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Essential Topic: Continuous cash flows
Chapters 2 and 3
The Mathematics of Finance: A Deterministic Approach
by S. J. Garrett
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Continuous payment streams
Example
Continuously paid annuities
Example
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C ONTINUOUS PAYMENT STREAMS
I
Continuously paid cash flows are an important theoretical
construction. Although they do not appear in practice, a
frequently paid discrete cash flow can be approximated by
a continuous payment stream over the long term.
I
For example, consider a pension paid weekly for an
extended time period, say, 20 years. The natural time unit
here is the week and we should use nominal rates i(52) (t)
per annum in the present-value calculation involving
20 × 52 = 1040 payments.
I
Recalling that
δ(t) = lim i(p) (t)
δ→∞
i(52)
we see that δ ≈
and the importance of the force of
interest in the approximation of very frequently paid cash
flows is clear.
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C ONTINUOUS PAYMENT STREAMS
I
We define the rate of payment of a continuously paid cash
flow stream, ρ(t), such that
ρ(t) =
dM(t)
for all t
dt
where M(t) is the total payment between time 0 and t.
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Between times t and t + dt with dt → 0, the total payment
is therefore such that
lim {M(t + dt) − M(t)} = ρ(t)dt.
dt→0
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P RESENT VALUE AND ACCUMULATION
I
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We consider the payment stream as a collection of payment
elements, ρ(t)dt, at each time t within the cash flow.
In its general form, each element has present value
Z t
t
ν ρ(t)dt = ρ(t) exp −
δ(s)ds dt
0
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We then integrate (i.e. sum) these to give the total present
value of the stream
Z t
Z ∞
Z ∞
ν t ρ(t)dt =
ρ(t) exp −
δ(s)ds dt
0
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0
0
The equivalent expression for the accumulation of the
stream to time t = n is
Z n
Z n
ρ(t) exp
δ(s)ds dt
0
t
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E XAMPLE
If the force of interest at time t is given by
(
0.02
for 0 ≤ t < 5
δ(t) =
0.02 × (t − 5) for t ≥ 5
calculate
a.) the value at time 0 of £1000 due at time t = 10,
b.) the accumulated value at time t = 20 of a payment stream
of rate ρ(t) = £1.5t paid continuously between t = 8 and
t = 10.
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E XAMPLE
a.) In this case we have a single payment and should evaluate
the present value by breaking the period into two
subintervals
1000
1000
=
A(0, 10)
A(0, 5) × A(5, 10)
We have
A(0, 5) = exp (0.02 × 5) = e0.10
Z 10
A(5, 10) = exp
0.02 × (t − 5)dt = e0.25
5
Therefore
1000
1000
= 0.10+0.25 = £704.69
A(0, 10)
e
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E XAMPLE
b.) Here we need to consider the payment stream, ρ(t) = 1.5t.
Similar arguments to those above can be used to construct
the accumulation as
Z 10
Z 10
δ(s)ds dt
ρ(t) exp
8
t
Z 10
Z 10
=
1.5t exp
0.02(s − 5)ds dt = £29.39
8
t
Note that here the piecewise nature of δ(t) does not matter
as the integrals are constrained within 8 ≤ t ≤ 10 where
δ(t) = 0.02 × (t − 5).
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C ONTINUOUSLY PAID ANNUITIES
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Consider the particular case that δ(t) = δ and the payment
stream is ρ(t) = 1 for 0 ≤ t ≤ n.
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This payment stream represents a continuously paid, unit
n-year annuity.
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In order to construct an expression for the present value of
the annuity we follow the procedure above
Z
0
I
n
Z t
1 − e−δn
1 × exp −
δds dt =
δ
0
It is usual to denote the present value of this annuity as
ān =
1 − νn
δ
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A CCUMULATED VALUE
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The accumulated value at time n of the continuously paid
annuity, s̄n , can be calculated in a number of ways.
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One approach is to construct the accumulation of the
payment stream as above
Z n
Z n
(1 + i)n − 1
s̄n =
1 × exp
δds dt =
δ
0
t
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Alternatively, one can simply accumulate the present value
at time t = 0 to time t = n
s̄n = (1 + i)n × ān =
(1 + i)n − 1
δ
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E XAMPLE
If the effective rate of interest is i = 4% per annum, calculate
the following quantities.
a.) 3ā10
b.) 10s̄5
Answers
a.) This is the present value of a continuous annuity paying 3
per annum for 10 years,
3ā10 = 3 ×
1 − ν 10
1 − (1.04)−10
=3×
= 24.816
δ
ln(1.04)
b.) This is the accumulated value at t = 5 of a continuous
annuity paying 10 per annum for 5 years,
10s̄5 = 10 ×
(1 + i)5 − 1
(1.04)5 − 1
= 10 ×
= 55.239
δ
ln(1.04)
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I
The present value and accumulation of a continuously
paid cash flow can be calculated by constructing integral
expressions in terms of ρ(t) and δ(t).
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A continuously paid unit n-year annuity is the particular
case that ρ(t) = 1 for 0 ≤ t ≤ n years. If δ(t) = δ, we denote
the present value of this stream as
ān =
1 − νn
δ
and the accumulated value at time t = n as
s̄n =
(1 + i)n − 1
δ