essential topic: the theory of interest · essential topic: the theory of interest chapters 1 and 2...

CONTENTS PAGE MATERIAL SUMMARY

Essential Topic: The Theory of InterestChapters 1 and 2

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


CONTENTS PAGE

MATERIAL

The types of interestSimple interestCompound interestThe time value of moneyThe principle of consistencyPiecewise constant iDiscountingInterest-rate quantities

SUMMARY


THE TYPES OF INTEREST

I Simple interest:I Interest is earned by the initial capital deposited. Interest

does not earn interest.I After n years at a rate of simple interest i, a deposit of

amount C will have grown to

C× (1 + ni)

I Compound interest:I Interest is earned on the capital and previously earned

interest.I After n years at a rate of compound interest i, a deposit of

amount C will have grown to

C× (1 + i)n


SIMPLE INTEREST

I Consider an initial deposit of amount C in an account thatpays simple interest at a fixed rate i per time unit. Thevalue of the account at t = 2 is

C× (1 + 2i)

I Consider instead that the investor withdraws his moneyfrom the account at t = 1 and immediately redeposits it. Att = 2, he has

C(1 + i)× (1 + i) = C(1 + 2i + i2

)I The two strategies lead to an inconsistency in the value of

the same initial deposit at t = 2.I Simple interest does not encourage long-term investment

and is inconvenient in practice.


COMPOUND INTEREST

I Consider an initial deposit of C in an account that payscompound interest at a fixed rate i per time unit. The valueof the account at t = 2 is

C× (1 + i)2 = C(1 + 2i + i2

)I Consider instead that the investor withdraws his money

from the account at t = 1 and immediately redeposits it. Att = 2, he has

C(1 + i)× (1 + i) = C(1 + 2i + i2

)I The two strategies do not lead to an inconsistency in the

value of the same initial deposit at t = 2.I Compound interest does encourage long-term investment

and is convenient in practice.


THE TIME VALUE OF MONEY

I It is clear that a deposit grows under the action of apositive interest rate. We call this growth accumulation andfocus on compound interest in all that follows.

I In general, A(t0, t0 + n) denotes the accumulation factor for aunit n-year deposit. In the simple case that i is constant

A(t0, t0 + n) = (1 + i)n

I For example, a deposit of £100 invested at t0 at 8% perannum compound accumulates like

£100× A(t0, t0 + n) = £100× (1.08)n

I Since i is assumed fixed, it is the period of investment, n,that determines the accumulation, not the start time t0.


THE PRINCIPLE OF CONSISTENCY

I As we have seen, compound interest does not lead toinconsistencies when funds are withdrawn and reinvested.

I Mathematically this is stated by the principle of consistency

A(t0, tn) = A(t0, t1)× A(t1, t2)× · · · × A(tn−1, tn)

for times t0 < t1 < · · · < tn.I Unless otherwise stated, one should always assume that

the principle of consistency holds.


PIECEWISE CONSTANT i

I The principle of consistency can be used to calculate theaccumulation of a deposit invested under a piecewiseconstant rate of interest.

I For example, if

i =

{5% for 0 ≤ t < 66% for t ≥ 6

the accumulation factor A(0, 10) is constructed as

A(0, 10) = A(0, 6)× A(6, 10) = (1.05)6 × (1.06)4

I This is easily generalized for any number of subintervals,each defined by the period of fixed i.


DISCOUNTING

I A deposit grows under the action of positive compoundinterest. However, we can look at this from the reverseperspective.

I For example, I have a liability of £1000 to pay in 5 years’time and access to an account paying compound interest at5% per annum. How much, X, should I invest now tocover the liability?

I It is clear that X should be such that

X × A(0, 5) = 1000 =⇒ X = 1000× (1.05)−5 = 783.53

I We refer to the result, £783.53, as the present value of £1000due in 5 years’ time.


DISCOUNTING

I It is useful to define the discount factor ν = (1 + i)−1 suchthat, under fixed i,

1A(t0, t1)

= (1 + i)−(t1−t0) = νt1−t0

I The present value of £1000 due in 5 years is thereforeexpressed as

1000ν5

I As with accumulations, present-value calculations areeasily extended to piecewise constant interest rates usingthe principle of consistency.


INTEREST-RATE QUANTITIES

I Formally, we refer to i as the effective rate of interest per unittime.

I In addition we use ih to denote the nominal rate of interestper unit time on transactions of term h. This is such that

A(t0, t0 + h) = 1 + hih

I In the particular case that h = 1/p, we use i1/p = i(p).I For example, if i(12) = 24% per annum, the effective rate is

i =i(12)

12= 2% per month

I Using the principle of consistency we can determine that

1 + i =

(1 +

i(p)

p

)p


INTEREST RATE QUANTITIES

I The limit that p→∞ (h→ 0) refers to transactions thatoccur over an increasingly small time scale.

I In general, we define the force of interest per unit time to belimit of the nominal rate on momentary transactions

δ(t) = limp→∞

i(p)(t)

I From this it is possible to derive that

A(t0, t1) = exp[∫ t1

t0

δ(s)ds]

and νt1−t0 = exp[−∫ t1

t0

δ(s)ds]

I It is then clear that for δ(t) = δ

1 + i = eδ


SUMMARY

I Interest can be simple or compound. Compound interest is moreimportant in practical situations and is our focus.

I The accumulation factor A(t0, t1) gives the value, at time t1, of aunit investment made at time t0 < t1.

I The discount factor 1/A(t0, t1) = νt1−t0 gives the value of thedeposit required at time t0 to have unit value at time t1 > t0.

I The nominal rate of interest on transactions of term 1/p, i(p), is suchthat

A(0, 1) = (1 + i) =(

1 +i(p)

p

)p

I The force of interest, δ(t) = limp→∞ i(p)(t), is such that

A(t0, t1) = exp[∫ t1

t0

δ(s)ds]

essential topic: the theory of interest · essential topic: the theory of interest chapters 1 and 2...

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