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Title: Cash Flows (For Mathematics in Finance and Investments)
Description: This will help university students studying Finance as well as Financial Mathematics, and will help them learn about how to calculate cash flows in real life situations that may arise in roles within the Finance sector.

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29/09/2015

AM4000/AM4100/AM5000
Lecture 5
Discounting and Accumulating Cash
Flows

Learning objectives


By the end of this lecture you should be
able to:
◦ Calculate the present value of a cash flow
series under the operation of specified rates
of interest where the payments received are
discrete or continuous;
◦ Value cash flow series at any given time;
◦ Calculate the accumulated value of a cash
flow series where the payments are made
continuously
...


Timing of payments may be known or
uncertain
...

 Cash flows may be discrete or
continuous
...

 The company will pay out money for
maintenance, debt repayment and other
management expenses
...
e
...
e
...

lecture 5 cash flows

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Cash flow example 2
A car insurance company receives
premium income
...

 From the company’s point of view:


◦ The premium income is a positive cash flow
...


lecture 5 cash flows

5

PV of a discrete cash flow series


The present value of the amounts:
C  t1  , C  t2  , , C  tn  due at times t1 , t2 , , tn

 where 0  t1  t2   tn  is:
C  t1  v  t1   C  t2  v  t2    C  tn  v  tn 
  C t j  v t j 
n

j 1

◦ where v(tj) is the present value of one unit of
money due at time tj
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PV of an infinite discrete cash flow
series


If the number of payments is infinite, then
the present value is:


PV   C  t j  v  t j 
j 1

◦ provided this series converges
...


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Discrete cash flows: example 2


Calculate the present value of the
following cash flow, assuming an effective
interest rate of 6% per annum:
 £500 due after 1 year, £800 due after 2 years,
£1,200 due after 3 years and £1,000 due after
5 years
...


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5

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Continuously payable cash flows


Suppose that:
◦ T > 0 and that between times 0 and T, an
investor will be paid money continuously
...
e
...

 e
...
a salary that depends on length of service
...
V
...

 Then by definition:


 t   M  t  


d
M t 
dt

Hence total amount received between
times t =  and t = , where 0 ≤  <  ≤ T,
will be:


M     M     M   t  dt     t  dt


lecture 5 cash flows



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The rate of payment continued


Hence:
◦ The rate of payment at any time is the
derivative of the total amount paid up to that
time, and
◦ The total amount paid between any two times
is the integral of the rate of payments over
the appropriate time interval
...

◦ Hence the P
...
of the money received in the
small interval may be considered as:
v(t)(t)dt

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PV of a continuously payable cash
flow series continued


Therefore, the P
...
of the entire cash flow
from time 0 to time T is:

 v  t    t  dt
T

0



If T is infinite, the present value of the
continuous cash flow is:


 v  t    t  dt
0

lecture 5 cash flows

18

PV of discrete plus continuous cash
flows


By combining the formulae for discrete
and continuous cash flows, the present
value of any total cash flow is:
PV   C  t j  v  t j    v  t    t  dt
n

T

j 1

0

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Example 3


The force of interest (t) = 0
...
A continuously payable cash
flow is paid for a period of 7 years
...
07t per annum, 0 < t 7

lecture 5 cash flows

21

Example 3 solution guidelines


PV of cash flow is:

  t v t  dt
7

0



Since the force of interest (t) is a
constant 0
...
09t for all t

For part (a) (t) = £450
 For part (b) (t) = £450e0
...
06 + 0
...
Find the
present value of a continuous cash flow
paid at the rate of:
£100 exp(0
...
08t + 0
...


lecture 5 cash flows

25

Example 4: solution guidelines


PV of cash flow is:

   t v  t  dt
6

0

◦ (t) = £100 exp(0
...
08t + 0
...
06  0
...
06 for t < 15
(t) = 0
...

(b) Find the PV of a continuous cash flow paid
at a constant rate of 10 per annum for 12
years
...
03t per annum for 30 years
...
04  0
...
05



Derive, and simplify as far as possible,
expressions for v(t) where v(t) is the
present value of a unit sum of money due
at time t
...

◦ Calculate the annual effective rate of discount
implied by the transaction above
...
01t units per annum
between t = 10 and t = 15
...


lecture 5 cash flows

34

Exercise 3 - Tutorial 5 Q1


The force of interest, (t), is a function of
time and at any time t (measured in years)
is given by:
◦ (t) = 0
...
005t for
◦ (t) = 0
...
1t paid from t = 10 to t = 18
...
76 and the PV of 1 due at time 4 is
0
...


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Discounted value of a continuous
cash flow series


For a continuous cash flow paid at rate
(t) received from time a to time b
...

◦ The discounted value at time a of this
payment stream is:



b

a





  t   exp    s  ds dt
t

a

◦ The accumulated value at time b of this
payment stream is:



b

a

  t   exp



b

t



  s  ds dt     t   A  t , b  dt
b

a

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Example 7
A continuous cash flow is received for 12
years and the rate of payment at time t is
10e0
...
The force of interest, (t) is
constant at 0
...

 Calculate the accumulated value of the
cash flow after 12 years
...
06
◦ (t) = 0
...
01t
◦ (t) = 0
...
04

a)

for
for
for

0t4
47
Calculate the discounted value at time
t=5 of £1,000 due for payment at time
t=10
...

c) Calculate the accumulated amount at
time t = 10 of a payment stream paid
continuously from t = 0 to t = 4 under
which the rate of payment at time t is:
b)

100e0
...
1t
payable from time t = 0 to t = 10, if the
force of interest is 5% per unit time
...
89455

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Lecture 5 summary: Present values


PV (value at time 0) of a discrete cash
n
flow series is:
PV   C  t j  v  t j 
j 1



PV (value at time 0) of a continuous cash
flow series paid at rate (t) at time t is:





PV     t   exp     s  ds dt
b

a

t

0

    t   v  t  dt
b

a

lecture 5 cash flows

56

Lecture 5 summary: discounted
values


For a continuous cash flow paid at rate
(t) received from time a to time b, the
discounted value at time a, is
...


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58

Lecture 5 summary: accumulated
amount


For a continuous cash flow paid at rate
(t) and received from time a to time b,
the accumulated value at time b is:

   t   exp     s  ds  dt
b

b

a

t

    t  A  t , b  dt
b

a

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19


Title: Cash Flows (For Mathematics in Finance and Investments)
Description: This will help university students studying Finance as well as Financial Mathematics, and will help them learn about how to calculate cash flows in real life situations that may arise in roles within the Finance sector.