The Time-Value of Money and the Money-Value of Tim

Electronic copy available at: http://ssrn.com/abstract=2613469

The Time-Value of Money and the Money-Value of Time: Duration, Roundaboutness, Productivity and Time-Preference in Finance and Economics

Peter Lewin

Naveen Jindal School of Management

University of Texas at Dallas

800W Campbell Road,

Richardson, TX 85080 [email protected]

Nicolás Cachanosky

Department of Economics

Metropolitan State University of Denver

Campus Box 77, P.O. Box 173362

Denver, CO 80217 [email protected]

mailto:[email protected]

mailto:[email protected]


The Time-Value of Money and the Money-Value of Time

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The Time-Value of Money and the Money-Value of Time: Duration, Roundaboutness, Productivity and Time-Preference in Finance and Economics

1. Introduction All human action takes place in time, and with an awareness of time. We make choices in the present with a view to the future. Choices imply valuation, and the relative value to us of the things we choose depend crucially on the their connection to time. We discount the value of things depending on how long we have to wait for them. This well-known phenomenon is called the time-value of money. When contemplating multiple receipts or payments distributed over time, if we need to judge the effect of the influence of time on the each of the values received or paid, we need some way to value the time periods involved. We may call this the money-value of time. A particular approach to capturing this idea has been developed in the finance literature, known as ‘duration’. Similar considerations can be found in the economics literature in the area of capital-theory. In this paper we explore the interaction between these two phenomena. In the next section we examine the familiar time-value of money equations and focus on the meaning and significance of the discount-rate in investment decision-making. In section 3 we examine the calibration of time in decision-making, especially in the face of potential changes in discount-rates and the value of assets. In section 5 we focus on the concept of duration and its uses and limitations, introducing a recent contribution to the precise measurement of the discount-rate-elasticity of present-value for discrete changes in discount-rates. We also consider the importance of non-flat yield-curves and differences in time-horizons across investors. In section 6 we consider the well-known, but little understood, debate in economics over ‘reswitching paradoxes’ and show that the paradoxes are spurious. In doing this we open a window into a better understanding of the concept of ‘capital’ and the role it plays in economic life. This understanding is facilitated by the knowledge gained earlier in exploring the interaction between the time-value of money and the money-value of time. Section 7 concludes.

2. Investment decisions and the time-value of money For investment decisions in a monetary economy we model the time-value of money in a familiar way.

(1) 𝐶𝑉 =

𝐶𝐹1

(1 + 𝑑)+

𝐶𝐹2

(1 + 𝑑)2+ ⋯ +

𝐶𝐹𝑛

(1 + 𝑑)𝑛= ∑

𝐶𝐹𝑡

(1 + 𝑑)𝑡

𝑛

𝑡=1

= ∑ 𝑓𝑡𝐶𝐹𝑡

𝑛

𝑡=1

Where:

CV = the capital-value1 of the investment, being the present value of the investment. In some contexts it is the original financial capital outlay. For a bond traded in a competitive financial market it is the market price of the bond.

1 The use of the term capital-value, CV, synonymous with net-present-value, NPV, is deliberate, though, we realize not without potential for confusion – because of the simultaneous use of the word ‘capital’ in the description of certain productive inputs as capital-goods. One of our goals, however, is to examine the meaning and nature of the



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CFt = the money-valued cash-flow expected from the investment in period t (t = 1, … n) - which is the net-value of earnings and outlays in that period and can be positive, negative or zero.

n = the time-horizon of the investment or the number of periods for which the investor is planning from now until the investment is considered to end. It is the planning period of the investor. For a fixed income investment like a bond it is called the term to maturity.

d = the rate of discount applied to any future-value to reduce it to present-value. As explained below, depending on the context, d, can be considered the to be the rate of time-preference of the investor, or it can be a market interest-rate that determines the market price of the investment (as in the case of a bond), or something similar. We will refer to it as the discount rate. As we shall see, the discount-rate is really a markup rate.

𝑓𝑡 = 1

(1+𝑑)𝑡 which we shall refer to as the discount-factor.

This equation expresses a universal arithmetic relating value and time as perceived by human actors. There are a large number of potential unknowns. For the equation to be of practical use information must be supplied for all but one of the unknowns. So, for example, in the case of a fixed coupon bond the equation would be:

(1a) 𝑃 =𝐶

(1 + 𝑦)+

𝐶

(1 + 𝑦)2+ ⋯ +

𝐶 + 𝐹𝑉

(1 + 𝑦)𝑛= 𝐶 [∑

1

(1 + 𝑦)𝑡

𝑛

𝑡=1] +

𝐹𝑉

(1 + 𝑦)𝑡

P = the market price of the bond

C = the fixed coupon payment over the life of the bond

n = the life of the bond

FV = the face-value (original price) of the bond, to be paid back upon its expiration

y is the yield to maturity (YTM) of the bond. 2 Everything except y is known. Barring default the bond-holder knows that the bond will pay C per period and FV at the end of the investment period, n. The price to purchase the bond, P, is given in the market. y is calculated given P, C, n and FV. It is that number that solves the equation, making the present-value of the stream of payments equal to the price. Other special cases, like premium bonds, discount bonds and perpetuities, are well known and need not be repeated here. The essential take-away point is the significance of y in connecting values over time. An investor purchasing the bond knows that each dollar of investment of P dollars will be marked-up by y percent in each sub-period of the investment period (Osborne, 2014). It is the essence of what is known as the time value of money. In a more general context, encompassing any kind of multi-period investment, y is known as the internal rate of return, i ( or IRR)– that rate that reduces the expected income stream of the investment to its

concept of ‘capital’ and to provide a clarification of any potential confusion along these lines and for this purpose the term capital-value is very appropriate. 2 This equation simplifies to

𝑃 = 𝐶 [1 −

1(1 + 𝑦)𝑡

𝑦] +

𝐹𝑉

(1 + 𝑦)𝑡


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current cost – the current financial capital outlay. At this rate of return, the cost of the project is equal to its present-value. Thus, i can be compared to current market interest-rates (yields) to see if the investment per dollar covers its opportunity cost.3 To gain a fuller understanding of this principle, a more detailed examination of the meaning and significance of d (or y) is necessary. Consider the following example:4 You are offered a choice. You can receive $1 with no conditions, or you can accept a wager to toss a coin. If it lands ‘heads’ you get $6, if it lands ‘tails’ must pay $4. If the coin is fair the expected value of the wager is $1 – it is a choice between two identical expected values. As we know, most people choose the certain $1, because the second alternative is risky and they are risk-averse. The prospect of losing $4 looms larger than that of winning $6. But, as we also know, there is some similar wager that most people will accept. If I increase the upside to $7 it will be more attractive (an expected value of $1.5). If this is the tipping point at which you accept this wager then your minimum required markup to make the risk acceptable is 50 cents on $1 or 50%, which is d, your discount-rate in this instance. Consider a second example. You must choose between receiving $1 today and $1 exactly one year from today. A no brainer? Why? We say it is because $1 is today is preferred to the promise of $1 tomorrow. We call this time-preference5. But what is the reason for this time-preference. Though it is not really necessary for our purposes to answer this question, we believe that it is reasonable to suppose that it is not merely ‘impatience’. The passage of time in and of itself is always connected to uncertainty. Something promised in the future, no matter how firm the promise, can never be as secure as the same thing available right now. That is the very nature of time as we experience it. So the two examples are actually closely related. Whether or not we can attach probabilities to future outcomes will determine whether we talk about risk or uncertainty (Knight 1921) and risk-aversion or uncertainty-aversion, but the difference will not matter for our purposes. This is related to the yield-curve to be discussed below. Considering further the second example we may ask how much must the one-year deferred payment be increased for it be deemed equivalent to the $1 now? Assume it is 20 cents, that is, the one-year deferred payment of $1.20 is deemed just sufficient for you to postpone the receipt. You thus require a markup of 20%. d is 20% in this instance. Let d have the specific value d1. Generally, and familiarly, for a payment of p and a future payment of fv1, p(1+d1) = fv1; or p = fv1/(1+ d1). Now imagine a third alternative. You can have the $1.20 in one-year hence or a higher amount in two-year’s time. What is the tipping point? Let’s assume it is 50%, so that fv2=$1.50. This 50% rate is a two-year discount or markup rate that is higher than the one year rate of 20%. Since we know the one-year rate we can write, fv2 = p(1+d1)(1+d2). d1 and d2 define a term-structure of two one-year and one two-year

3 It is well known that the IRR criterion is inferior to using the magnitude of NPV (net present-value) when deciding among exclusive investment projects and that there are instances when the two criteria give different rankings. Among available investments that cover the (the opportunity) cost of capital the investor should choose the one with the highest NPV at that cost. This does not affect our discussion. 4 The line of reasoning and the examples that follow are taken from Osborne 2014, 2-3. 5 We banish any consideration of being able to invest the $1 during the year and thus accumulate more than $1. We imagine this choice to be made in an environment where the market rate of interest that can be earned by investing this $1 is zero.


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discount rates. More generally, fvn = p(1+d1)(1+d2) … (1+dn) for an n-period choice considered to have been revealed by a series of one-period choices. This logic suggests that for every multi-period investment choice there is an implicit logic of discount-rates that can be expressed in this way. The values of the discount rates are dependent on the time-preference of the investor and the decision framework supplied, namely, the alternatives presented to the investor. The latter includes the way time is measured and attached to the payments involved – the periodic rate of markup (or discount). There is nothing in the logic that compels the periodic discount rates to be equal to one another. In fact a more general version of equation 1, considering a sum of future values, CFt, for each period 1 … n, can now be written as,

(1b)

𝐶𝑉 =𝐶𝐹1

(1 + 𝑑1)+

𝐶𝐹2

(1 + 𝑑1)(1 + 𝑑2)+ ⋯ +

𝐶𝐹𝑛

(1 + 𝑑1)(1 + 𝑑2) … (1 + 𝑑𝑛)= ∑ [

𝐶𝐹𝑡

∏ (1 + 𝑑𝑗)𝑡𝑗=1

𝑛

𝑡=1]

It is apparent that a particular CV is compatible with an infinite number of different values for = d1,d2 … dn – there are an infinite number of versions of this equation. Unless we have access to the type of experiment described above, there is no way to determine an individual’s detailed time preference structure. To be sure, in a financial market of individuals, the term structure of market interest-rates (the yield curve) gives us an estimate of the time preferences of the marginal traders, and, as we know, it is not, in general, flat. Though the ‘typical’ shape of the yield curve is upward sloping, representing higher interest-rates for longer investment terms, short-term interest-rates ‘typically’ fluctuate over time. indeed, the prospect of interest-rate fluctuations, and the attendant fluctuations in capital-values that this implies, motivates much discussion in financial economics. Comparing equation 1 with equation 1b, the former is clearly much more tractable and more useful in most practical situations. For any investment of term n, of the form in equation 1, it is possible to calculate a discount rate d, that is in some sense an average of all the different discount rates for all the sub-periods involved. This invokes the metaphor of a common markup (discount) per dollar within each sub-period over the life of the investment. But, it is important to emphasize, this is a matter of convention. There is nothing ‘natural’ about assuming that d = d1 = d2 = … = dn. It is a very convenient convention used to summarize an important aspect of any investment, especially given the number of alternative possible ways of doing so. To illustrate, consider an investment in a start-up business with an initial outlay, followed by fluctuating earnings and expenses. Perhaps the large initial outlay is expected to be followed by a few years of negative cash flows, after which, positive cash flow will materialize for an extended period, say ten years, after which a year of restructuring will become necessary, etc. The calculated rate of return for this venture will be very sensitive to the time-horizon presumed. If the investment is considered to terminate before the advent of positive cash flows, the rate of return will appear to be negative. Waiting a few more years may substantially turn this around. Terminating the investment before the ten-year restructuring may yield a higher return than waiting for it. But staying the course beyond than may again raise the return, etc. To ask at what rate a dollar is ‘actually’ growing within any sub-period of a chosen investment period, is rather irrelevant metaphysics. What matters to the investor with a chosen time horizon is the calculated rate of return (and implied NPV) over that time horizon –which will be compared to his time-preference or cost of capital in the making of his decision.


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These are the essentials of the time-value of money. Money earned over time is valued according to the influence of time on its value. The units are dollars, or dollars/time.

3. Investment decisions and the money-value of time Because of positive marginal time-preferences (including impatience and risk-uncertainty-aversion), investments will be made only if they promise pay a premium. In a growing economy this implies creating value. Resources are marshalled and combined in ways that promise to produce outcomes that consumers value enough to cover the costs of doing so. In common terms, the transformation of resources into more valuable uses is known as production, and the more value added the more productive this process is considered to be. At least since Adam Smith economists have considered this phenomenon to be at the heart of the creation of the wealth of nations. Some economists have paid more attention to the role of time in this. The Austrian School of economics (pioneered by Carl Menger) in particular is associated with an examination of the role of time in production, giving rise to a body of work known as the Austrian Theory of Capital (ATC).6 Most explicitly in the work of Eugen von Böhm-Bawerk, the role of time in production is manifest. Since production takes time, the relationship between value and time must be considered. Time has to be ‘spent’ in order to get results in the form of products that are useful to consumers, that are valued more highly than the combined value of what went into them over time. This suggests that if ‘more’ time is to be taken to produce anything, there must be a reward. This comes in the form of a higher valued product. In Böhm-Bawerk’s terms, wisely-chosen roundabout production processes are more productive. But what does it mean to take ‘more’ time? Consideration of this leads one very quickly into difficult territory. To attempt to quantify the time-taken raises a whole host of difficult questions. When does the ‘time-period’ begin – or end? It is not time per se that is taken. Rather it is work-time – the application of effort over time by different kinds of resources. So it is input-time that is relevant and must be measured. In what units? And so on. In order to simplify the matter, and hopefully make it tractable, Böhm-Bawerk suggested the concept of the ‘average period of production’ (APP)7 – a conceptual measure of the ‘average

6 Not only ‘Austrians’ have contributed key insights. Important names are Carl Menger, Eugene von Böhm-Bawerk, William Stanly Jevons, Knut Wicksell, Frank Fetter, Ludwig von Mises, John Hicks, Friedrich Hayek, and Ludwig Lachmann. For a summary see Lewin [1999] 2011, chapter 4 and the references therein. 7 The APP may succinctly express as follows:

𝑇 =∑ (𝑛 − 𝑡)𝑙𝑡

𝑛𝑡=0

∑ 𝑙𝑡𝑛𝑡=0

= 𝑛 −∑ 𝑡 ∙ 𝑙𝑡

𝑛𝑡=0

𝑁

where T is the APP for a production process lasting n calendar periods; t, going from 0 to n, is an index of each sub-period; It is the amount of labor expended in sub-period t, and N = ∑ 𝑙𝑡

𝑛𝑡=0 is the unweighted labor sum (the total

amount of labor-time expended). Thus T is a weighted average that measures the time on average that a unit of labor l is “locked up” in the production process. The weights (n-t) are the distances in time from the emergence of the final output. T depends positively on n, the calendar length of the project, and on the relation of the time pattern of labor applied (the points in time t at which labor inputs occur) to the total amount of labor invested N. Except for


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amount of time’ taken in the production of any product. Several scholars picked up on this aspect of Böhm-Bawerk’s work and made it the basis of criticism. But the APP has refused to die. Over the decades it has reappeared in various guises, explicitly or implicitly, in a series of ‘capital controversies’8. This is considered further below. While Böhm-Bawerk admittedly used a concept to capture the role of time in production that is very limited in its applicability to real-world processes, the essential idea is incredibly important and is a precursor to much work on the nature of production in the modern world. Böhm-Bawerk tried to capture in quantitative terms the average amount of time taken in any production project. As can be easily shown, except for the most simple of cases, this is impossible. As soon as one considers the relationship between capital and time, value enters the analysis and a purely physical (quantitative) measure is impossible. Böhm-Bawerk’s essential error lies not in his attempt to take account of time considerations in the mind of the investor/entrepreneur as expressed in some simple formulation, but, rather, in his attempt to do so by confining his attention to a strictly physical measure. As John Hicks (1939, p. 186) pointed out as early as 1939 a valid form of the APP does exist – he called it the average period (AP). It is exactly that same construct developed by Frederick Macaulay (1938)9 in 1938 that is known as ‘duration’. Duration (D) is most easily understood as ‘the average amount of time for which one has to wait for $1’ in any investment. It is a measure of the ‘length’ of the project – or, at least, some aspect of the length. It captures an important aspect of what is in the investor’s mind as he contemplates his investment. Specifically,

(2) 𝐷 = ∑ (

𝑓𝑡𝐶𝐹𝑡

𝐶𝑉)

𝑛

𝑡=1𝑡

where the terms are as previously defined. Note D is a weighted average of the time-units involved in the project, starting from 1, the earliest, to n, the last, where the weights are the proportions of the present value of the investment received (or paid) in the time period (ftCFt/CV). It is the (present) value weighted amount of time involved in the investment. As such it is a money-value of time measure. The logic is simple. The economic significance of the time involved in the investment, the amount of time for which one has to wait for payments to be made or received, is dependent on the relative size of payments involved in each of the periods involved. The simple size of the calendar time, n, is not very informative. The same n can have very different significance to the investor depending on whether the payments occur sooner or later and in what proportions. The value-significance of the time involved must be considered. Given time-preference, other things constant, a longer average period (duration) should carry a higher markup. These are the essentials of the money-value of time. the amount of time involved in any investment is valued according to the influence of value on time.

very simple processes, this formulation is not theoretically sound, and is not at all helpful for real-world decision-making. This bears some resemblance to the discussion below concerning the question of capital ‘reswitching’. 8 For a recent summaries and assessments see Cohen and Harcourt (2003a, 2003b and Garrison 2006). 9 See also P. Lewin and N. Cachanosky, 2014.


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4. Discount rates changes and time Financial markets fulfil the vital task of facilitating the flow of resources from those who have claim to them to those who can make best (better) use of them. in a well-oiled market economy this works automatically, but not seamlessly. Considerable judgement is called for. Financial institutions and financial specialists profit by supplying some of this judgement (to augment that of the investing entrepreneur). Financial specialists and entrepreneurs are faced with the task of appraising investment prospects of a wide variety of types. Any investment has multiple dimensions, one of which is the size of the return promised, as discussed in the previous section. Also relevant are various kinds of risk, including interest-rate risk (from fluctuating interest-rates) and the risk of default or bankruptcy. We are concerned here mainly with the former (the latter is also reflected in the interest-rate required). Using equation 1 to characterize investments (where d is presumed equal for all sub-periods), the sensitivity of the CV to changes in interest-rates (more specifically to the rate of discount applied to the investment) is a key factor in investment appraisal. And financial specialists have long worked to develop tools to mitigate, if not completely immunize, investments from this risk. It turns out, as first indicated by Hicks (1939) that D is also a measure of the elasticity of the (present) value of the project with respect to the discount ft. It measures how any estimate of net present value changes with a change in the discount factor, for small changes. Hicks’s formulation (1939 [1947]: 186) proceeds as follows: The capital-value (CV) of any stream of n payments (cash flows) is given as before by

(1) 𝐶𝑉 = ∑𝐶𝐹𝑡

(1 + 𝑑)𝑡

𝑛

𝑡=1= ∑ 𝑓𝑡𝐶𝐹𝑡

𝑇

𝑡=1

We may calculate the elasticity of this CV with respect to the ft’s, as

𝐸𝐶𝑉,𝑓𝑡=

𝐸(𝐶𝑉)

𝐸(𝑓𝑡)=

1

𝐶𝑉[1 ∙ 𝑓1𝐶𝐹1 + 2 ∙ 𝑓2𝐶𝐹2 + ⋯ + 𝑛 ∙ 𝑓𝑇𝐶𝐹𝑇]

or

(2’) 𝐸𝐶𝑉,𝑓𝑡= ∑ (𝑡

𝑛

𝑡=1)

𝑓𝑡𝐶𝐹𝑡

𝐶𝑉

where E is the elasticity (or d log) operator. This follows from the rule that the elasticity of a sum is the weighted average of the elasticities of its parts. 𝐸𝐶𝑉,𝑓𝑡

turns out to have a number of interesting

interpretations.

Firstly, and obviously, 𝐸𝐶𝑉,𝑓𝑡

provides a measure of the sensitivity of the value of the project (investment)

to changes in the rate of discount, or (inversely) in the discount factor. So, anything that affects the discount rate applied to investments will affect their relative valuations. Significantly, the perceived values


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of investment projects that constitute the components of the capital-structure10 will be unevenly affected by monetary policy that systematically affects discount rates. Those components of existing production processes that have a higher 𝐸𝐶𝑉,𝑓𝑡

will be relatively more affected – for example, a fall in the discount

rate (perhaps provoked by a fall in the federal-funds and other interest-rates) will produce a rise in the value of high-𝐸𝐶𝑉,𝑓𝑡

projects relative to those with lower ones.

But, secondly, 𝐸𝐶𝑉,𝑓𝑡

is Hicks’s AP which is duration11. Equation 2’ is identical to equation 2, though 2 is

interpreted as a proportional CV-weighted-average of time-elapsed, and 2’ is interpreted as a time-weighted-average of proportional changes in overall CF. D thus serves the dual purpose of measuring both ‘roundaboutness’ and the sensitivity of capital-value to changes in the discount rate (factor). The two aspects of D may be seen as a time-weighted-average of proportional responses in CV and as a value-weighted-average of the time involved in any investment respectively. It is the same formula with different components respectively being considered to be weights for summing the other components. It is only in the second aspect – as a measure of response, as first formulated by Macaulay and subsequently further developed, that has been of interest to the finance discipline, but both are of interest to economists and have wider application for them. We will discuss both aspects of D in what follows.

5. The uses and limitations of D Modified duration and immunization Macaulay’s (1938) initial discovery of D was motivated by the search for a useful measure of the ‘length’ of an investment, and, though discovered again by Hicks (1939) who pointed out that it was also a measure of the sensitivity of the CV to a change in the rate of discount (also Samuelson 1945, who independently offered similar insights), it was only the work of Redington (1952) that brought this to the attention of the actuarial community, and then a further 19 years before it was picked up by the field of financial analysis through the work of Fisher and Weil (1971).12 Redington sought to use D to find a way to immunize the value of a bond portfolio from interest-rate risk. D as expressed above is the elasticity of the bond price (its CV) with respect to the discount factor 𝑓𝑡 =

1

(1+𝑦)𝑡. Modified duration, MD is closely related - it is a linear approximation of the sensitivity of the price

of the bond to changes in the discount rate = the yield to maturity, y. MD is measured as the percent change in the price of the bond when y changes by one unit. MD is, then, the semi-elasticity of the bond price P with respect to y. MD and its relation to D can be represented by the following expression:13

(3) 𝑀𝐷 =𝑑𝑙𝑜𝑔𝐶𝑉

𝑑𝑦=

− 𝐷

1 + 𝑦

where D is the duration of the investment evaluated at y. Thus, in general, while D is a measure of the elasticity of CV with respect to the discount factor f, MD is a measure of the (semi) elasticity of CV with

10 In the finance literature the term ‘capital structure’ signifies the debt-equity ratio. It is used here, as in capital-theory economics, to signify the structure of physical-capital projects. 11 “… when we look at the form of this elasticity we see that it may be very properly described as the Average Period [AP] of the stream [of earnings]; for it is the average length of time for which the various payments are deferred from the present, when the times of deferment are weighted by the discounted values of the payments.” (Hicks 1939: 186, italics in original, see also 218-22).. 12 A number of useful surveys exist, including Bierwag et. al. 1983 and Bierwag and Fooladi, 2006. 13 For any CV, 𝐷 = 𝐸𝐶𝑉,𝑓𝑡

=𝐸(𝐶𝑉)

𝐸(𝑓𝑡)=

𝐸(𝐶𝑉)

−𝐸(1+𝑦)=

𝐸(𝐶𝑉)

−𝑑

1+𝑦.dy

𝑦

= − 𝑀𝐷(1 + 𝑦). We have equation 3.


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respect to y. The essence of immunization consists of structuring the portfolio (containing known revenues to pay off known outlays) in such a way that the present-value and duration of the outlays equals the present-value and duration of revenues. Then the overall effect on the CV from a change in the discount rate will be zero. Though a very appealing idea of great potential use for financial practitioners, matters are not quite that simple. Convexity Duration (either D or MD) has a number of moving, interacting parts, and, thus, like any elasticity, is a valid measure of response only for very small changes. Duration measures the effect of a change in the discount rate on present-value. But the present-value, in turn, is used to calculate duration. The money-value of time interacts with the time-value of money. So there are second-order and higher effects. Used as a practical tool for dealing with discernable discrete changes, and needing precise magnitudes of responses (for example, for immunization purposes), these interaction effects must be taken into account. One method is to use a measure of the second-order effect known as convexity.

We know that 𝑀𝐷 = −𝑑𝑙𝑛(𝑃)

𝑑𝑦= −

1

𝑃∙

𝑑𝑃

𝑑𝑦 and therefore that

𝑑𝑃

𝑑𝑦= −𝑀𝐷 ∙ 𝑃, where P is the price (present

value) of a bond discounted at rate y. Convexity is defined as Č =1

𝑃∙

𝑑2𝑃

𝑑𝑦2.

A bond with a larger convexity (Č) has a price that changes at a higher rate when there is a change in y, than a bond with a lower convexity. Č is important in portfolio management because two bonds can have the same MD but different values of Č. A sinking fund bond with a shorter maturity can have the same MD as a zero coupon bond with a longer maturity. Therefore, the same change in the discount rate affects to a different extent the price of each bond because these two bonds have a different Č even if they have the same MD. The financial literature has known methods for dealing with this (Bierwag, et. al. 1983, pp. 19-20).14 Taking account of convexity will provide a more accurate estimate of the effect on the bond price of changes in the yield to maturity. It does not take into account still higher-order effects. Recent work by Michael Osborne (2005, 2014) has suggested a fascinating, more sophisticated approach, based on the fundamental mathematics of the time-value of money equation. Using polynomial roots The bond-price equation with known cash-flows (coupon payments and principle) can be written as a polynomial of degree n, the term to maturity. To understand this recall the expressions fv2 = p(1+d1)(1+d2) for a two-period investment, and fvn = p(1+d1)(1+d2) … (1+dn) for an n period investment, leading up to equation 1b. In addition to being an intuitive useful financial convention, equation 1 is also actually a very important special case of equation 1b, where the same discount (markup) rate applied to each sub-period is a ‘solution’ to the equation. It is that value of d (equal for all sub-periods) that makes the expression a valid equation – it is the root of the equation considered as a polynomial expression. Taking the two-period case (we borrow again from Osborne 2005, pp. 3-8)

14 The matter is similar to the situation facing an economist trying to estimate the response of the amount demanded

to a discrete change in price in a real-world setting. The elasticity of demand (estimated for example from a simple linear regression) is a rough linear approximation to the desired result. It is less accurate the greater the curvature of the demand curve.


11

fv2 = p(1+d1)(1+d2) we seek the solution d = d1 = d2

f = p(1+d1)(1+d2) = p(1+d)2 or p = f/(1+d)2

In terms of the numbers provided earlier, $1 accumulates (is marked up) to $1.50 in two years. What is the rate of markup that applies to both years equally? Clearly it is the geometric mean (in this case square root) of the two-year markup. Using the numbers considered earlier, (1+d) = (1.5).5 = 1.2247, so that d is 22.47% which is the geometric average of the two one-year rates, 20% and 25%. (1 +d) = (1+d1)(1+d2) = (1.2)(1.25)=1.2247. In general (1+d) = [(1+d1)(1+d2) … (1+dn)]1/n for the n-period case. Now, p = f/(1+d)2 or p(1+d}2 – f = 0 is an instance of the second-degree polynomial equation ax2 +bx +c = 0, where x= (1+d), a = p, b = 0 and c = -f. and it is well-known that this equation has two solutions of the form

𝑥 = −𝑏 ± √𝑏2 − 4𝑎𝑐

2𝑎

or (1 + 𝑑) = ± √−4𝑝(−𝑓)

2𝑝. For p = 1 and f = 1.5, the values for (1+d) are 1.227 and -1.2247 implying values

for d of 0.2247 and -0.2247. The first solution confirms our earlier result. The existence of two roots is well-known, and, since it is only the positive root that appears to have economic meaning, the negative root is routinely ignored. This is not only understandable, it appears to be the only defensible approach. Someone investing a dollar today in return for a promise of a dollar and a half in two years has a clear understanding that the in two years the amount received will be worth more than the dollar is worth today (assuming no inflation), that is, will be exchangeable for more or high-valued goods and services. To say that the dollar will be worth less than a dollar in two years if we apply the second solution appears to be irrelevant. No one would choose that option. Thus for purposes of borrowing and lending money (valuable things) the existence of multiple roots is not evidently significant. Where, however, one wishes to use the mathematics to provide practical solutions to the problem of managing, mitigating or avoiding the risk to capital values from interest (discount) rate fluctuations, it might be more relevant. Such is the claim made by Michael Osborne. It is also implicit in the famous debate between the two ‘Cambridges’ about the meaning of capital that we will consider briefly below. To understand Osborne’s claim, we need to consider cases of more than two roots, higher-order polynomials. Again following Osborne, take the case of a three period investment, where in exchange for $1 today you are promised $1.60 in three years, so that, f = p(1+d)3= 1.6.


12

For the equation p(1+d)3- f = 0, there are three solutions (available from suitable software programs for finding the solutions for nth-order equations). (1+d1) = 1.1696 (the one obtainable in the usual ‘orthodox’

way). The other two are (1+d2) = -.5848-1.012291i and (1+d3) = -.5848+1.012291i. i is √−1. These two

roots are complex numbers, numbers involving imaginary numbers (multiples of √−1 ) and real numbers. Though they have widespread use in engineering, their possible relevance in economics is not at all obvious. In short, the roots of any polynomial can be real numbers, positive or negative, and complex numbers, with positive or negative elements (the complex roots always come in conjugate pairs, so that when multiplied together result in a real number) – there will always be at least one positive real root. Economists and others, who have known this, have suggested that the only economically meaningful roots are those that are real and positive.15 Osborne claims, however, that this is a mistake and that all of the other roots considered together contain important, economically significant information. One example of this is the formula for duration, D.16 As discussed above, for discrete changes, the logarithmic derivative of NPV with respect to the log of the discount factor (or with respect to the discount rate) may be a poor linear approximation to the actual proportional change in the NPV that results. Adding in the effect of convexity (the second term in the relevant Taylor series expansion for the NPV equation) brings one closer to the real change. In essence, Osborne’s claim is that all the roots taken together include all the information available for all higher-order effects (all of the terms in the relevant Taylor series expansion), so that his alternative, corrected formula, for D is precise. Osborne shows that every polynomial has a dual equation, so that the time-value of money equation for the capital-value of an n-period investment, CV, can be writer as,

(4) 𝐶𝑉 = ∑

𝐶𝐹𝑡

(1 + 𝑑)𝑡

𝑛

𝑡=1=

∑ 𝐶𝐹𝑡𝑛𝑡=1

1 − (−1)𝑛 ∏ 𝑑𝑡𝑛𝑡=1

where, (1+d1), (1+d2) , … , (1+dn) are the n roots of the equation and which for most cases (patterns of cash-flows) can be more simply written,

(4’) 𝐶𝑉 =

∑ 𝐶𝐹𝑡𝑛𝑡=1

1 + ∏ |𝑑𝑡|𝑑1𝑛𝑡=2

This ‘dual’ expression tells us that the CV (the present-value) of a cash-flow equation can be written in terms of its cash-flows divided by (1+) the product of all the roots of the equation; and that for most cases involving financial assets the denominator reduces to (1+) the absolute value of the product all of the

15 Osborne quotes Kenneth Boulding (1936) as an example: “Now it is true that an equation of the nth degree has n roots of one sort or another, and that therefore the general equation for the definition of a rate of interest can also have n solutions, where n is the number of ‘years’ concerned. … Nevertheless, in the type of payments series with which we are most likely to be concerned, it is extremely probable that all but one of these roots will be either negative or imaginary, in which case they will have no economic significance.” 16 Readers requiring more information as to the meaning and significance of imaginary and complex numbers, and the basis of Osborne’s claim are referred to Osborne, 2005, 2014.


13

‘unorthodox’ roots multiplied by the ‘orthodox’ root. The orthodox root, is the one we use normally as the yield to maturity (see Osborne 2005, 2014). This result has an interesting interpretation.

Every nth degree TVM (time-value of money) equation solves for n interest-rates. We select one of these interest-rates. … the product of the (n-1) non-selected interest-rates solving the TVM equation enumerates the number of times the selected rate is applied to an invested dollar during amortization of the equation’s cash flows. This statement is true no matter which of the n interest-rates solving he TVM equation is selected. If the rate selected is the orthodox rate (d1), … as it is in financial practice, then the product of the unorthodox interest-rates enumerates the applications of the orthodox rate (d1) to an invested dollar during amortization. (Osborne 2014, pp. 20-21, italics added and notation changed to ours).

Consider a (positive or negative) shift in the discount rate from d to d’ such that (1+d) = (1+d’)(1+m) or m = (d – d’)(1+d’) = Δd/(1+d’). Given that there are n original values of dj for the roots, and one new d’. We can then construct n markups of all of the roots to d’ to get n values mj, mj = (dj – d’)(1+d’) = Δdj/(1+d’). Applying this to an analysis of duration, Osborne shows that duration can be written as

𝐷∗ = ∏|𝑚𝑡|

𝑛

𝑡=2

and that this is a precisely accurate measure of duration for any change in the (selected) discount rate to m1. (Osborne 2014, pp, 85-86). As written, however, the expression is unsuitable for practical use to the extent that it requires the calculation of all of the roots, some of which will be complex numbers. Osborne shows, however, that there is an exact equivalent expression that does not require this, namely,

Compare this with equation 2 reproduced below

(2) 𝐷 = ∑ 𝑡(𝑓𝑡𝐶𝐹𝑡

𝐶𝑉)

𝑛

𝑡=1

The difference between D and D* is that for the latter each of the CFt is marked up by the sum ∑ [1 + (1 + 𝑚1)]𝑗𝑡

𝑗=0 . Only real numbers are involved, so, while involving quite a lot of computations, this

formula is eminently calculable. Writing out the equation for n = 4 is illustrative.

𝐷∗ = 1

𝐶𝑉[(𝑓1(𝐶𝐹1[1]) + (𝑓2(𝐶𝐹2[1 + (1 + 𝑚1)]) + (𝑓3(𝐶𝐹3[1 + (1 + 𝑚1) + (1 + 𝑚1)2])

+ (𝑓4(𝐶𝐹4[1 + (1 + 𝑚1) + (1 + 𝑚1)2 + (1 + 𝑚1)3])]

(5) 𝐷∗ = ∑ 𝑡(

𝑓𝑡𝐶𝐹𝑡 ∑ [1 + (1 + 𝑚1)]𝑗𝑡𝑗=0

𝐶𝑉)

𝑛

𝑡=1


14

𝐷 = = 1

𝐶𝑉[(𝑓1(𝐶𝐹1[1]) + (𝑓2(𝐶𝐹2[2]) + (𝑓3(𝐶𝐹3[3]) + (𝑓4(𝐶𝐹4[4])]

According to Osborne D* will yield precisely accurate measures for the change in the bond price for any discrete change in the yield to maturity whereas all other measures are approximations. Once programed there should be no extra computational effort to worry about. This work is very recent and has not yet received the attention and review it deserves, so it is too early to tell if it will be accepted and integrated into common practice. All of the discussion about duration so far has applied mainly to bonds for which, barring default, the coupon payments are known in advance and the yield to maturity is calculated assuming that the one-period interest-rates are all equal, in other words, that the yield curve is flat. If this is not the case, there are more unknowns and the matter is more complicated. Too many unknowns? The most general form of the TVM equation, equation 1b above, reveals that at any point of time the market price of a financial asset “is the discounted value of the expected stream of future payments, where the discount rates for each payment are the one-period rates expected in each future period up to the date of payment. Given this price, the yield to maturity is computed as that single rate that equates the price and the value of the future cash payments” (Bierwag et. al. 1983, p.18). In well arbitraged markets the term-structure of interest-rates (the yield curve) reflects the pattern of expectations of traders regarding future short term rates (though, as we argued above, preference for liquidity, aversion to risk, which may be presumed to rise with term, also plays a part). Thus, for coupon bonds, the yield to maturity will equal the simple average of the one-period discount rates, or the holding-period yield, only if all the rates are equal – that is, only when the term-structure is flat. Where it is not, changes in discount rates are not uniquely related to changes in yields to maturity. Any given change in the pattern of discount rates is consistent with a variety of changes in the yield to maturity, and vice versa. Duration as measured above thus does not provide an accurate measure of the proportional change in bond prices for all changes in the pattern of relevant interest-rates, which may produce different yields to maturity. The response to this in the literature is to consider how knowledge of changes to the term-structure may be obtained and connected to changes in bond prices to provide better measures of duration. It is posited that the pattern of interest-rates is generated by an invariant stochastic process, so that “derivation of a correct duration measure requires knowledge of or assumptions about the actual stochastic process driving interest-rate changes (Bierwag, et. al., 1983, p.18, italics added). Under a variety of such assumptions different, more complicated, measures of duration have been developed (Bierwag, et. al., 1983, p.18, appendix, pp.34-35). The assumption of a stochastic process mechanically generating interest-rates may appear problematic to many. The pattern of interest-rates in financial markets at any point in time, is, in large part, a distillation of the subjective preferences and expectations of the many trading individuals in the market and not generated by any given and fixed stochastic process. Realization of the role played by the structure of interest-rates – as opposed to simply the level of interest-rates – adds a significant degree of uncertainty to the reliability of predictions based on the idea of duration. How important this is in practice depends on just how much additional uncertainty is added in practice, that is, on whether the use of simple and/or complex measures of duration still continue to provide measures that are useful to real-


15

world investors contemplating the various risks they face and how these risks can affect value of their particular investments. Thus, even for a fixed coupon bond, where the coupon-rate, face-value, and investment-period is known, the effects of changes in the pattern of interest-rates adds to the unknowns. In more general cases, for example where the future payments are not known, but must be estimated, these too must be added to the list of unknowns. investments in business projects are of this variety. In these cases, the CFt are not contractually fixed, but are dependent upon a variety of complex phenomena, including the productivity and effectiveness of the business organization in question and the future market for the goods and services being produced. Finally, as we discussed earlier, duration is crucially dependent on the time-horizon of the investor which will, in general, not be the same as the term to maturity of particular financial assets. Consideration of this has prompted the following ‘drastic’ judgments from Bierwag, et. al.1983.

A risk measure unique to all securities and all investors may not exist. A security that appears riskless to an investor with an investment horizon equal to the security’s duration would appear risky to investors with either longer or shorter horizons (23). It is unlikely that all or even most investors have single, well-defined investment horizons (23). [and] To the extent that investors have different planning horizons, this finding, casts doubt on the uniqueness of … any risk measure of a security (15).

We may thus ask, after considering the extent of what is not known, or not possible to know, what is left of the idea of measuring and reacting to the relationship between time and value in investments? In what follows it will be suggested that, regardless of our ability to ‘objectively’ measure this relationship, it cannot be denied that individual investors, at some level, must be conscious of it, of the importance of time in their investments, and will react differently depending on the extent of the ‘time-involved’. What matters is their perception of the value of their investment, given their particular time-horizons, preferences, expectations, etc. and how these change with circumstances, including changes in interest-rates.

6. Capital theory, discounting, time and reswitching Background In the traditional neoclassical economics framework, that is widely taken for granted by most finance and economics scholars, capital and labor are considered to be categories of productive resources that are combined in production to add value. The prices of the services of the various kinds of labor and capital-goods employed in production are presumed to be determined by the marginal additions of these services to the revenue of the productive firm – their marginal value products. There is a compelling logic of choice involved. If, in the judgement of the employer/entrepreneur, the additional value of the labor or capital-good’s service is not at least equal to the estimated addition to revenue, it will not be purchased – the factors will not be employed. In this way, individual earnings, of workers or owners of capital-goods are determined by their functions in production. It is a functional account of the distribution of earnings.


16

An historical, but on-going, challenge to the very concept of capital as a factor of production that earns a marginal product has been somewhat influential. Consideration of the relationship between capital-value and the discount-rate is a key factor in this challenge. The veracity of the functional distribution of earnings is denied in favor of an explanation based on the spending propensities of the various “social classes” (deriving from an aspect of the work of the classical economist David Ricardo). In support of their case, these neo-Ricardians mounted an attack on neoclassical production theory based on the identification of certain theoretical ‘paradoxes’. These paradoxes consist of cases in which it is alleged, for example, that a fall in the interest-rate (equated to the ‘rate of profit’ earned by ‘capital’ or the ‘price’ of the services of capital-goods) which, according to neoclassical production theory is expected to bring about an increase in the demand for capital in production, can lead to the exact opposite at some interest-rates. More specifically, a fall in the interest-rate may first lead to the adoption a more ‘capital-intensive’ productive technique, and then switch, paradoxically, to a less ‘capital-intensive’ technique, and then switch back again as the interest-rate continues to fall. Among many alternative techniques, characterized by their physical capital-labor ratios, there may occur switches, re-switches and reversals (moving between 3 or more techniques in paradoxical fashion). In short, there is no monotonic relationship between capital and its price, the rate of interest. These anomalies are taken to be devastating to the entire neoclassical edifice, based as it is on a quantity of capital earning a marginal product. The whole notion of a ‘production function’ is rendered by their account meaningless. This refers to the so-called ‘Cambridge capital controversy’ between MIT (Cambridge U.S.) and Cambridge University (U. K.). during the 1960’s and 1970’s.17 Though the neoclassicals (Cambridge U.S.) conceded the logic of the neo-Ricardian (Cambridge U.K.) attack, they doubted its relevance and have continued to use the neoclassical framework undeterred. The neo-Ricardians thus continue to attack, to this day, indignant at the intransigence of the neoclassicals in refusing not only to abandon the neoclassical framework but to embrace the neo-Ricardian alternative. It is not possible, nor relevant, to examine here the details of this episode or its on-going developments.18 Our interest is solely in the light that it sheds on notions of capital, and, especially on the relationship between capital and time. For that purpose it will be illustrative to analyze certain aspects of the so-called paradoxes identified. A simple illustrative example To do this we borrow the example used by Garrison (2006, pp, 190-196). This example is similar to the one used by Yeager (1976), which he in turn adopted from a seminal article by Samuelson (1966). The neo-Ricardians refer to a ‘technique’ of production as a method of producing a particular output of a given product with given amounts of labor input in specific periods. Any non-labor inputs are implicit in the analysis. The techniques are fixed, and the output they produce is fixed (Yeager refers to bottles of

17 For a recap of the debate see Cohen and Harcourt (2003) and the references therein. 18 It is our conviction that the assertions of the neo-Ricardians were decisively debunked by Yeager (1976) – who clarified the source of the so-called paradoxes and provided an alternative perspective for understanding the nature of capital and its earnings, in which no such paradoxes appear. In effect, Yeager shows that the entire case of the neo-Ricardians is based on a fundamental category mistake in regards to what capital is and what it earns. We shall make use of this in what follows. In a more recent contribution, Garrison (2006) returns to Yeager’s analysis in order to explain why the neo-Ricardians summarily dismissed Yeager’s argument without even considering it. We shall make use of this too.


17

Champagne), and no reference is made to either the prices of the inputs or the output which, presumably, are also fixed (unchanging). Two such techniques are given in table 1 below.

Table 1: Labor Requirement by Technique.

Labor required

Time Period Technique A Technique B

1 100

2 210

3 110.16

Total 210.16 210

The two techniques are distinguished mainly by the timing of their (original) labor inputs (units of labor-service), and only minimally by the amount of labor required. Technique B is identified as the more ‘capital-intensive’ because it requires less labor to produce the same output. ‘Capital-intensity’ is a purely physical matter.19 If we now consider the cost of financing each technique we get a paradoxical result. Imagine each unit of labor costs $1, then the cost of financing each technique at a 5% interest-rate is given in table 2.

Table 2: Cost by Technique (interest-rate = 5%)

Cost

time period Technique A Technique B

1 $100.00 $205.88

2 $105.00 $210.00

3 $220.41 $220.50

The general expression is in table 2A, where r is the rate of interest used to compound labor inputs. Note, although, for concreteness, we are using money values (dollars) to value the labor inputs by assuming each unit to be worth $1, this assumption is unnecessary. The analysis applies whatever metric is used to value the labor inputs as long as it is constant. The analysis would work even if we used elementary labor-units and count the interest-rate as the mechanism by which such labor is augmented. Metaphorically the interest-rate ‘grows’ the labor inputs in the production process.

Table 2A: Cost by Technique for interest-rate r

Cost

time period Technique A Technique B

1 $100.00

2 $100.00(1+r) $210.00

3 $100(1+r)2+110.16 $210(1+r)

19 The neo-Ricardians identify all ‘capital’ as intermediate goods, like machines, tools, or raw-materials. They are goods-in-process from the original labor that constructed them, to the emergence of the final consumer good. So all capital-goods (can be) and are reduced to dated-labor. In this way, we get a purely physical measure of ‘capital’, one that, by construction, does not vary with the interest-rate.


18

At 5% technique A is the cheaper to finance, hence the one that will be chosen. But, this is not true for all interest-rates as can be shown by repeating the process for various interest-rates according to the information in table 2A. Techniques A and B can be described by the expressions 100(1+r)2 + 110.16 and $210(1+r), respectively. Calculating their NPVs at various interest-rates yields table 3. The output produced by both techniques is identical and invariant and thus can be ignored in this analysis.

Table 3: NPV by technique

Relative Costs

Interest-rate NPV (A) NPV (B) NPV (A)/ NPV (B)

0% 210.16 210 1.0008

1% 212.17 212.1 1.0003

2% 214.2 214.2 1.00000

3% 216.25 216.3 0.99977

4% 218.32 218.4 0.99963

5% 220.41 220.5 0.99959

6% 222.52 222.6 0.99964

7% 224.65 224.7 0.99978

8% 226.8 226.8 1.00000

9% 228.97 228.9 1.00031

10% 231.16 231 1.00069

At interest-rates below 2% technique B is adopted. Between interest-rates of 2% and 8% technique A is adopted but a reswitch occurs at interest-rates higher than 8%, where, paradoxically, the more ‘capital-intensive’ technique B is chosen. See figure 1. This example reveals the essence of the neo-Ricardian case.

This bears a close relationship to our earlier discussion of multiple interest-rates. Another way to tell the story is to consider techniques A and B as aspects of a single decision, with the option not-chosen seen as

0.999

0.9992

0.9994

0.9996

0.9998

1

1.0002

1.0004

1.0006

1.0008

1.001

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

Figure 1: NPV(A)/ NPV(B) - costs by discount-rate


19

the opportunity cost of the decision. In this way we can combine the two configurations to yield the equation, for the rate of return (r=i) of the project, 100(1+r)2 - 210(1+r) + 110.16-= 0 This is a second-order polynomial which has two roots, (1+r) = 1.02 and (1+r) = 1.08, no paradox there. Evaluation In this stark form, and given the number of restrictive assumptions, including the lack of substitutability within techniques, and the invariance of all relative-prices, it may seem as if the paradoxes identified are merely theoretical curiosities without much practical significance. This may be true. However, the discussion does raise interesting issues concerning categories. The neo-Ricardian paradoxes occur because of their insistence that ‘capital’ to be a useful category as a factor of production, must be physically measurable, on a par with physical labor. And what our discussion shows is that a purely physical measure of capital is impossible. Indeed, this has been well-known at least since pointed out by Wicksell ([1911] 1934). Any attempt to ‘count’ the ‘amount’ of capital units encounters the interest-rate, which is a price that introduces value into the exercise and contaminates its pure physical-ness. Therefore, according to the neo-Ricardians, capital should not be considered as a factor of production exhibiting scarcity, having a price and a demand curve. And this is so much the worse for all neoclassical constructs that depend on it, most notably the aggregate production function that is used to explain the earnings of ‘labor’ and ‘capital’. The fallacious nature of this criticism lies in its understanding of capital. The discussion above clearly suggests that capital, by its very nature, is a value-construct. The collection of things that we, perhaps unfortunately, call capital-goods do not constitute the economy’s capital. Rather, capital refers to the value of any such collection (in terms of its potential to produce useful things). There is, moreover, no essential difference between the capital-value of labor employed and capital-goods employed. The fact that capital-goods can conceptually be traced back all the way to the input of ‘pure’ labor (and nature) is entirely irrelevant for investment decisions in a market economy, which are, as are all such decisions, necessarily forward-looking. An investor contemplating the financing of any given project, for example as described techniques A and B considered above, cares only about the relative cost to him of each (or alternatively, the NPV of their profitability, their relative capital-values). Capital in this context refers to the accumulated value to be expected from the investment, or at any moment to the accumulated value up to that moment. The is no reason to expect ‘capital-intensity’ to be an invariant property of any technique. In terms of the amount of pure labor to non-labor inputs the ‘capital-intensity’ of a technique will fluctuate with the interest-rate as its capital-value changes. There is no paradox. Furthermore, and even more important, the neo-Ricardians identify the price of capital as the rate of interest which they regard as synonymous with the rate of profit. But neither is the correct. The interest-rate is, indeed, the price of capital as we understand it, it is the cost of borrowing ‘capital’ or any valuable resource. It is the price of credit and is determined as we have discussed, by the time-preferences of borrowers and lenders and the production possibilities available. (The neo-Ricardians have no discussion of what determines interest-rates). The interest-rate is not the price paid for the services of capital-goods, and it is not the rate of profit. The price of the services of capital-goods is a rental-rate on capital-goods. It is well-understood. For example, a firm renting a copy-machine, pays a monthly fee its services. If it owns the copy machine, sound accounting dictates that it must charge itself the rental-rate for its services – which may be the basis of a depreciation fund. It is dimensionally equivalent to labor, conceived as


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human-capital. Labor services cannot be alienated from its owner (in the absence of slavery), so they must be rented. The rental rate on labor (human-capital) is what we call wages. And, profits are the residual value left over after interest and all factor-costs (wages and capital-goods’ rents) are paid. Profits are the reward for being right in an uncertain world. Consider the example above in table 1. If, instead of labor-inputs, we imagine these to be the inputs of mechanical robots (not so fanciful in today’s economy) that are rented by the firm. Then the rental paid for the services of any robot is its “wage”. By the reckoning of the neo-Ricardians, both techniques have 100% capital-intensity, or, more accurately, the capital-intensity cannot be figured out until we know how much pure labor was used – or, since theirs is a strictly static equilibrium exercise – needs to be used to construct the robots. From the perspective of real-world investment decision-making, however, this is indeed irrelevant. The functional distribution of earnings remains intact. At any moment in time there exists a set a technical possibilities for the production of useful things. We can call these production functions. The inputs into these production functions are the physically homogeneous productive resources, of whatever kind, that are available. Because resources are very heterogeneous in nature, capital-goods more than labor, there will be very many of categories of inputs. But for each kind, there will be a price and a demand-curve implicit from the production function. There is no reswitching in terms of physical inputs. To be sure, in a general-equilibrium setting, factor-prices may appear to act perversely, because of complicated complementary relationships in production, but this is hardly news. There is an elementary distinction between demand-curve shifts and movements along them. Is capital (as opposed to a capital-good) a factor of production? Yes, if you consider it necessary for production to occur.20 Physical inputs have to be financed, in the broadest sense of that term, because they have to be deployed over time, and can be so deployed in various configurations. That is the function of capital, a valuable function, for which there is a scarcity-price, the rate of interest. But it is not physically measurable. As explained above, it is the result of the interplay between the time-value of money and the money-value of time so nicely captured in the concept of duration.

7. Conclusion The colloquial understanding of capital as financial capital is after-all close to the mark, at least closer than thinking of capital as a collection of physical things. The latter is perhaps responsible for more confusion and controversy than clarity. A consideration of the role of time in production and investment decisions, as explored in this paper, brings one to the realization that capital is the result of a process of evaluation. Capital is the result of ‘capital-accounting’. It is the ability to use capital accounting that is in large part responsible for the phenomenal success of capitalism. Productive physical resources, whether natural or produced, are either never capital in this sense or else are all capital. They are never capital in their pure physical nature. But they are all different types of capital, human or non-human, when considered as a stock of potential value over time. 21

20 Yeager (1976) suggests that it makes sense to think of capital as a stock of ‘waiting’ whose price is the interest-rate. As Yeager points out, this line of thought goes back at least to the Swedish economist Gustav Cassel (1903). 21 Productive resources are all capital in that they ‘know how” to do certain useful things, they ‘embody useful knowledge’. See Lewin and Baetjer, 2011.


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