rate of interest rates, actuarial discount rate

Updateand financial asset pricesJean-Marcel DalbaradeMethods of measurement of financial assets have recentlymore complex, the pace of financial innovation and developmentQuick derivatives: futures, options or swaps. The techniquebase remains the discounted based itselfon conventional calculations of rates, which focuses on thefirst section of this article. Is applied in the second section,these general methods to the case made by the instrumentsthe most important financial and representative.1. Definitions and principlesThe easiest financial transaction consists of two streamspayment of opposite direction, on two separate dates. This operation bysuch a loan and its repayment or purchase of an asset u zerocoupon ", and resale (Figure 1) is the main support forpresentation of basic definitions, all related to vocabularyrate.Figure 1 - The basic operation "zero-coupon" F,F0 t F (t)1.1. The interest rateIn the elementary operation to two flows, the difference between thesecond stream of the first flow and repayment of the loan is calledinterest. The amount of interest paid by the borrower, is calculatedgenerally with an annual percentage of the amount ofthe operation, the rate of interest i, applied to the time, expressed inyears.Numerous conventions, in particular related to the calculation of thistime t, corresponding to as many assessment methods that amount,or which allows to deduce the second stream from the first or,Conversely, the initial payment from the refund.Formulas [1] and [2] are two classic examples whererefund is calculated from the amount borrowed, either bysimple interest calculation proportional to time or by calculatingcompound of interest, the growth of the second flow with respect to timethen being exponentiallyFt = Fo (1 + it) [1]Fo = ft (1 + i) [2]This second method of calculation has a significant advantage over thefirst, that of allowing the composition to different periodssuccessive (hence its name) flows received, placed at the same rateannual interest. If, indeed, after having taken an amount for Fàa time tl, it reinvests the first payment for a Fltime t2, we get a second stream of reimbursement F2 as:F1 F2 = (1 + i) t2 = Fo (1 + i) t1 + (1 + i) t2 = Fo (1 + i) tl + t2result [3] which is strictly equivalent to the placement of the initial flux Fo ~the lengthTotal tl + t2The calculation of simple interest obviously leads to the same result,since the term (1 + iti) (1 + it2) is not equal to i + 1 (t + t2). Thedefinitiona simple interest rate is thus fundamentally related to the durationits application (regardless of any complication related to thecurve).For operations whose duration is longer than one year, severalPayments usually take place between the beginning and the end ofreimbursement. These payments are in practice either flowinterest intermediaries, or partial repayments ofcapital. It is useful to understand these calculations, decomposethe overall operation several elementary operations (two-stream)sometimes by chaining them in time, or by superimposingthe original date, several simultaneous operations zero-coupon.1.2. The continuous rateThe interest rate continuously, denoted r, is a simple interest rate,annual applicable over a much shorter rlt, whereassumes reinvestment of the interest continuous.Growth of an initial flow FL is such that, at each instant t:
1.3. The actuarialThe actuarial rate (or yield to maturity) of an operationfinancial rate of interest, compounded annual, which allowsreplenish its cash flows.In practice, the date of the first stream is chosen as the origin, itis always possible to assume that all flows are furthereach flow to repay a zero-coupon transaction whosefirst stream is paid to the original date. Equation [5] then expressedknown that the initial flow Fo is the sum of all the first stream,calculated from the repayment flows, also known, andCAGR unknown.Figure 2 - An operation to any multiple streamsF0 = [F1 / (1 + i) exp t1] + [F2 / (1 + i) exp t2] + ... + [Fn / (1 + i) exp tn]This equation [5] we deduce, by iterative methods, the valuethe only unknown, the actuarial rate i.1.4. The rate of returnFinancial transactions are not the only ones to berepresented by a stream pattern offset in time. Ofbusiness investment, with durable goods orstocks, followed by operating revenues, are operations thatrespond to the same logic and can therefore besubject to the same calculations.Rate of return, also known as "internal rate of return"Is thus equivalent among economists or managersbusiness, the yield to maturity of the financial statements.1.5. The discount rateThe discount rate measures the degree of preference for the present(Or conversely, the cost of waiting) of economic agent, thea priori preference being expressed subjectively by comparingchanges in income over time by definition, if a is theannual discount rate, received around 1 franc, equivalent to (1+ A) francs a year, equivalent to itself of hui is EQ1 + a) 2francs in two years of dansitivité urea preferences of a rateconstant at (1 + a) t francs received bConversely, and this is the normal use of the discount rate, amountafter a time t is now equivalent to 1 / (i + a) talso called "1 franc updated."we can evaluate the equivalent unee date today (even nonewhole) byFt any amount received at a future date by the formula formula:_ F0 = Ft / (1 + a) tIn this equation, Fo is paid today to say pourc'estupdated the amount rece Ft. This may be the price p in the future.