essential topic: continuous cash flows -...
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CONTENTS PAGE MATERIAL SUMMARY
Essential Topic: Continuous cash flowsChapters 2 and 3
The Mathematics of Finance: A Deterministic Approachby S. J. Garrett
CONTENTS PAGE MATERIAL SUMMARY
CONTENTS PAGE
MATERIAL
Continuous payment streamsExampleContinuously paid annuitiesExample
SUMMARY
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CONTINUOUS PAYMENT STREAMS
I Continuously paid cash flows are an important theoreticalconstruction. Although they do not appear in practice, afrequently paid discrete cash flow can be approximated bya continuous payment stream over the long term.
I For example, consider a pension paid weekly for anextended time period, say, 20 years. The natural time unithere is the week and we should use nominal rates i(52)(t)per annum in the present-value calculation involving20× 52 = 1040 payments.
I Recalling thatδ(t) = lim
δ→∞i(p)(t)
we see that δ ≈ i(52) and the importance of the force ofinterest in the approximation of very frequently paid cashflows is clear.
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CONTINUOUS PAYMENT STREAMS
I We define the rate of payment of a continuously paid cashflow stream, ρ(t), such that
ρ(t) =dM(t)
dtfor all t
where M(t) is the total payment between time 0 and t.I Between times t and t + dt with dt→ 0, the total payment
is therefore such that
limdt→0{M(t + dt)−M(t)} = ρ(t)dt.
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PRESENT VALUE AND ACCUMULATION
I We consider the payment stream as a collection of paymentelements, ρ(t)dt, at each time t within the cash flow.
I In its general form, each element has present value
νtρ(t)dt = ρ(t) exp[−∫ t
0δ(s)ds
]dt
I We then integrate (i.e. sum) these to give the total presentvalue of the stream∫ ∞
0νtρ(t)dt =
∫ ∞0
ρ(t) exp[−∫ t
0δ(s)ds
]dt
I The equivalent expression for the accumulation of thestream to time t = n is∫ n
0ρ(t) exp
[∫ n
tδ(s)ds
]dt
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EXAMPLE
If the force of interest at time t is given by
δ(t) =
{0.02 for 0 ≤ t < 50.02× (t− 5) for t ≥ 5
calculatea.) 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 andt = 10.
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EXAMPLE
a.) In this case we have a single payment and should evaluatethe present value by breaking the period into twosubintervals
1000A(0, 10)
=1000
A(0, 5)× A(5, 10)
We have
A(0, 5) = exp (0.02× 5) = e0.10
A(5, 10) = exp(∫ 10
50.02× (t− 5)dt
)= e0.25
Therefore1000
A(0, 10)=
1000e0.10+0.25 = £704.69
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EXAMPLE
b.) Here we need to consider the payment stream, ρ(t) = 1.5t.Similar arguments to those above can be used to constructthe accumulation as∫ 10
8ρ(t) exp
(∫ 10
tδ(s)ds
)dt
=
∫ 10
81.5t exp
(∫ 10
t0.02(s− 5)ds
)dt = £29.39
Note that here the piecewise nature of δ(t) does not matteras the integrals are constrained within 8 ≤ t ≤ 10 whereδ(t) = 0.02× (t− 5).
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CONTINUOUSLY PAID ANNUITIES
I Consider the particular case that δ(t) = δ and the paymentstream is ρ(t) = 1 for 0 ≤ t ≤ n.
I This payment stream represents a continuously paid, unitn-year annuity.
I In order to construct an expression for the present value ofthe annuity we follow the procedure above∫ n
01× exp
(−∫ t
0δds)
dt =1− e−δn
δ
I It is usual to denote the present value of this annuity as
an =1− νn
δ
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ACCUMULATED VALUE
I The accumulated value at time n of the continuously paidannuity, sn , can be calculated in a number of ways.
I One approach is to construct the accumulation of thepayment stream as above
sn =
∫ n
01× exp
(∫ n
tδds)
dt =(1 + i)n − 1
δ
I Alternatively, one can simply accumulate the present valueat time t = 0 to time t = n
sn = (1 + i)n × an =(1 + i)n − 1
δ
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EXAMPLEIf the effective rate of interest is i = 4% per annum, calculatethe following quantities.a.) 3a10
b.) 10s5
Answersa.) This is the present value of a continuous annuity paying 3
per annum for 10 years,
3a10 = 3× 1− ν10
δ= 3× 1− (1.04)−10
ln(1.04)= 24.816
b.) This is the accumulated value at t = 5 of a continuousannuity paying 10 per annum for 5 years,
10s5 = 10× (1 + i)5 − 1δ
= 10× (1.04)5 − 1ln(1.04)
= 55.239
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SUMMARY
I The present value and accumulation of a continuouslypaid cash flow can be calculated by constructing integralexpressions in terms of ρ(t) and δ(t).
I A continuously paid unit n-year annuity is the particularcase that ρ(t) = 1 for 0 ≤ t ≤ n years. If δ(t) = δ, we denotethe present value of this stream as
an =1− νn
δ
and the accumulated value at time t = n as
sn =(1 + i)n − 1
δ