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Multiple Interest Rate Analysis
Theory and Applications
M. Osborne
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eBook - ePub
Multiple Interest Rate Analysis
Theory and Applications
M. Osborne
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About This Book
This book is an analysis of all possible interest rates. Dual expressions are used to solve long-standing puzzles, eliminate anomalies and draw conclusions about best practice and sound policy advice in areas of economics and finance. Topics include retail and corporate finance, capital budgeting and investment appraisal, bond risk management.
An on-line model demonstratingideas from the book is available in the Wolfram Demonstrations Project (WDP) bysearching "multiple interest rate analysis" in the WDP search engine.A 'computable document' containing the model and the model's code are alsoavailable as free downloads from the site.
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1
Multiple-Interest-Rate Analysis: What It Is and Why It Is Important
Abstract: This chapter provides a simple summary of multiple-interest-rate analysis, both what it is and what it is not.
Keywords: Bond risk, capital budgeting, capital theory, complex number, consumer credit, corporate finance, interest rate, investment appraisal, time value of money
JEL classifications: B1, B16, B2, C00, C02, C60, G0
Osborne, Michael J. Multiple Interest Rate Analysis: Theory and Applications. Basingstoke: Palgrave Macmillan, 2014. DOI: 10.1057/9781137372772.
Multiple-interest-rate analysis sheds new light on the TVM equation. The analysis is important because the concept of the time value of money and its application via the TVM equation underlie most of finance and much of economics. This chapter summarizes multiple-interest-rate analysis, both what it is and what it is not.
I offer you a choice: you receive $1 with no conditions attached or you accept a wager. The wager is to toss a coin; if it lands ‘heads’ I pay you $6 and if ‘tails’ you pay me $4. Assuming the coin is fair the expected outcome from the wager is (0.5 × $6) – (0.5 × $4) = $1. The expected values are the same: which do you prefer – certainty or wager?
Most people accept the certain dollar. When asked why, they speak of risk: the two outcomes have the same expected value but do not have the same risk. The prospect of receiving $6 makes the wager attractive but the prospect of losing $4 makes it so unattractive that they opt for certainty. Other things being equal, the certain outcome trumps the risky one.
How should I alter the wager so that you are persuaded to accept it? Assume you receive $7 if the coin lands heads and you pay $4 if it lands tails. The expected outcome is now (0.5 × $7) – (0.5 × $4) = $1.5. Your choice is between a certain dollar and a risky $1.5. If you accept the wager then 50 cents is the value placed on the risk. This additional 50 cents is conveniently expressed as a percentage mark-up on the certain dollar, that is, $1.5/$1.0 = 1.5 = (1 + r) where r = 0.5, or 50 percent.
I now make another kind of offer, one involving time. You receive one dollar today or a dollar on the same day next year. Which do you choose? Most people choose a dollar today because much can happen during a year, and therefore a future dollar carries risk. I could have an accident, suffer amnesia (do I really owe you a dollar?), your employer sends you to work in a far-away location and we never meet again, or inflation reduces the purchasing power of the dollar. For reasons like these, most people prefer the dollar today.
How much must the deferred payment be for you to accept it? If you propose $1.20 then the value you place on risk during the year is 20 percent. This mark-up, or rate of interest (r), measures the time value of money. When the payment today is p and the future payment is f, the appropriate mathematical expression is the compounding equation p(1 + r) = f. The element (1 + r) is a compound factor, converting value today to value tomorrow. The equation can be converted to the discounting equation p = f [1/(1 + r)] in which the element in square brackets is a discount factor, converting value tomorrow to value today.
The choice changes yet again: you can have $1.20 next year or a higher amount in two years’ time. How much must you be offered to accept payment after two years instead of one? A figure of $1.50 implies an interest rate of 50 percent over two years. Comparing 50 percent over two years with 20 percent over one year is not straightforward, therefore financial markets adopt a convention – rates of interest are usually annualized. The future $1.50 is analyzed by splitting the two years into two one-year periods, each having its own annual rate of interest.
The second year is farther away in time and therefore more risky, in which case a higher rate of interest could be demanded for the second year. This situation, having two different one-year compound factors, is expressed in the following equation.
Since p = $1 and f = $1.50 and we know that r1st = 0.2 then r2nd is calculated from (1.2).
The interest rate r2nd is 0.25 or 25 percent.
The pattern of interest rates through time, such as the simple two-year pattern of r1st followed by r2nd, is the yield curve. A non-flat yield curve, in which every future period has a different interest rate, presents financial markets with some interesting problems, and therefore the yield curve is the subject of much study. It will appear again in a later chapter. This example is quoted here in order to highlight a potential confusion: the existence of multiple interest rates along a non-flat yield curve is not the subject of this monograph. The term ‘multiple interest rates’ in this work means something different.
In order to explain the term ‘multiple interest rates’ the problems presented by non-flat yield curves are eliminated by assuming interest rates are equal in all periods, that is, r1nd = r2nd = r. Equation (1.1) becomes the compounding equation (1.3).
As before, we note the compound factor, in this instance (1 + r)2, and the fact that the compounding equation is reversible, that is, equation (1.3) becomes the discounting equation p = f [1/(1 + r)2] in which the element in square brackets is the discount factor.
At this point a useful question can be asked. When the values of p and f are known (p = $1 and f = $1.5) what is the rate of interest (r) applying to both years? Equation (1.3) is easily transformed into (1.4) from which r is calculated to be 22.47 percent.
This rate is also calculated from the geometric mean of the two one-year compound factors, that is, (1 + r) = [(1 + r1st)(1 + r2nd)] 0.5 = [(1.2)(1.25)]0.5 = 1.2247.
This apparently simple question-plus-answer is interesting for the following reason. The quadratic equation (1.5) can be solved for the variable x, given values for the coefficients a, b, and c.
Equation (1.6) is the well-known quadratic formula solving the quadratic equation, and it gives two solutions for x, not one.
Equation (1.7) is the time value of money equation (1.3) recast into the format of the quadratic equation (1.5) in which the variable is not x but (1+r), and the coefficients are not a, b, and c, but the cash flows p and f.
The quadratic formula solving equation (1.7) for the two solutions to (1+r) is (1.8).
Given the values of the coefficients p = 1 and f = 1.5, the values for (1 + r) are 1.2247 and minus 1.2247, implying values for the interest rate (r) of 0.2247 (or 22.47 percent, which is known already) and minus 2.2247 (or minus 222.47 percent).The orthodox interest rate of 22.47 percent per year for two years is readily understood, but the deeply negative interest rate presents a puzzle because such a rate is never found in the real world, and therefore it appears to have no use or meaning.
The puzzle deepens if the number of periods increases. When present value is $1 and future value is $1.60 after three years then equation (1.9) applies....