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TOPICS:
1.
2.
3.
4.
What is financial mathematics?;
What is annuity?;
Time line;
Kinds of annuity;
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1. WHAT IS FINANCIAL MATHEMATICS?
Mathematical finance is a field of applied mathematics, concerned with financial markets.
The subject has a close relationship with the discipline of financial economics, which is
concerned with the underlying theory. Generally, mathematical finance derives and
extends the mathematical or numerical models suggested by financial economics.
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2. WHAT IS ANNUITY?
The term annuity is used in financial mathematics to refer to any terminating sequence of regular
fixed payments over a specified period of time. Loans are usually paid off by an annuity. If
payments are not at regular (irregular) periods, we are not working with an annuity. We get
two types of annuities:
i.
The ordinary annuity :This is an annuity whose payments are made at the end of each
period. (At the end of each week, month, half year, year, etc.) Paying back a car loan, a
home loan, etc…;
ii.
The annuity due: This is an annuity whose payments are made at the beginning of each
period. Deposits in savings, rent payments, and insurance premiums are examples of
annuities due.
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3. ON THE TIMELINE…
If we look at the timeline, we clearly see that if we are looking at investing money into an
account, then we will be working with the future value of these payments. This is so because
we are saving up money for some use one day in the future. If we want to consider the
present value of a series of payments, then we will be looking at a scenario where a loan is
being paid off. This is so because we get the money today, and pay that money with interest
back to the financing company some time in the future.
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GRAPHIC …
Future Value works forwards
Present Value works backwards
T
T1
T2
T3
T4
T5
T6
X
X
X
X
X
X
X
T7
X
Regular periodic payments x
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4. KINDS OF ANNUITY:
We have four kinds of annuity; these are:
A. future values of ordinary annuities;
B. Future value of due annuity;
C. Present value of annuity;
D. Present value of due annuity;
E. Perpetuities annuity;
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A. FUTURE VALUE OF ORDINARY ANNUITY
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The Future Value of Ordinary Annuity (FVoa) is the value that a stream of
expected or promised future payments will grow after a given number of
periods at a specific compounded interest.
The Future Value of an Ordinary Annuity could be solved by calculating the
future value of each individual payment in the series using the future value
formula and then summing the results. A more direct formula is:
Fvoa= = PMT [((1 + i)n - 1) / i]
Where:
FVoa = Future Value of an Ordinary Annuity
PMT = Amount of each payment
i = Interest Rate Per Period
n = Number of Periods
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EXAMPLE…
What amount will be accumulated if we deposit $5,000 at the end of each year for the
next 5 years? Assume an interest of 6% compounded annually.
Fvoa =b is future value of annuity.
Pmt = 5,000.00
i = 0.06
n = 5 years
FVoa = 5,000*[(1.06)^5-1/0.06] =
5,000 *[ (1.3382255776 - 1) /0.06 ] =
5,000* (5.637092) = 28,185. 46
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Let's watch the graphic: ……
Future Value works forwards
0
Pmt
Pmt
Pmt
Pmt
FVoa
Pmt
1
2
3
4
5
6
7
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B. FUTURE VALUE OF DUE ANNUITY
The Future Value of due Annuity is identical to an ordinary annuity except that
each payment occurs at the beginning of a period rather than at the end. Since
each payment occurs one period earlier, we can calculate the present value of
an ordinary annuity and then multiply the result by (1 + i).
FVad = FVoa *(1+i)
Where:
 FVad = Future Value of an Annuity Due
 FVoa = Future Value of an Ordinary Annuity
 i = Interest Rate Per Period
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EXAMPLE…
What amount will be accumulated if we deposit $5,000 at the beginning of
each year for the next 5 years? Assume an interest of 6% compounded
annually.
PV = 5,000
i = 0.06
n=5
FVoa = 28,185.46* (1.06) = 29,876.59
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Let's watch the graphic: ……
0
Pmt
Pmt
Pmt
Pmt
Pmt
1
2
3
4
5
FVoa
6
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C. PRESENT VALUE OF AN ORDINARY ANNUITY
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The Present Value of an Ordinary Annuity (PVoa) is the value of a stream of expected or
promised future payments that have been discounted to a single equivalent value today. It is
extremely useful comparing two separate cash flows that differ in some way.
PV-oa can also be thought of as the amount you must invest today at a specific interest rate so
that when you withdraw an equal amount each period, the original principal and all accumulated
interest will be completely exhausted at the end of the annuity.
The Present Value of an Ordinary Annuity could be solved by calculating the present value of
each payment in the series using the present value formula and then summing the results. A
more direct formula is:
PVoa = PMT [(1 – (1 + i)-n) / i]
Where:
PVoa = Present Value of an Ordinary Annuity
PMT = Amount of each payment
i = Discount Rate Per Period
n = Number of Periods
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EXAMPLE…
Mario buying a car, paying € 5000 per year for 7 years, starting next
year. Applying the assessment rate del15% per year, what is the
price of the car?
PVoa = PMT [(1 – (1 + i)-n) / i]
Pvoa= present value of ordinary annuity
PMT= 5000€
i= 0.15
N= 7 years
Pvoa= 5000*[(1-(1+0.15)^-7)/0.15] = 20802,10€
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we now see the graph, how to do .....
PVoa
Pmt
0
1a
Pmt
2a
Pmt
3a
Pmt
4a
Pmt
5a
Pmt
Pmt
6a
7a
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D. PRESENT VALUE
OF DUE ANNUITY
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The Present Value of due annuity is identical to an ordinary annuity except that
each payment occurs at the beginning of a period rather than at the end. Since
each payment occurs one period earlier, we can calculate the present value of an
ordinary annuity and then multiply the result by (1 + i).
PVoad= PVoa*(1+i)
Where:
Pvoad= present value of an ordinary annuity due;
Pvoa= present value of an ordinary annuity.
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EXAMPLE…
Mario buying a car, paying € 5000 per year for 7 years, starting next
year. Applying the assessment rate del15% per year, what is the
price of the car?
PVoad = Pvoa*(1 + i)
Pvoad= present value of ordinary annuity due
PMT= 5000€
i= 0.15
N= 7 years
Pvoad= (5000*[(1-(1+0.15)^-7)/0.15] )*(1.15)= 23922.42€
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we now see the graph, how to do .....
Pmt
PVoa
0
Pmt
1a
Pmt
2a
Pmt
3a
Pmt
4a
Pmt
5a
Pmt
6a
7a
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E. PERPETUITY ANNUITY
A perpetuity is an annuity in which the periodic payments begin on a fixed date and continue
indefinitely. It is sometimes referred to as a perpetual annuity. Fixed coupon payments on
permanently invested (irredeemable) sums of money are prime examples of perpetuities.
Scholarships paid perpetually from an endowment fit the definition of perpetuity.
The value of the perpetuity is finite because receipts that are anticipated far in the future have
extremely low present value (present value of the future cash flows). Unlike a typical bond,
because the principle is never repaid, there is no present value for the principal. Assuming
that payments begin at the end of the current period, the price of a perpetuity is simply the
coupon amount over the appropriate discount rate or yield, that is:
PV= A/i
Where:
 PV = Present Value of the Perpetuity;
 A = Amount of the periodic payment;
 i = Discount rate or interest rate.
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1. Present value of perpetuity annuity
An example:
calculate the present value of the following perpetuity: € 1254 per
year, postponed to the 11% rate.
PV= A/i
PV= present value of perpetuity annuity;
A= 1254€
i= 0.11
PV= 1254/0.11= 12400€
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we now see the graph, how to do .....
PV
Pmt
Pmt
Pmt
Pmt
Pmt
Pmt
+ infinite
0
1a
2a
3a
4a
5a
6a
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2. Present value of due perpetuity annuity
An example:
calculate the present value of the following perpetuity: € 1254 per
year, postponed to the 11% rate.
PVD= (A/I)*(1+i)
PV= present value of perpetuity annuity;
A= 1254€
i= 0.11
PV= 1254/0.11= 12400€
PVD= (1254/0.11)*(1.11)= 12654€
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we now see the graph, how to do .....
Pmt
PVD
Pmt
Pmt
Pmt
Pmt
Pmt
+ infinite
0
1a
2a
3a
4a
5a
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GLOSSARY
Annuity
= Rendite;
Future
values of ordinary annuities = Montante di una rendita anticipata;
Future
value of due annuity= montante di una rendita posticipata;
Present value of ordinary annuity = Valore attuale di una renditya anticipata;
Present value of due annuity= valore attuale di una rendita posticipata;
Present value of Perpetuity annuity = rendita perpetua anticipata
Present value of due perpetuity annuity= rendita perpetua posticipata
Discount rate= interessi.
N= numero di rate.
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