Module I - A - tvm
Transcript of Module I - A - tvm
-
8/10/2019 Module I - A - tvm
1/37
Time Value of Money
Next
-
8/10/2019 Module I - A - tvm
2/37
Financial Management by Paresh Shah Biztantra NextBack
Time Value of Money
Prof. (Dr.) Paresh ShahFCMA., Ph.D. (Finance),
F.D.P. (IIM-Ahd.)
-
8/10/2019 Module I - A - tvm
3/37
Financial Management by Paresh Shah Biztantra NextBack
Introduction to Financial
Mathematics
Financial mathematics deals with the problem of
investing money, or capital. If a company (or an
individual investor) puts some capital into an
investment, it (or he/she) will want a financial return
for it.
-
8/10/2019 Module I - A - tvm
4/37
Financial Management by Paresh Shah Biztantra NextBack
Time lines show timing of cash flows.
CF0 CF1 CF3CF2
0 1 2 3i%
Tick marks at ends of periods, so Time 0is today; Time 1 is the end of Period 1;or the beginning of Period 2.
-
8/10/2019 Module I - A - tvm
5/37
Financial Management by Paresh Shah Biztantra NextBack
Time line for a Rs.100 lump sumdue at the end of Year 2.
100
0 1 2 Yeari%
-
8/10/2019 Module I - A - tvm
6/37
Financial Management by Paresh Shah Biztantra NextBack
Notation Subscripts
In both compounding and discounting, subscripts
are used to show the period of time to which a
sum of money relates.
-
8/10/2019 Module I - A - tvm
7/37Financial Management by Paresh Shah Biztantra NextBack
Interest
Interest is the amount of money, which a capital
investment earns when it is invested for a certain
length of time.
-
8/10/2019 Module I - A - tvm
8/37Financial Management by Paresh Shah Biztantra NextBack
Simple Interest
Simple interest is the amount of interest, which is earned inan equal amount every year (or month), and which is a givenproportion of the (original investment or principal).
If a sum of money is invested for a period of time than theamount of simple interest, which accrues is equal to thenumber of accounting periods x the interest rate X theamount invested, i.e.
S n= P + nr P
Where P = Original sum invested;
r = Interest rate (Expressed as a proportion, e.g.
10% = 0.1)
n = Number of periods (Normally years);
S = Sum after n periods, consisting of the originalcapital plus interest earned
-
8/10/2019 Module I - A - tvm
9/37Financial Management by Paresh Shah Biztantra NextBack
Compound Interest
Interest is normally calculated by means of
compounding. If a sum of money, the principal, is
invested at a fixed rate of interest and the interest is
added to the principal and no withdrawals are made,
then the amount invested grows at an increasing rate
as time progresses, because interest earned in earlier
periods itself also earns interest later on.
-
8/10/2019 Module I - A - tvm
10/37Financial Management by Paresh Shah Biztantra NextBack
Whats the FV of an initial Rs.100after 3 years if i = 10%?
FV = ?
0 1 2 3
10%
Finding FVs (moving to the right
on a time line) is called compounding.
100
-
8/10/2019 Module I - A - tvm
11/37Financial Management by Paresh Shah Biztantra NextBack
After 1 year: FV1 = PV + INT1 = PV + PV (i)= PV(1 + i)
= Rs.100(1.10)
= Rs.110.00.
After 2 years: FV2 = PV(1 + i)2
= Rs.100(1.10)2= Rs.121.00.
-
8/10/2019 Module I - A - tvm
12/37Financial Management by Paresh Shah Biztantra NextBack
After 3 years: FV3 = PV(1 + i)3
= Rs.100(1.10)3
= Rs.133.10.
In general,
FVn = PV(1 + i)n.
-
8/10/2019 Module I - A - tvm
13/37Financial Management by Paresh Shah Biztantra NextBack
Will the FV of a lump sum be larger orsmaller if we compound more often, holding
the stated I% constant? Why?
LARGER! If compounding is more
frequent than once a year--for
example, semiannually, quarterly,
or daily--interest is earned on interestmore often.
-
8/10/2019 Module I - A - tvm
14/37Financial Management by Paresh Shah Biztantra NextBack
0 1 2 3
10%
0 1 2 3
5%
4 5 6
134.01
100 133.10
1 2 30
100
Annually: FV3= Rs.100(1.10)3= Rs.133.10.
Semiannually: FV6= Rs.100(1.05)6= Rs.134.01
-
8/10/2019 Module I - A - tvm
15/37Financial Management by Paresh Shah Biztantra NextBack
We will deal with 3 differentrates:
iNom = nominal, or stated, orquoted, rate per year.
iPer = periodic rate.
EIR = .
effective annualrate
-
8/10/2019 Module I - A - tvm
16/37Financial Management by Paresh Shah Biztantra NextBack
iNomis stated in contracts. Periods per year
(m) must also be given.
Examples:
8%; Quarterly
8%, Daily interest (365 days)
-
8/10/2019 Module I - A - tvm
17/37Financial Management by Paresh Shah Biztantra NextBack
Periodic rate= iPer= iNom/m, where m is numberof compounding periods per year. m = 4 for
quarterly, 12 for monthly, and 360 or 365 for
daily compounding.
Examples:
8% quarterly: iPer= 8%/4 = 2%.
8% daily (365): iPer
= 8%/365 = 0.021918%.
-
8/10/2019 Module I - A - tvm
18/37Financial Management by Paresh Shah Biztantra NextBack
The annual rate which causes PV to grow tothe same FV as under multi-periodcompounding.
Example: EFF% for 10%, semiannual:FV = (1 + iNom/m)
m
= (1.05)2 = 1.1025.
EFF%= 10.25%because(1.1025)1= 1.1025.
Any PV would grow to same FV at 10.25%
annually or 10% semiannually.
Effective Annual Rate (EAR =
EFF%):
-
8/10/2019 Module I - A - tvm
19/37
Financial Management by Paresh Shah Biztantra NextBack
An investment with monthly payments
is different from one with quarterly
payments. Must put on EIR% basis to
compare rates of return. Use EIR% only
for comparisons.
Banks say interest paid daily. Same
as compounded daily.
-
8/10/2019 Module I - A - tvm
20/37
Financial Management by Paresh Shah Biztantra NextBack
How do we find EFF% for a nominal rate of 10%,
compounded semiannually?
Or use a financial calculator.
EIR% = - 1(1 + )iNomm
m
= - 1.0(1 + )0.102
2
= (1.05)2- 1.0= 0.1025 = 10.25%.
-
8/10/2019 Module I - A - tvm
21/37
Financial Management by Paresh Shah Biztantra NextBack
EAR = EFF% of 10%
EARAnnual = 10%.
EARQ = (1 + 0.10/4)4- 1 = 10.38%.
EARM = (1 + 0.10/12)12- 1 = 10.47%.
EARD(360) = (1 + 0.10/360)360- 1 = 10.52%.
-
8/10/2019 Module I - A - tvm
22/37
Financial Management by Paresh Shah Biztantra NextBack
FV of Rs.100 after 3 years under 10%
semiannual compounding? Quarterly?
= Rs.100(1.05)6 = Rs.134.01.
FV3Q = Rs.100(1.025)12= Rs.134.49.
FV = PV 1 .+i
mn
Nom
mn
FV = Rs.1001 +0.10
2
3S
2x3
-
8/10/2019 Module I - A - tvm
23/37
Financial Management by Paresh Shah Biztantra NextBack
Can the effective rate ever be equal
to the nominal rate?
Yes, but only if annual compounding isused, i.e., if m = 1.
If m > 1, EFF% will always be greater than
the nominal rate.
-
8/10/2019 Module I - A - tvm
24/37
Financial Management by Paresh Shah Biztantra NextBack
When is each rate used?
iNom: Written into contracts, quoted
by banks and brokers. Notused in calculations or shown
on time lines.
-
8/10/2019 Module I - A - tvm
25/37
Financial Management by Paresh Shah Biztantra NextBack
Changes in the Rate of Interest
If the rate of interest changes during the period of aninvestment, the compounding formula would be amended
slightly as follows
Sn = P(1+ r1)x (1+r2)(n-x)
Where r1= Initial rate of interest
x = Number of years in which the interest rate r1
applies
r2= Next rate of interest
n
x =Balancing number of years in which theinterest rate r2applies
-
8/10/2019 Module I - A - tvm
26/37
Financial Management by Paresh Shah Biztantra NextBack
Increasing the Sum Invested
Each Year
There is an alternative formula for calculating thevalue of an investment at the end of n years, which
you may prefer to use, or which you may be
required to use in an examination.
-
8/10/2019 Module I - A - tvm
27/37
Financial Management by Paresh Shah Biztantra NextBack
Discounting
The discounting is the reverse of compounding.
Its major application in business is in the evaluation ofthe capital expenditure projects, to decide whether theyoffer a satisfactory return to the investor.
This technique of discounting is known as DiscountedCash Flow, (DCF), and this will be described later.
Wh t th PV f R 100 d i 3
-
8/10/2019 Module I - A - tvm
28/37
Financial Management by Paresh Shah Biztantra NextBack
10%
Whats the PV of Rs.100 due in 3 years
if i = 10%?
Finding PVs is discounting, and its
the reverse of compounding.
100
0 1 2 3
PV = ?
-
8/10/2019 Module I - A - tvm
29/37
Financial Management by Paresh Shah Biztantra NextBack
Solve FVn= PV(1 + i )nfor PV:
PV =FV
1+ i = FV
1
1+ i
n
n n
n
PV = 100 11.10= 100 0.7513 = Rs.75.13.
3
-
8/10/2019 Module I - A - tvm
30/37
Financial Management by Paresh Shah Biztantra NextBack
Present Values
The term present valueis often used in
discounting cash flow. It simply means the amount
of money, which must be invested now for n years
at an interest rate of r %, to earn a future sum of
money at the end of year n.
-
8/10/2019 Module I - A - tvm
31/37
Financial Management by Paresh Shah Biztantra NextBack
What is the PV of this uneven cash
flow stream?
0
100
1
300
2
300
310%
-50
4
90.91
247.93
225.39-34.15
530.08 = PV
-
8/10/2019 Module I - A - tvm
32/37
Financial Management by Paresh Shah Biztantra NextBack
Annuities
An annuity is a stream of equal annual cashflows.
A more general definition of an annuity would be a
constant sum of money each year for a given number
of years.
-
8/10/2019 Module I - A - tvm
33/37
Financial Management by Paresh Shah Biztantra NextBack
Ordinary Annuity
PMT PMTPMT
0 1 2 3
i%
PMT PMT
0 1 2 3i%
PMT
Annuity Due
Whats the difference between an
ordinary annuity and an annuity due?
PV
FV
-
8/10/2019 Module I - A - tvm
34/37
Financial Management by Paresh Shah Biztantra NextBack
Whats the FV of a 3-year ordinary annuity of
Rs.100 at 10%?
100 100100
0 1 2 310%
110
121FV = 331
-
8/10/2019 Module I - A - tvm
35/37
Financial Management by Paresh Shah Biztantra NextBack
Whats the PV of this ordinary annuity?
100 100100
0 1 2 310%
90.91
82.6475.13
248.69 = PV
-
8/10/2019 Module I - A - tvm
36/37
Financial Management by Paresh Shah Biztantra NextBack
Cash Flows of Perpetuity
The present value of an annuity for every year in
perpetuity is PV = a / r where r is the cost of
capital as proportion.
-
8/10/2019 Module I - A - tvm
37/37
Exit the Show
Restart the Show