+0.07)
 FV1 = $1,070
 Compound interest:
 You earned $70 interest on your $1,000 deposit over the first
year. This is the same amount of interest you would earn
under simple interest.
 The difference between the future value with compounded
interest and that with simple interest is the interest-on-
interest.

22. Future value single deposit:
 FV1 = Po (1+i)1
 = $1,000 (1+0.07)
 = $1,070
 $1,000 FV1 = $1,070 FV2
 FV2 = FV1 (1+i)1
 = Po (1+i)(1+i)
 = $1,000(1.07)(1.07)
 = Po (1+i)2
 = $1,000(1.07)2
 = $1,144.90
 You earned an extra $4.90 in year 2 with compound over the simple interest.
 This $4.90 is the interest earned on the 7% interest of $1,000, i.e.
 1000 x 7% = 70
 70 x 7% = 4.9

23. Types of annuities:
 Annuity:
 A series of equal payments or receipts of an equal amount at fixed
intervals occurring for a specified number of periods. We see examples
of annuities all around us in business. Interest payments on a fixed-
income security (like a bond) are an annuity. Monthly payments on a
loan are an annuity. Lease payments on a car or apartment are an
annuity. The amount doesn't change from period to period.
 Ordinary (deferred) annuity:
 In an ordinary annuity payments or receipts occur at the end of each
period.
 Annuity due:
 In an annuity due payments or receipts occur at the beginning of each
period.

24. Ordinary annuity (FV):
 FVAn = PMT( FVIFAi,n )
 FVAn = PMT[(1+i)n – 1/i]
 Where;
 FVAn = future value of an annuity over n periods
 PMT = payments
 FVIFAi,n = annuity future value interest factor
 i = interest rate
 n = number of periods

25. Example:
 If PMT=$1,000, i=8%, n=3 years. Calculate future value
of an ordinary annuity?
 FVAn = PMT( FVIFAi,n )
 FVAn = PMT[(1+i)n – 1/i]
 FVAn = $1,000 [(1.08)3 – 1/(0.08)]
 FVAn = $3,246.

26. Ordinary annuity (PV):
 PVAn = PMT( PVIFAi,n )
 PVAn = PMT [1-[1/(1+i)n ]/i ]
 Where;
 PVAn = present value of an annuity of n periods
 PMT = payments
 PVIFAi,n = present value interest factor annuity
 i= interest rate
 n= number of years

27. Example:
 Periodic receipts of $1,000 at the end of each year,
discount rate = 8%, n = 3 years. Calculate present value
ordinary annuity?
 PVAn = PMT( PVIFAi,n )
 PVAn = PMT [1-[1/(1+i)n ]/i ]
 PVAn = $1,000 [1-[1/(1.08)3 ]/(0.08)]
 PVAn = $2,577.

28. Annuity due (FV):
 FVAD = PMT (FVIFAi,n)(1+i)
 FVAD = PMT [(1+i)n – 1/i](1+i)
 Where;
 FVAD = future value annuity due
 PMT = payment
 FVIFAi,n = future value interest factor of an annuity
 i= interest rate
 n= number of years

29. Example:
 PMT = $1,000, i= 5%, n= 5 years. Calculate future value
of an annuity due?
 FVAD = PMT [(1+i)n – 1/i](1+i)
 = $1,000 [1-[1/(0.05)5 ]/(0.05)](1.05)
 = $5,802.3

30. Annuity due (PV):
 PVAD = (1+i)(PMT)(PVIFAi,n)
 PVAD = (1+i) (PMT) [1-[1/(1+i)n ]/i]
 Where;
 PVAD = present value annuity due
 PMT = payments
 PVIFAi,n = present value interest factor of an annuity
 i= interest rate
 n= number of years

31. Example:
 PMT=$1,000, i=8%, n= 3 years. Calculate present value
of annuity due?
 PVAD = (1+i)(PMT)(PVIFAi,n)
 PVAD = (1+i) (PMT) [1-[1/(1+i)n ]/i]
 = (1.08) (1,000) [1-[1/(1.08)3 ]/(0.08)
 = $2783.16

32. Perpetuity:
 An ordinary annuity whose payments or receipts
continue forever.
 PVA00 = PMT/I
 where;
 PVA00 = perpetuity
 PMT = payment or receipts
 I = interest rate

33. Example:
 PMT= $1,000, I = 5%. Calculate perpetuity annuity?
 PVA00 = PMT/I
 = $1,000/0.05
 = $2,000

34. Extra notes:
 Definition of 'Rate Of Return'

The gain or loss on an investment over a specified period,
expressed as a percentage increase over the initial investment
cost. Gains on investments are considered to be any income
received from the security plus realized capital gains.
 Definition of 'Real Rate Of Return'

The annual percentage return realized on an investment, which
is adjusted for changes in prices due to inflation or other external
effects. This method expresses the nominal rate of return in real
terms, which keeps the purchasing power of a given level of
capital constant over time.

35.  Inflation premium:
 Return on an investment over its normal rate of
return. Investors seek this premium to compensate for
the erosion in the value of their capital due to inflation.
 Definition of 'Nominal Interest Rate'

The interest rate before taking inflation into account. The
nominal interest rate is the rate quoted in loan and deposit
agreements. The equation that links nominal and real interest
rates is:
(1 + nominal rate) = (1 + real interest rate) (1 + inflation rate).
It can be approximated as nominal rate = real interest rate +
inflation rate.

36.  Definition of 'Market Risk'

The possibility for an investor to experience losses due to factors that affect the overall
performance of the financial markets. Market risk, also called "systematic risk," cannot
be eliminated through diversification, though it can be hedged against. The risk that a
major natural disaster will cause a decline in the market as a whole is an example of
market risk. Other sources of market risk include recessions, political turmoil, changes in
interest rates and terrorist attacks.
 Maturity Risk
 Fixed-income investment securities -- primarily bonds -- typically pay a fixed rate of
interest and the face or principal amount when a bond matures. Available maturities
range from 30 days to 30 years. Maturity risk is the potential for interest rates to change
while your money is tied up in a bond until it matures. Buying a bond with a longer time
to maturity increases the likelihood that interest rates could rise over that period. The
maturity risk premium is the extra yield you will earn from buying a bond with a longer
time to maturity.

37.  Definition of 'Liquidity Risk'

The risk stemming from the lack of marketability of an investment that cannot
be bought or sold quickly enough to prevent or minimize a loss. Liquidity risk
is typically reflected in unusually wide bid-ask spreads or large price
movements (especially to the downside). The rule of thumb is that the smaller
the size of the security or its issuer, the larger the liquidity risk.
 Definition of 'Default Risk'

The event in which companies or individuals will be unable to make the
required payments on their debt obligations. Lenders and investors are
exposed to default risk in virtually all forms of credit extensions. To mitigate
the impact of default risk, lenders often charge rates of return that correspond
the debtor's level of default risk. The higher the risk, the higher the required
return, and vice versa.