Download Present and Future Values

Time Value of
Money
A. Caggia – M. Armanini
Financial Investment & Pricing
2016-2017
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Investments & Cashflows
Simple viewpoint …
Investment = the present commitment of money for the
purpose of receiving more money later.
General viewpoint …
Investment = a set of cashflows of expenditures (negative
cashflows) and receipts (positive cashflows) spanning a
period of time
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Objective(s) of Investment
The objective of investment is to tailor the pattern of
cashflows over time, in order to fit our needs.
This could be done in order to maximise returns.
Often future cash flows have a degree of uncertainty,
and part of the design of a cash flow stream may be
concerned with controlling that uncertainty, perhaps
reducing the level of risk.
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Cashflow stream
Usually cashflow stream (either positive or negative) occur
at known specific dates, such as at the end of each quarter
of a year or at the end of each year.
The stream can then be described by listing the value and
the date of each cashflow.
The stream can be deterministic (known in advance) or
random (unknown in advance)
Cashflow stream
Example:
Imagine that you will receive 10 EUR in six months time,
another 10 EUR in one year and you will spend 20 EUR in
two years time. This cashflow stream ca be represented by
two vectors denoting the values of each individual cashflow
and when these will happen.
Cashflows:
Payment dates:
C = (+10,+10,-20)
D = (0.5, 1.0, 2.0)
cashflow graphical representation
Example
+
−
T0= 0
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
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T1
T2
T3
T4
time
cashflow graphical representation
Example
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+
−



10
10
10

time
cashflow graphical representation
Example
30
+
−



10
10
10

time
cashflow graphical representation
Example
+
−
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3
3
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time
Present and Future Values




Every financial operation generates cash inflows and/or
outflows over a certain time horizon
These cashflows represent the amount of money that
are expected to be received or paid over time on the
back of an investment/debt decision
If the cashflows are scheduled on different maturities,
their value can’t be directly compared
To be compared they need to be expressed on same
timing conventions:
PRESENT VALUE and FUTURE VALUE
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Present and Future Values
DF
PV
PV
FV
=
DF
× FV
time
Present and Future Values
DF1
FV2
FV1
PV
time
DF2
PV
=
DF1
×FV +
1
DF2
×FV
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Present and Future Values
KNOWN CASHFLOWS
UNKNOWN CASHFLOWS
Time Value and Interest Rates

Money has a time value because of the opportunity for
investing money at some interest rate (which is the
“compensation” for not spending now)

The main reasons that concur to explain the time value
of time (as well as the level of interest rates) are:
 Individual preferences on consumption and savings
 Inflation
 Uncertainties
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Interest Rate Determinants

The main determinants of the level of interest rates are:
 a real part: compensate the investor for the choice of
postponing his/her consumption in terms of higher
expected spending power
 a nominal part: is added to compensate for the loss
of spending power due to the consequences of
positive inflation rates
 a risk premium part: depends of the uncertainties
associated to investing (e.g. likelihood of debtor not
repaying the loan)
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Contents

Compounding

From Present Value to Future Value

Simple Interest

Compound Interest

Continuous Compounding
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Compounding

Compounding = to put together into a whole; to combine.

Interest Rate Compounding = to apply interest rates to
investments with a certain frequency

Therefore, in order to define accurately the amount to be
paid under a legal contract with interest, the frequency of
compounding (yearly, half-yearly, quarterly, monthly,
daily, etc.) and the interest rate must be specified.
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Types of Compounding

Simple Compounding = interest rate is applied on a
single period between today (t=0) and maturity (t=T).

Compounding N Periods = interest rate is applied
sequentially N times between today (t=0) and maturity
(t=T).

Continuous Compounding = interest rate is applied
continuously (i.e. infinite number of periods) between
today (t=0) and maturity (t=T).
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Present to/from Future Values
Present
Value
Present
Value
CF
PV
FV
DF
Present
Value
time
Future
Value
Present to/from Future Values
FV
=
CF
PV
FV
PV
DF
time
DF
PV
=
DF
×FV
Simple Compounding
CF
PV
FV
time
Compounding: N periods
CF
PV
CF
FV
CF
FV
FV
time
Compounding: Continuous
CF
PV
FV
time
Simple Compounding
A is the amount invested at t=0
r is the interest rate applied (annualized)
T is the maturity of the investment (annualized)
V is the final value of the investment at t=T
Compounding: N periods
N is the number of equally periods per year, NT is then
the total number of periods between today (t=0) and
maturity (t=T)
Compounding: N periods
r’ is the effective interest rate, i.e. the equivalent yearly
interest rate that would produce the same result without
compounding
Compounding: Continuous
Conversion Formulas

If Rc is a continously compound interest
rate, while Rm is the equivalent m times
per year compound rate, the formulas
relating Rc and Rm are:
 Rm 
Rc  mln1 

m 



Rm  m e Rc / m  1
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Annuities


The present value of future cashflows, for which
different discount factors apply, can be manually
calculated adding the present value of each
individual cashflow
There are formulas that allow to simplify the
calculation of present and future values,
depending on the shape of cashflows. In
general, the typical format is the one of an
annuity: a stream of inflows of outflows to be due
on future dates
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Annuities

Constant: fixed
cashflows, all equal

Variable: variable
cashflows, change over
time

Ordinary: cashflows are
paid at regular intervals
for a fixed time horizon

Perpetuity: stream of
cashflows never ending

In Arrears: cashflows are
exchanged at the end of
each relevant period

In Advance: cashflows
are exchanged at the
beginning of each
relevant period
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Amortization

Amortizing a debt means to repay it back over
time. There are 3 main types of models:
 Reimburse
S(1 + i)t of capital S at the maturity n of
the debt
 Repay periodic interest rate accrues over capital S (Si
at any given period) and repay capital S at the
maturity n of the debt
 Repay both interest rate accrues and capital S over a
certain number of periodic installments
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