Essential Topic: The Theory of Interest

Transcription

1 Essential Topic: The Theory of Interest Chapters 1 and 2 The Mathematics of Finance: A Deterministic Approach by S. J. Garrett

2 CONTENTS PAGE MATERIAL The types of interest Simple interest Compound interest The time value of money The principle of consistency Piecewise constant i Discounting Interest-rate quantities SUMMARY

3 THE TYPES OF INTEREST Simple interest: Interest is earned by the initial capital deposited. Interest does not earn interest. After n years at a rate of simple interest i, a deposit of amount C will have grown to C (1 + ni) Compound interest: Interest is earned on the capital and previously earned interest. After n years at a rate of compound interest i, a deposit of amount C will have grown to C (1 + i) n

4 SIMPLE INTEREST Consider an initial deposit of amount C in an account that pays simple interest at a fixed rate i per time unit. The value of the account at t = 2 is C (1 + 2i) Consider instead that the investor withdraws his money from the account at t = 1 and immediately redeposits it. At t = 2, he has C(1 + i) (1 + i) = C ( 1 + 2i + i 2) The two strategies lead to an inconsistency in the value of the same initial deposit at t = 2. Simple interest does not encourage long-term investment and is inconvenient in practice.

5 COMPOUND INTEREST Consider an initial deposit of C in an account that pays compound interest at a fixed rate i per time unit. The value of the account at t = 2 is C (1 + i) 2 = C ( 1 + 2i + i 2) Consider instead that the investor withdraws his money from the account at t = 1 and immediately redeposits it. At t = 2, he has C(1 + i) (1 + i) = C ( 1 + 2i + i 2) The two strategies do not lead to an inconsistency in the value of the same initial deposit at t = 2. Compound interest does encourage long-term investment and is convenient in practice.

6 THE TIME VALUE OF MONEY It is clear that a deposit grows under the action of a positive interest rate. We call this growth accumulation and focus on compound interest in all that follows. In general, A(t 0, t 0 + n) denotes the accumulation factor for a unit n-year deposit. In the simple case that i is constant A(t 0, t 0 + n) = (1 + i) n For example, a deposit of 100 invested at t 0 at 8% per annum compound accumulates like 100 A(t 0, t 0 + n) = 100 (1.08) n Since i is assumed fixed, it is the period of investment, n, that determines the accumulation, not the start time t 0.

7 THE PRINCIPLE OF CONSISTENCY As we have seen, compound interest does not lead to inconsistencies when funds are withdrawn and reinvested. Mathematically this is stated by the principle of consistency A(t 0, t n ) = A(t 0, t 1 ) A(t 1, t 2 ) A(t n 1, t n ) for times t 0 < t 1 < < t n. Unless otherwise stated, one should always assume that the principle of consistency holds.

8 PIECEWISE CONSTANT i The principle of consistency can be used to calculate the accumulation of a deposit invested under a piecewise constant rate of interest. For example, if i = { 5% for 0 t < 6 6% 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 This is easily generalized for any number of subintervals, each defined by the period of fixed i.

9 DISCOUNTING A deposit grows under the action of positive compound interest. However, we can look at this from the reverse perspective. For example, I have a liability of 1000 to pay in 5 years time and access to an account paying compound interest at 5% per annum. How much, X, should I invest now to cover the liability? It is clear that X should be such that X A(0, 5) = 1000 = X = 1000 (1.05) 5 = We refer to the result, , as the present value of 1000 due in 5 years time.

10 DISCOUNTING It is useful to define the discount factor ν = (1 + i) 1 such that, under fixed i, 1 A(t 0, t 1 ) = (1 + i) (t 1 t 0 ) = ν t 1 t 0 The present value of 1000 due in 5 years is therefore expressed as 1000ν 5 As with accumulations, present-value calculations are easily extended to piecewise constant interest rates using the principle of consistency.

11 INTEREST-RATE QUANTITIES Formally, we refer to i as the effective rate of interest per unit time. In addition we use i h to denote the nominal rate of interest per unit time on transactions of term h. This is such that A(t 0, t 0 + h) = 1 + hi h In the particular case that h = 1/p, we use i 1/p = i (p). For example, if i (12) = 24% per annum, the effective rate is i = i(12) 12 = 2% per month Using the principle of consistency we can determine that ( ) p 1 + i = 1 + i(p) p

12 INTEREST RATE QUANTITIES The limit that p (h 0) refers to transactions that occur over an increasingly small time scale. In general, we define the force of interest per unit time to be limit of the nominal rate on momentary transactions δ(t) = lim p i (p) (t) From this it is possible to derive that [ t1 ] A(t 0, t 1 ) = exp δ(s)ds t 0 It is then clear that for δ(t) = δ [ t1 ] and ν t 1 t 0 = exp δ(s)ds t i = e δ

13 SUMMARY Interest can be simple or compound. Compound interest is more important in practical situations and is our focus. The accumulation factor A(t 0, t 1 ) gives the value, at time t 1, of a unit investment made at time t 0 < t 1. The discount factor 1/A(t 0, t 1 ) = ν t 1 t 0 gives the value of the deposit required at time t 0 to have unit value at time t 1 > t 0. The nominal rate of interest on transactions of term 1/p, i (p), is such that ) p A(0, 1) = (1 + i) = (1 + i(p) p The force of interest, δ(t) = lim p i (p) (t), is such that [ t1 ] A(t 0, t 1 ) = exp δ(s)ds t 0