Mesure et calcul du taux d’intérêt

Chapter 1 : Mesure et calcul du taux d’intérêt

1. Interest Accumulation and Effective Rates of Interest

• With an interest rate of i per annum and interest credited annually, an initial deposit of C will earn interest of Ci for the following year. The accumulated value or future value at the end of the year will be C + Ci = C (1+i). The account will continue to grow by a factor of 1 + i per year, resulting in a balance of C (1+i)n at the end of n years.

• Note that ‘’year t’’ starts at time t – 1 and ends at time t.

1.1.1 Effective Rates of Interest

• At interest rate j per compounding period, an initial deposit of amount C will accumulate to C (1+j)n after n compounding periods.

• D1.1 : The effective annual rate of interest earned by an investment during a one-year period is the percentage in change in the value of the investment from the beginning to the end of the year, without regard to the investment behavior at intermediate points in the year.

• D1.2 : Two rates of interest are said to be equivalent if they result in the same accumulated values at each point in time.

1.1.2 Compound Interest

• D1.3 : a(t) is the accumulated value at time t of an investment of 1 made at time 0 and defined as the accumulation factor from time 0 to time t. The notation A(t) will be used to denote the accumulated amount of an investment at time t, so that if the initial investment amount is A(0), then the accumulated value at time t is A(t) = A(0) x a(t). A(t) is the accumulated amount function.

• Compound interest rate at rate i per period is defined with t as any positive real number.

• D1.4 : At effective annual rate of interest i per period, the accumulation factor from time 0 to time t is : a(t) = (1+i)t.

• A fraction of a year is generally described in terms of either an integral number of m months, or an exact number of d days. In the case that time is measured in months, it is common in practice to formulate the fraction of the year t in the form t = m/12, even though not all months are exactly 1/12 of a year. In the case that time is measured in days, t is often formulated as t = d/365 (some investments use a denominator of 360 days instead of 365 days, in which case t = d/360.

• When considering the equation X (1+i)t = Y, given any three of the four variables X, Y, i, t, it is possible to find the fourth. If the unknown variable is t, then solving for the time factor results in t = . If the unknown variable is the interest rate i, then solving for i results in i = 1/t – 1. [pic 1][pic 2]

1.1.3 Simple Interest

• D1.5 : The accumulation function from time 0 to time t at annual simple interest rate i, where t is measured in years is : a(t) = 1 + it.

• As in the case of compound interest, for a fraction of a year, t is usually either m/12 or d/365.

1.1.5 Accumulated Amount Function

• A(t) is the value of the investment at time t, with t usually measured in years. Time t = 0 usually corresponds to the time at which the original investment was made. The amount by which the investment grows from time t1 to time t2 is often regarded as the amount of interest earned over that period, and this can be written as A(t2) – A(t1). Also, with this notation, the effective annual interest rate for the one-year period from time u to time u+1 would be iu+1, where A(u+1) = A(u)(1+iu+1), or equivalently,

Iu+1 =  .[pic 3]

The subscript ‘’u+1’’ indicates that we are measuring the interest rate in year u+1.  Accumulation can have any sort of pattern. This relationship for iu+1 shows that the effective annual rate of interest for a particular one-year period is the amount of interest for the year as a proportion of the valur of the investment at the start of the year, or equivalently, the rate of investment growth per dollar invested. In other words :

Effective annual rate of interest for a specified one-year period

=  [pic 4]

1. Present Value

• If we let X be the amount that must be invested at the start of a year to accumulate to 1 at the end of the year at effective annual interest rate i, then X (1+i) = 1, or equivalently, X = . The amount 1/(1+i) is the present value of an amount of 1 due in one year.[pic 5]

• D1.6 : If the rate of interest for a period is i, the present value of an amount of 1 due one period from now is 1/(1+i). The factor 1/(1+i) is often denoted v in actuarial notation and is called a present value factor or discount factor.

• When a situation involves more than one interest ratem the symbol vi may be used to identify the interest rate i on which the present value factor is based.

• Accumulation under coumpound interest has the form A(t) = A(0)(1+i)t. This expression can be rewritten as

A(0) =  = A(t)(1+i)-t = A(t)vt[pic 6]

Thus Kvt is the present value at time 0 of an amount K due at time t when investment growth occurs according to compound interest. This means that Kvt is the amount that must be invested at time 0 to grow to K at time t, and the present value factor v acts as a ‘’compound present value’’ factorin determining the present value. Accumulation and present value are inverse-processes of one another.

• Present value of 1 due in one period as a function of i : 1/(1+i)

• Present value of 1 due in t periods as a function of t : 1/(1+i)t = vt

• If simple interest is being used for investment accumulation, then A(t) = A(0)(1+it) and the present value at time 0 of amount A(t) due at time t is A(0) = A(t)/(1+it).

1. Canadian Treasury Bills

• Given an accumulated amount function A(t), the investment grows from amount A(t1) at time t1 to amount A(t2) at time t2 > t1 . Therefore an amount of A(t1)/A(t2) invested at time t1 will grow to amount 1 at time t2. In other words, A(t1)/A(t2) is a generalized present value factor from time t2 back to time t1.

1. Equation of value

• In order to formulate an equation of value for a transaction, it is first necessary to choose a reference time point or valuation date. At the reference time point the equation of value balances, the following two factors :

1. The accumulated value of all payments already disbursed plus the present value of all payments yet to be disbursed, and

1. The accumulated value of all payments already received plus the present value of all payments yet to be received.

• In general, when a transaction involves only compound interest, an equation of value formulated at time t1 can be translated into an equation of value formulated at time t2 simply by multiplying the first equation by (1+i)t2-t1. For most transactions there will often be one reference time point that allows a more efficient solution of the equation of value than any other reference time point.

1. Nomial Rates of Interest

• D1.7 : A nominal annual rate of interest compounded or convertible m times per year refers to an interest compounding period of 1/m years.

Interest rate for 1/m period = [pic 7]

1. Actuarial Notation for Nominal Rates of Interest

• In actuarial notation, the symbol i is generally reserved for an effective annual rate, and the symbol i(m) is reserved for a nominal annual rate with interest compounded m times per year. The notation i(m) is taken to mean that interest will have a compounding period of 1/m years and compound rate per period of (1/m) x i(m) = i(m)/m.

• The general relationship linking equivalent nominal annual interest rate i(m) and effective annual interest rate i is

1 + i = [pic 8]

• The comparable relationships linking i and i(m) can be summarized in the following two equations

Note that (1+i)1/m is the 1/m year growth factor, and (1+i)1/m – 1 is the equivalent effective 1/m year comound interest rate.

1. Effective and Nominal Rates of Discount

1. Effective Annual Rate of Discount

• The rate of discount is the rate used to calculate the amount by which the year end value is reduced to determine the present value.

• D1.8 : In terms of an accumulated amount function A(t), the general definition of the effective annual rate of discount from time t = 0 to time t = 1 is

1. Equivalence Between Discount and Interest Rates

• Equation (1.6) can be rewritten as A(0) = A(1) x (1-d), so we see that 1 – d acts as a present value factor. The value at the start of the year is the principal amount of A(1) minus the interest payable in advance, which is d x A(1). On the other hand, on the basis of effective annual interest we have A(0) = A(1) x v. We see that for d and i to be equivalent rates, present values under both representations must be the same, so we must have 1/(1+i) = v = 1-d, or equivalently, d = i/(1+i), or i = d/(1-d).

• The relationships between equivalent interest and discount rates for periods of other than a year are similar. Suppose that j is the effective rate of interest for a period of other than one year. Then dj = j/(1+j) where dj is the equivalent effective rate of discount for that period.

• The present value of 1 due in n years can be represented in the form vn = (1-d)n, so that present values can be represented in the form of compound discount.

1. Simple Discount and Valuation of US T-Bills

• D1.9 : With a quoted annual discount rate of d, based on simple discount the present value of 1 payable t years from now is 1 – dt. Simple discount is generally only applied for periods of less than one year.

1. Nominal Annual Rate of Discount

• D1.10 : A nominal annual rate of discount compounded m times per year refers to a discount compounding period of 1/m years,

Discount rate for 1/m period = quoted nominal annual discount rate / m

• In actuarial notation, the symbol d is generally used to denote an effective annual discount rate, and the symbol d(n) is reserved for denoting a nominal annual discount rate with discount compounded (or convertible) m times per year. The notation d(m) is taken to mean that the discount will have a compounding period of 1/m years and compound rate per period of

(1/m) x d(m) = d(m)/m

• With d(m) in effect, there would generally be m compounding periods during the year, so the equivalent effective annual present value factor would be (1 – d(m)/m)m. If d  is the equivalent effective annual rate of discount, then we have the relationship

1-d = (1- d(m)/m)m

1. The Force of Interest

• Suppose that the accumulated value of an investment at time t is repreented by the function A(t), where time is measured in years. The interest rate earned by the investment for the 1/m year period from time t to time t + 1/m is . This rate can be described in terms of a nominal annual rate of interest compounded m times per year. The nominal annual rate would be found by scaling up the 1/m year rate by a factor of m so that i(m) = m x . Again, although described as an annnual rate, i(m) is the quoted annual rate of interest based on the investment performance from time t to time t + 1/m.[pic 9][pic 10]

• If m is increased, the time interval [t, t+ 1/m] decreases, and we are focusing more and more closely on the investment performance during an interval of time immediately following time t. Taking the limit of i(m) as m → infinite results in :

I(m) =