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Essential Topic: Stochastic interest ratesChapter 12

Mathematics of Finance: A Deterministic Approachby S. J. Garrett

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CONTENTS PAGE

MATERIAL

MotivationVarying interest rate modelAccumulation of a single investmentExampleAccumulation of annual investmentsExampleLog-normal distributionExample

SUMMARY

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MOTIVATION

I We have so far been concerned with fixed, piecewiseconstant, or term-structured interest rates.

I In each case, the rates have been known in advance and allcash-flow analyses have been deterministic.

I However, in reality interest rates are not known inadvance. This motivates the use of stochastic models.

I A simplification is to assume the varying interest rate modelwith i.i.d. rates across the years.

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VARYING INTEREST RATE MODEL

I Under the varying interest rate model, the single rate thatapplies in each year is unknown in advance. However,once it has been determined (at the start of the year) it isfixed throughout that year.

I Furthermore, we assume that the interest rates that applyin each year are independent and identically distributed (i.i.d.)across all years.

I Essentially we are assuming that the interest rate in eachyear takes an unknown realization of a known distribution.

I Properties of the distribution of the random rate I areknown in advance and

E[I] = j var(I) = s2.

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ACCUMULATION OF A SINGLE INVESTMENT

I Consider a unit investment made at time t = 0 which willaccumulate under the i.i.d. varying interest rate model forn years.

I The accumulation after n years is denoted Sn.I As the interest rate that applies in each year is random {it},

the accumulation Sn will also be random.I We seek to determine the mean and variance of Sn in terms

of properties of the interest rate distribution.

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ACCUMULATION OF A SINGLE INVESTMENT

I The expected value of Sn is calculated as

E[Sn] =E [(1 + I1)× (1 + I2)× · · · × (1 + In)]

=E [1 + I1]× E [1 + I2]× · · · × E [1 + In] (ind.)

=

n∏t=1

(1 + E[It])

=⇒ E[Sn] =(1 + j)n (if identically distributed)

I This gives the expected value of the accumulation underthe i.i.d. varying interest rate model.

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ACCUMULATION OF A SINGLE INVESTMENTI The variance of Sn is calculated from the standard

expression

var(Sn) = E[S2

n]− E [Sn]

2

I We know that E[Sn]2 = (1 + j)2n

I Further

E[S2n] =E

[(1 + I1)

2 × (1 + I2)2 × · · · × (1 + In)

2]=E[(1 + I1)

2]× E[(1 + I2)

2]× · · · × E[(1 + In)

2] (ind.)

=

n∏t=1

(1 + 2E[It] + E[I2

t ])

=(1 + 2j + s2 + j2)n (if identically distributed)

I The expression for the variance of the accumulation underthe model is then

var(Sn) =((1 + j)2 + s2)n − (1 + j)2n.

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EXAMPLE

The annual interest earned on an investment each year is acollection of i.i.d. random variables with mean 0.05 andstandard deviation 0.15. Calculate the expected value andstandard deviation of the accumulation of £10 after 5 years.

Answer

We have that j = 0.05 and s2 = 0.152. The expected value andstandard deviation of the accumulation are then

E [10S5] =10× 1.055 = £12.76

var (10S5) =102 ×((1.052 + 0.152)5 − 1.0510

)= (4.16)2

=⇒ s.d. =£4.16

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ACCUMULATION OF ANNUAL INVESTMENTS

I We now consider the n-year accumulation of a unitinvestment made at the start of each year for n years underthe i.i.d. varying interest rate model.

I This accumulation is a random variable, An, such that

An = (1 + I1)(1 + I2)(1 + I3) . . . (1 + In)+(1 + I2)(1 + I3) . . . (1 + In)

+(1 + I3) . . . (1 + In)...

+(1 + In).

I To make progress towards expressing properties of An interms of properties of I, we note the recursive relationship

An = (1 + In)(1 + An−1).

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ACCUMULATION OF ANNUAL INVESTMENTSI We take the expectation of the recursive relationship to

find a recursive relationship for the expected value of then-year accumulation, µn

µn = E[An] = E[(1 + In)(1 + An−1)] (ind.)

I For identically distributed rates, this reduces to

µn =(1 + j)(1 + µn−1)

=s̈n at rate j

I Similarly the second moment, mn = E[A2n], can be formed in

terms of a recursive relationship

mn =[var(1 + In) + E[1 + In]

2]E[1 + 2An−1 + A2

n−1]

(ind.)

=[s2 + (1 + j)2] [1 + 2µn−1 + mn−1] (iden. dist.)

I The variance of the n-year accumulation is then

var(An) = mn − µ2n.

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EXAMPLEThe annual rate of interest earned on a particular fund is arandom variable with i.i.d. annual rates. In all years theexpected value of interest is 5% and the standard deviation 8%.Calculate the mean and standard deviation of the totalaccumulated amount at the end of 2 years if £1 is invested atthe start of the first and second years.

Answer

We have that j = 0.05 and s2 = 0.082 and require E[A2] and√var(A2).

E [A2] =µ2 = s̈2 = £2.15

var (A2) =m2 − µ22 =

(s2 + (1 + j)2) (1 + 2µ1 + m1)− 2.152

with m1 =(1 + j)2 + s2 = 1.1089 and µ1 = 1.05

=⇒ var (A2) =(0.082 + 1.052) (1 + 2× 1.05 + 1.1089)− 2.152

=(£0.21)2.

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LOG-NORMAL DISTRIBUTIONI An important distribution for the i.i.d. varying interest

rate model is the log-normal distribution

1 + I ∼ ln N(µ, σ2)

where µ and σ are parameters of the distribution.I The properties of the distribution are such that

ln Sn =

n∑t=1

ln(1 + It)

∼N(nµ,nσ2)

I The distribution parameters, µ and σ2, are linked to j ands2 via

1 + j = eµ+σ22 and s2 = e2µ+σ2 ×

(eσ

2 − 1).

I These form two simultaneous equations for thedistribution parameters in terms of j and s.

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EXAMPLE

A single investment of £2000 is made into a fund that paysinterest under the i.i.d. varying interest rate model with alog-normal distribution. The annual rates have mean 5% andstandard deviation 10%.a.) Calculate the expected value of the accumulation after 10

years.b.) Calculate the probability that the accumulated value after

10 years will be less than 90% of the expected value froma.).

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EXAMPLEAnswer

a.) We have j = 5% and so

E[2000S10] = 2000× (1.05)10 = £3257.79

b.) We can determine that µ = 0.0443 and σ2 = 0.0952.Therefore

P(

S10 < 0.90× 1.0510)=P (S10 < 1.4660)

=P (ln S10 < ln 1.4660)

=P(

ln S10 − 10µ√10σ2

<ln 1.4660− 10µ√

10σ2

)=P(

Z <ln 1.4660− 10× 0.0433√

10× 0.0952

)=P (Z < −0.201) ≈ 42%

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SUMMARY

I Under the i.i.d. varying interest rate model the interest rate in anyyear is unknown in advance but is a realization of the randomvariable I with known distribution. We have E[I] = j andvar(I) = s2.

I Sn denotes the n-year accumulation of a unit investment under thei.i.d. varying interest rate model. We have

E[Sn] = (1 + j)n and var(Sn) =((1 + j)2 + s2)2 − (1 + j)2n.

I An denotes the n-year accumulation of an annual unit investmentmade in advance. We have

E[An] =µn = s̈n and var(An) = mn − µ2n

where mn =[s2 + (1 + j)2

][1 + 2µn−1 + mn−1].

I The log-normal distribution is an important example of adistribution for I

1 + I ∼ ln N(µ, σ2) =⇒ ln Sn ∼ N(nµ,nσ2).