FM I - Chapter 3, Time Value of Money (TVM).ppt

Financial management chapter 3:- time value of money, interest rate determinants,future/present value of single sum/annual,rate of return, amortization

1. 6-1
Chapter 3
Time Value of Money and
Security valuation
Concept of time vale of money
Interest rate determinants
Future/present value of single sum
Future/Present value Annuities
Rates of return
Amortization

2. 6-2
Concept of TVM
Time value of money represents the fact that $1 today is more
valuable than $1 in the future, say one year from now.

The factors that make money to have time value:

Consumption preference: - Individuals prefer current
consumption to future consumptions. So people would have to
be offered more in the future to give up current consumption.

Inflation: - general increase in prices (inflation) erodes the
purchasing power of money. Hence, the value of money
decreases with time when there is inflation.

Uncertainty (Risk): - As compared to today’s money future
cash flows have risks (default risk) Hence, delaying cash
collection means assuming greater risks. Individuals want to be
rewarded for this additional risk assumed n future cash flows.

Investment opportunities: - cash received today could be
invested and fetch additional income.

3. 6-3
Concept of TVM…

Therefore, as money has time value, the same cash
flows in different periods have different values. This
makes aggregation and comparison of cash flows at
different times illogical unless adjusted for the above
factors. This adjustment is made using discount rates.

Discount rate: - rates at which present and future cash
flows are traded-off. It incorporates the above factors.

Greater present consumption….higher discount rates.
Individuals with greater present consumption require
higher rewards for giving it up.

Higher expected inflation ….. . higher discount rates.

Higher risks …….. higher discount rates.

Higher expected return on the present cash flows……
higher discount rates.

4. 6-4
Concept of TVM…

Discounting: - the process of moving cash flows that
are expected to occur in the future back to the present
using discount rates. It converts future cash flows to
present value terms.

Compounding: - the process of moving present cash
flows to future using discount rates.

5. 6-5
Determinants of Interest rate

Interest Rate Levels

Refers to the shift in demand and supply for funds and
the related change in the interest rate level
(equilibrium).Like any commodity capital is allocated
by its price (interest rate) which, in a pure market
economy, is determined by the forces of demand and
supply. Increase in the demand of debt capital pushes
interest rates up, and decreases in the demand of
capital, in times of recession, pulls it lower. Increase in
the total supply of debt capital reduces interest rates.
A decrease in the supply creates shortage of funds in
the market and those firms with profitable investment
tend to attract capital away from less profitable firms
by paying higher interest rates.

6. 6-6
Determinants of Interest rate

Determinants of Interest Rates

The quoted interest rate on a debt security, K, is
composed of a real risk free rate of interest plus
premiums for inflation, risk and liquidity.

K =K* + IP + DRP + LP + MRP

Where:

K*= the real risk free rate.

IP = inflation premium (the average expected inflation
rate over the life of the security)

DRP = Default risk premium

LP = Liquidity or marketability premium

MRP = maturity risk premium (the risk related to price
declines)

7. 6-7
Determinants of Interest rate

The Real Risk-Free Rate (K*)

This is the interest rate that would exist on a risk less
security if no inflation is expected. This is a no risk no
inflation interest rate. This rate changes over time
depending on economic conditions:

Rate of return expected by businesses and other
borrowers expect to earn on a productive assets. This
is the upper limit borrowers can afford.

People’s time preferences for current versus future
consumptions. Determines the amount of income
savers are ready to defer, hence the amount of funds
available.(Supply of funds)

8. 6-8
Future values and present
values and the Time lines

Show the timing of cash flows.

Tick marks occur at the end of periods, so
Time 0 is today; Time 1 is the end of the
first period (year, month, etc.) or the
beginning of the second period.
CF
0
CF
1
CF
3
CF
2
0 1 2 3
i%

9. 6-9
Drawing time lines:
$100 lump sum due in 2 years;
3-year $100 ordinary annuity
100 100100
0 1 2 3
i%
3 year $100 ordinary annuity
100
0 1 2
i%
$100 lump sum due in 2
years

10. 6-10
Drawing time lines:
Uneven cash flow stream; CF
0 = -$50,
CF
1
= $100, CF
2
= $75, and CF
3
= $50
100 50 75
0 1 2 3
i%
-50
Uneven cash flow stream

11. 6-11
What is the future value (FV) of an initial
$100 after 3 years, if I/YR = 10%?

Finding the FV of a cash flow or series of
cash flows when compound interest is
applied is called compounding.

FV can also be solved by using the
arithmetic, financial calculator, and
spreadsheet methods.
FV = ?
0 1 2 3
10%
100

12. 6-12
Solving for FV:
The arithmetic method

After 1 year:

FV
1 = PV ( 1 + i ) = $100 (1.10)
= $110.00

After 2 years:

FV
2
= PV ( 1 + i )
2
= $100 (1.10)
2
=$121.00

After 3 years:

FV
3 = PV ( 1 + i )
3
= $100 (1.10)
3
=$133.10

After n years (general case):

FV
n = PV ( 1 + i )
n

13. 6-14
PV = ? 100
What is the present value (PV) of $100
due in 3 years, if I/YR = 10%?

Finding the PV of a cash flow or series of
cash flows when compound interest is
applied is called discounting (the reverse
of compounding).

The PV shows the value of cash flows in
terms of today’s purchasing power.
0 1 2 3
10%

14. 6-15
Solving for PV:
The arithmetic method

Solve the general FV equation for PV:
PV = FV
n / ( 1 + i )
n
PV = FV
3
/ ( 1 + i )
3
= $100 / ( 1.10 )
3
= $75.13

15. 6-16
Solving for PV:
The calculator method

Solves the general FV equation for PV.

Exactly like solving for FV, except we
have different input information and
are solving for a different variable.
INPUTS
OUTPUT
N I/YR PMTPV FV
3 10 0 100
-75.13

16. 6-18
Annuities: an ordinary annuity and
an annuity due?
Ordinary Annuity
PMT PMTPMT
0 1 2 3
i%
PMT PMT
0 1 2 3
i%
PMT
Annuity Due

17. 6-19
The arithmetic method for
Annuities

Future value Formulae for annuities:
1. Ordinary annuities

FV = PMT [(1+i)
n
- 1)]

i

Where: PMT is the periodic cash flows,

i is the discount rate, and

n is the number of periods

Example : ABC co has planned to acquire machinery after
five years. To that end, the company deposits Birr $
3000.00 at the end of each year at a deposit rate of 12%.
How much is the terminal (future value) of the deposits at
the end of the fifth year? How much is the terminal value if
deposits are made semi-annually?

i

18. 6-20
The arithmetic method for
Annuities
2. Annuity Due

FV = PMT (1+ r) [(1+i)
n
- 1]

I

Example : How much will the terminal value of the
a cash flow we used to illustrate ordinary annuity
if the deposit is made at the beginning of each
year?

19. 6-21
The arithmetic method for
Annuities

Present value Formulas for annuities:
1. Ordinary annuities

PV = PMT [1- (1/(1+i)
n
]

i

Example 5: If in previous example , ABC co. has the
option of paying $10,000.00 at the time of purchase,
which option will ABC take at the discount rate of 12%.

PV = $3,000.00[1-1/(1+12%)
5
]

12%

= $10,814.00

20. 6-22
The arithmetic method for
Annuities

Present value Formulas for annuities:
2. Annuity Due

PV = PMT + PMT (1-1/(1+i)
n-1

i

21. 6-23
The arithmetic method for
Perpetuities

Present value Formulas for perpetuities:
A. Perpetuity: - a constant cash flow that is paid (received)
at a regular time interval forever.
PV = A/i
Where: A is the periodic cash flow, and
i is the discount rate
B. Growing perpetuities: - a cash flow that is expected to
grow at a constant rate forever.
 PV = CF
1

r-g
Where: CF
1 is cash flow after one period,
r is the discount rate, and
g is the growth rate

22. 6-28
What is the PV of this uneven cash
flow stream?
0
100
1
300
2
300
3
10%
-50
4
90.91
247.93
225.39
-34.15
530.08 = PV how about FV?

23. 6-31
The Power of Compound
Interest
A 20-year-old student wants to start saving for
retirement. She plans to save $3 a day. Every day, she
puts $3 in her drawer. At the end of the year, she
invests the accumulated savings of $1,095 (3*365days)
in an online stock account. The stock account has an
expected annual return of 12%.
How much money will she have when she is 65 years
old? Note it is ordinary annuity of 1095 every year.
she will have $1,487,261.89 when she is 65.

24. 6-35
Will the FV of a lump sum be larger or
smaller if compounded more often,
holding the stated I% constant?

LARGER, as the more frequently compounding
occurs, interest is earned on interest more
often.
Annually: FV
3
= $100(1.10)
3
= $133.10
0 1 2 3
10%
100 133.10
Semiannually: FV
6 = $100(1.05)
6
= $134.01
0 1 2 3
5%
4 5 6
134.01
1 2 3
0
100

25. 6-36
Classifications of interest
rates

Nominal rate (i
NOM
) – also called the quoted or
state rate. An annual rate that ignores
compounding effects.

i
NOM is stated in contracts. Periods must also be
given, e.g. 8% Quarterly or 8% Daily interest.

Periodic rate (i
PER) – amount of interest charged
each period, e.g. monthly or quarterly.

i
PER = i
NOM / m, where m is the number of
compounding periods per year. m = 4 for
quarterly and m = 12 for monthly compounding.

26. 6-37
Classifications of interest
rates

Effective (or equivalent) annual rate (EAR =
EFF%) – the annual rate of interest actually
being earned, taking into account
compounding.

EFF% for 10% semiannual investment
EFF%= ( 1 + i
NOM / m )
m
- 1
= ( 1 + 0.10 / 2 )
2
– 1 = 10.25%

An investor would be indifferent between
an investment offering a 10.25% annual
return and one offering a 10% annual
return, compounded semiannually.

27. 6-38
Why is it important to consider
effective rates of return?
An investment with monthly payments is
different from one with quarterly payments.
Must put each return on an EFF% basis to
compare rates of return. Must use EFF% for
comparisons. See following values of EFF%
rates at various compounding levels.
EAR
ANNUAL10.00%
EAR
QUARTERLY
10.38%
EAR
MONTHLY
10.47%
EAR
DAILY (365)
10.52%

28. 6-39
Can the effective rate ever be
equal to the nominal rate?

Yes, but only if annual
compounding is used, i.e., if m = 1.

If m > 1, EFF% will always be greater
than the nominal rate.

29. 6-40
When is each rate used?

i
NOM
written into contracts, quoted by
banks and brokers. Not used in
calculations or shown on time lines.

i
PER Used in calculations and shown on
time lines. If m = 1, i
NOM = i
PER = EAR.
EAR Used to compare returns on
investments with different payments per
year. Used in calculations when annuity
payments don’t match compounding
periods.

30. 6-41
What is the FV of $100 after 3 years
under 10% semiannual compounding?
Quarterly compounding?
$134.49 (1.025) $100 FV
$134.01 (1.05) $100 FV
)
2
0.10
1 ( $100 FV
)
m
i
1 (PV FV
12
3Q
6
3S
32
3S
nmNOM
n







31. 6-42
What’s the FV of a 3-year $100
annuity, if the quoted interest rate is
10%, compounded semiannually?
Payments occur annually, but
compounding occurs every 6 months.
Cannot use normal annuity valuation
techniques.
0 1
100
2 3
5%
4 5
100 100
6
1 2 3

32. 6-43
Method 1:
Compound each cash flow; or
method 2 is using financial
calculator
110.25
121.55
331.80
FV
3
= $100(1.05)
4
+ $100(1.05)
2
+ $100
FV
3
= $331.80
0 1
100
2 3
5%
4 5
100
6
1 2 3
100

33. 6-46
Loan amortization

Amortization means retiring a debt in a given length of
time by equal periodic payments that include compound
interest. After the last payment, the obligation ceases to
exist-it is dead-and it is said to have been amortized by
the payments.

In amortization our interest is to determine the periodic
payment, R, so as to amortize (retire) a debt at the end of
the last payment. Solving the PV of ordinary annuity
formula for R in terms of the other variables, we obtain
the following amortization formula:

34. 6-47
Loan amortization

Where:

R = periodic payment

P = PV of loan

i= interest rate per period

n = number of payment periods

i
R = P
-n
1 - 1 + i
 
 
 
 

35. 6-48
Loan amortization
Amortization tables are widely used for home
mortgages, auto loans, business loans,
retirement plans, etc.
Financial calculators and spreadsheets are
great for setting up amortization tables.
EXAMPLE: Construct an amortization
schedule for a $1,000, 10% annual rate loan
with 3 equal payments.
note: first find periodic payment of 402 per year using
PV of ordinary annuity formula

36. 6-53
Constructing an amortization table:
Repeat steps 1 – 4 until end of loan

Interest paid declines with each payment
as the balance declines. What are the tax
implications of this?
YearBEG BALPMT INT PRIN END
BAL
1 $1,000 $402 $100 $302 $698
2 698 402 70 332 366
3 366 402 37 366 0
TOTA
L
1,206.3
4
206.341,000 -

37. 6-54
Illustrating an amortized payment:
Where does the money go?
Constant payments.
Declining interest payments.
Declining balance.
$
0 1 2 3
402.11
Interest
302.11
Principal Payments

38. 6-55
Mortgage Payments

In atypical house purchase transaction, the home-buyer pays part of the
cost in cash and borrows the remained needed, usually from a bank or a
savings and loan association. The buyer amortizes the indebtedness by
periodic payments over a period of time. Typically, payments are monthly
and the time period is long-30 years is not unusual.

 

Mortgage payment and amortization are similar. The only differences are

The time period in which the debt/loan is amortized/repaid

The amount borrowed.

In mortgage payments m is equal to 12 because the loan is repaid from
monthly salary, but in amortization m may take other values.

39. 6-56

In Mortgage payments we are interested in the determination of
monthly payments.

 Taking A = total debt

R = monthly mortgage payment

r = stated nominal rate per annum

n = 12 x t

 R can be determined as follows:
  
r/12 i
R = A (OR) R = A
-n -n
1 - 1 + r/12 1 - 1 + i
   
   
   
   
Similarly Similarly Similarly Similarly

40. 6-57

Similarly,

Example

Mr. X purchased a house for Birr 115,000. He made a
20% down payment with the balance amortized by a
30 yr mortgage at an annual interest of 12%
compounded monthly.

What is the amount that Mr. X should pay monthly so as to
retire the debt at the end of the 30
th
yr?

Find the interest charged.


Similarly









i
i
RA
n
11