*Gerald Bierwag is the Ryder Systems Chair in Business Administration, Department of Finance, College of Business, Florida International University, Miami Florida, 33199. Iraj Fooladi holds the Douglas C. Mackay Chair in Finance, School of Business Administration, Dalhousie University, 6152 Coburg Road, Halifax, Nova Scotia, Canada, B3H 3J5, telephone: (902) 494-1843, Fax: (902) 494-1107, e-mail: Iraj.Fooladi@dal.ca. Gordon Roberts is CIBC Professor of Financial Services, Schulich School of Business, York University, 4700 Keele Street, North York, Ontario, Canada, M3J 1P3, telephone (416)736-2100 (ext.77953), Fax: (416) 736-5687, e-mail:groberts@ssb.yorku.ca. The Social Sciences and Humanities Research Council of Canada provided partial financial support for this research. Iraj Fooladi also received support from the Mackay fund at Dalhousie.

1. INTRODUCTION

As one of the great success stories of academic finance, duration analysis enjoys widespread applications by practitioners in many different areas. Duration analysis is most well known for its use in bond portfolio management where billions in fixed income funds are managed using duration as a measure of interest rate sensitivity. Risk managers in banks, insurance companies, and other financial institutions have utilized duration analysis to control or measure their institution’s exposure to interest rate risk. Many investment banks and other financial institutions have used duration measurements in value –at- risk analyses, as in Risk Metrics, to estimate the possible losses that may result from adverse interest movements. Traders in fixed income securities use duration notions in their everyday evaluations of bond price movements. The practical uses of duration techniques has, indeed, permeated the financial services industry

In this paper we further consider duration analysis as a risk management tool for government organizations. Many government entities have a considerable amount of physical assets under their control and the values of these assets, as well as their financial assets, may be sensitive to interest rate fluctuations. We wish to highlight and to indicate some of the potential limitations of duration analysis as a risk management tool for such organizations.

We begin by introducing duration and briefly explaining how traders and portfolio managers use this measure in speculative and hedging strategies. We continue with a brief discussion of the application of this tool in risk management for financial institutions and explain some of the difficulties that arise from it. Next, we focus on the application of duration strategies for governments emphasizing the special features as well as the limitations of duration strategies in risk management. Using examples from a real case, we discuss the legitimate questions that arise from applying these strategies to hedge non-financial assets of government, such as roads and hospitals. We conclude by providing possible answers to some of these questions, and suggesting directions for possible future research.

2. DURATION APPLICATIONS: A REVIEW

Invented by Macaulay [1938] as an alternative to term to maturity, duration represents a more precise measure of the maturity profile of the promised cash flows of a bond or other fixed income security. Macaulay’s duration is defined as a weighted average of the times at which cash flows from an asset are obtained. Duration is calculated as

D = = (1)

where C(t) is the cash flow received at time t, W(t) = C(t)/P₀(1+ r)^t is the weight attached to the time t cash flow, r is the discount rate for the cash flows, and P₀ is the current price of the bond.

There are several different versions of duration but Macaulay’s is the one that is the simplest and most commonly used in practice. The Macaulay formula , equation (1) simplifies duration analysis by using the bond’s yield to maturity to calculate all the present values.

Duration also represents the elasticity of an asset price with respect to the discount factor (1+ r)^-1. First developed by Hicks [1939], this property has applications for the management of active bond portfolio strategies and for evaluating value at risk---a measure of how much the value of a portfolio or financial position will change for a specified change in interest rates. As in Hopewell and Kaufman [1973], this elasticity can also be written as:

Duration = D = - [ ] = - x , (2)

and it can be rearranged as

= - D[D r / (1 + r)] P (3)

This means that if interest rates decrease (increase) slightly, the price change is proportional to duration. The link between bond duration and price volatility has important practical applications in managing risky positions. By knowing the duration of the assets, risk managers can compute the possible price changes for the assets by using equation (3). If the risk exposure, as measured by equation (3), falls within the institution’s guidelines for permissible maximal changes in interest rates, the risk manager need not undertake any action. If, however, the risk is greater than permitted, the prudent manager would examine strategies to hedge the interest rate risk faced by the institution. As indicated in equation (3), the risk can be reduced by decreasing the dollar exposure (the value of the assets) or by lowering the duration of the assets.

Duration hedging or immunization draws on the following key property of the relationships underlying the development of equations (1) to (3): By maintaining portfolio duration equal to the amount of time remaining in a planning horizon, the investment manager can lock in (immunize) the initially promised return on a portfolio. This proposition can be demonstrated as follows. If P is the initial value of a bond or a portfolio that has a yield to maturity of r per year, and if a planning horizon of q years is given, then the promised value of the portfolio at the end of q years is V(r) = (1+r)^qP. Taking the derivative of V(r) with respect to r, using equation (2), then

V’(r) = q(1+r)^q-1P + (1+r)^q{-DP/(1+r)} = (q-D)(1+r)^q-1P (4)

so that when q = D, V’(r) = 0, and no change can take place in the promised return, V(r), as long as the duration of the portfolio is maintained at q, the length of the planning horizon. Immunization has been called a passive strategy because it requires no forecast of future rates. It is a strategy that balances re-investment and capital risks forming a hedge against the effects of interest rate changes. The strategy is particularly attractive to some investors when interest rates are viewed as volatile and having an uncertain trend.

Redington [1952] and Samuelson [1945] developed the earliest versions of the immunization result. Fisher and Weil [1971] point out that the flat term structure (in which r, the discount rate, is the same for all maturities) as used in the Macaulay duration is unrealistic. They assume a more general (non-flat) term structure of interest rates and prove that a bond portfolio is immune to interest rate shifts, if its duration is maintained equal to the investor’s remaining planning period, but where duration is now defined with r replaced by r_t, a discount rate that is defined for the tth period. The immunization result is thus a very general proposition.

The immunization strategy provides the risk manager with a powerful tool for managing interest rate risk. Following this strategy, the manager can be assured of maintaining at least a pre-specified value at the end of the investment horizon.The balance sheet or financial position of the risk manager can be stipulated as A(r_A) – L(r_L) = K, where K is the net worth position of the manager. Treating K as the net asset which is a function of the interest rates r_A and r_L, one can derive a measure of the sensitivity of K to changes in the interest rates. This gives rise to the familiar expression for DGAP:

[D_A - KD_L $ ] = DGAP (5)

where K = L/A, $ = ) r_L / ) r_A, A denotes assets. L denotes liabilities, r_A and r_L denote the rates of return on the assets and liabilities respectively, and D_A and D_L denote the durations respectively of the assets and liabilities. It can be shown, as in Fooladi and Roberts [1999] who assume certain conditions regarding convexities of asset and liability values, that this duration gap is related to changes in the value of equity (or net worth K) by the formula

) E - (DGAP)A . (6)

This formula shows how a change in interest rates impacts the market value of a financial institution’s equity. The DGAP plays the same role as that of duration in the corresponding formula for fixed income securities in equation (3). When the duration gap is zero, the equity value of the financial institution will not be affected by interest rate shocks. When the DGAP is is positive (negative), the equity value will fall (increase) when rates increase.

In a highly innovative study, the government of New Zealand in 1993-94 explored how duration analysis might help guide the restructuring of its liabilities. The purpose of the study was to develop a possible optimal procedure for measuring and controlling interest rate risk (and other risks) involved in the management of the government’s assets and liabilities including its physical assets. The study involved finding appropriate methodologies for measuring durations for various classes of the assets under the government control.1 Thus the focus of the study was on quantifying durations.

In the early 1990s, New Zealand became one of the first countries to engage an accounting firm to state its balance sheet according to Generally Accepted Accounting Principles. A major privatization program in which the government sold some of its assets to the private sector provided the government with cash that could be used to reduce its outstanding debt. This raised the question of how the remaining debt could be structured so as to hedge the government’s balance sheet against changes in interest rates and exchange rates.

Although the use of duration principles for hedging or controlling the value of financial assets involves well documented procedures, the use of the same principles to control the volatility of the value of physical assets is a complete novelty that the government was willing to explore. Implementing any "on balance sheet macrohedging" policy for the government’s assets and liabilities would require obtaining duration estimates for such assets as roads, national parks, and equity investments in state-owned enterprises like schools and hospitals. As with a security, the duration of a government asset like a road system depends on the flows of cash and non-cash benefits, the discount rate, and the asset life. Unlike a bond or mortgage, a road system brings social benefits (regional economic development) that are difficult to quantify. To avoid the problem of putting exact dollar values on the flows, it was found that the duration estimates could be based on the growth rates of the benefit flows instead, as we show in this paper.

A main concern in the exploratory study was the derivation of real durations for the government’s physical assets. Real durations are measures of the sensitivity of the corresponding real value of the assets to changes in real interest rates. Using these durations, the government may be able to devise hedges that prevent its net worth from fluctuating with real interest rates. However, hedging could also include developing nominal durations. These durations are measures of the sensitivity of the corresponding value of assets to changes in nominal interest rates induced by changes in expected inflation rates. In the present paper, we focus on real durations for two reasons. First, inflation was under control in New Zealand at the time of the exercise. Second, for many of the government’s assets, it was reasonable to assume that the flow of benefits was indexed to inflation. In this case, nominal duration goes to zero.

Whether or not the government should establish a policy to control the volatility of all of its assets and liabilities is a question of some concern because such policies may or may not involve social benefits. Still, without doubt, if a government has a policy of entering markets to buy and sell assets whether physical or not and of restructuring its liabilities as a consequence or for other reasons there may be social benefits that derive from macrohedging. If a macrohedging policy is to be based on duration analyses, it then becomes important to find ways to estimate the durations of various classes of assets under the government’s control.1 To estimate the durations, one may need (1) to project the growth rates of various service benefits, (2) to estimate the remaining economic life of various assets, and (3) to make appropriate estimates of the relevant discount rates.

In order to apply duration analysis to government controlled assets, a sound procedure for estimating the real durations of various relevant accounts must be devised. Because many assets have cash flows and other benefits that are not routinely measured or estimated, many of the durations must be estimated with a paucity of data. For many government assets, obtaining data comparable to what is required to calculate bond durations would be extremely difficult. Often, however, the actual cash flows or other benefits derived from an asset over time are not needed for the computation of duration. For example, if an asset has constant cash flows, one does not need to know their actual values in each period to derive the durations; it is enough to know that the flows are constant over time. Similarly, if the flows grow at a certain rate per period, the only information needed is the growth rate, the discount rate, and the maturity. The real duration can then be derived as a function of the growth rate.

The durations derived on the basis of the characteristics of the asset flows may be viewed as providing a long-term or secular basis for real duration. Changes in the real durations because of month to month or year to year fluctuations in national or regional income may not be captured under the various growth rate assumptions. For many of the government’s assets, the flows represent benefits and costs that are not wholly realized as actual dollar cash flows, but represent the value of benefits and costs in a social sense. Many of the fluctuations in asset value from month to month are self-correcting. Heding of these fluctuations is not necessary. A longer term or secular hedge is not only more appropriate but easier to manage.

If cash flows, C_t, t = 1,2,….., grow at a constant rate so that C_t = C₀(1+g)^t , the value of the real asset can be denoted as:

P = + B P(1+r)^-N, (7)

where B P is the terminal value of the asset which is assumed to be proportional to the initial

value of P. The Macaulay duration can then

be expressed as

D = =

given that r g. For an asset with infinite life, provided r > g, equation (6) reduces to

If the asset has a finite life with no terminal value, then the first two terms of (8) constitute the the duration.

The flows in (8) and (9) are assumed to be independent of interest rate movements. These durations do not work well for securities or assets for which this is not the case. Various methods, however, have been developed to adapt to these cases in which the flows depend on the interest rates. Examples are: "Cash Flow Adjusted Durations", "the Ratio Method", "Options Adjusted Spreads", and "Historical or Empirical Durations."

Our methodology for estimating the duration of a major real government asset is well illustrated in the case of a highway system. A real duration measure for highway systems must take indirect costs and benefits into account. The economic development and enhanced economic improvements in a region that was previously unserved by quality roads is an example of indirect benefits. The environmental damage resulting from a road's construction are examples of indirect costs. External diseconomies and economies of this kind would not normally be counted as part of the service flows emanating a network of highways. Such benefits and costs are not likely to accrue to a private owner of such a system of roads either. The benefits are social dividends and the costs are social costs.

In order to derive real duration measures, we do not need to include indirect costs and benefits from externalities in the service flows. Reasonable assumptions regarding the relation between the growth rate of these indirect costs and benefits and the growth rate of the service flows enable us to capture their effects in the real duration measures of the highway system. For example, if the benefits were constant over time, or if they grew at the same rate as national income, then the method that uses growth rate patterns instead of the actual service flows permits the calculation of a real duration measure that can be viewed as taking external economies and diseconomies into account.

Real Discount Rate

Computing a duration measure requires an appropriate discount rate. Using the Fisher equation, we can derive a real discount rate, r, from a nominal rate, R. One simply adjusts the nominal yearly rate, R, by the rate of inflation. That is, r = [(1+R)/(1+h)] –1 , Fisher’s adjustment equation. Historical estimates of the inflation rate can be made from the Consumer Price Index (CPI) or the GDP deflator. For small countries with lots of imports, the CPI is likely a better measure of inflation as it captures the total basket of goods and services consumed by the population. In contrast, the GDP deflator only reflects inflation on goods and services produced within the country. For publicly financed highways, it is reasonable to assume a low risk premium, because funding for highways comes from general obligation debt instruments. Highway revenues are not linked to servicing of the supporting debt and, therefore, highway financing differs only marginally from general government debt. One benchmark for the expected real rate may come from government bonds which are indexed by inflation rates.

Real Growth Rate for Road Services

One can project the real growth rate for road services on the basis of expectations and and these estimates can then be compared with the growth rates contained in historical data. These projections can employ information on historical and projected growth in Gross Domestic Product (GDP) as a proxy for growth in road services. However, although historical estimates are useful, it is important to consider forward-looking figures that tend to capture the impact of future policy shifts not reflected in historical data.

Asset Life for Roads

The economic life of an asset should be defined as the length of time for which the stream of services obtained from that asset is economically viable. For a highway system, the end of the economic life could be the point at which repairs become no longer economical. We envisage repairs being made every year (to offset depreciation) at increasing expense as the system ages. Eventually, it becomes uneconomical to continue repairs and, on the assumed maturity date, a significant new investment is needed to upgrade the system.

In order to estimate the asset life for a typical road, it is necessary to recognize that roads physically consist of three parts each with a different economic life-- the land, formation (the subgrade and landscaping under the road) and the structure (the subbase/basecourse and the surface seal).

It may be reasonable to assume that the land and formation have infinite lives and do not depreciate. Highway depreciation is limited to the structure and is a combination of surface and subbase/basecourse deterioration. Resealing corrects surface deterioration and a more comprehensive process called rehabilitation corrects deterioration of the subbase and subcourse. The frequency with which resealing or rehabilitation is undertaken depends on the type and location of the highways.

For the sake of providing examples, in this paper we consider a typical section of highway which may require reseals every 9 years and rehabilitation every 25 years. It follows that a steady state, optimal maintenance policy would result in the average seal being half way through its design life with an age of 4.5 years. Similarly, the average half-life for the subbase/basecourse would be 12.5 years.

Table 1 summarizes our discussion of the lives of the road components. The Table also assumes that subbase/basecourse and seal are equal parts of structure. Additionally, Table 1 shows that land and formation combined represent 58.5 percent of the total replacement cost of roads with structure making up the remaining 41.5 percent. These percentages come from the breakdown in replacement costs from a real case.

Since the land and formation have an infinite life, it is not possible to calculate a weighted average life for roads. Instead we calculate the durations separately for land/formation and structure and then take a weighted average of the durations in order to compute an over-all duration. This is consistent with established practice in forming portfolios of interest sensitive securities.

Duration Estimates for Highways

An example of duration estimates for highways is presented in Table 2. The first four columns show the calculations for the duration of the structure using equation (8) based on:

The resulting durations (DUR Structure) for the subbase/subcourse range from 4.71 to 4.90 years. The next column, labelled DUR Land g=0, present the durations for the land and formation. In general, the formula for duration when N is infinite reduces to Equation (9). Assuming that land values will likely grow at the rate of inflation, the real growth rate in the value of land is estimated to be zero. Column 5 in Table 2 shows different estimates for the duration of land and formation (DUR Land g=0) for various values of r.

The weighted overall duration for roads is given at the right in Table 2. The weights are from Table 1: 41.5 percent for structure and 58.5 percent for land and formation. To illustrate the calculation of this column we use the shaded line in Table 2.

DUR = .415 x DUR Structure + .585 x DUR Land

= .415 x 4.81 + .585 x 21.0 = 14.28 years.

Governments typically own or control a vast amount of property in the form of land and buildings that are utilized as hospitals, schools, and government administration buildings of all sorts. The development of a measure of real duration for real property in this paper is similar to the way it was done for highways. The duration of a class of buildings or other property depends on the appropriate risk-adjusted real discount rate, the economic life of the physical asset, and on the growth rate of the stream of services generated by the asset.

Given that buildings depreciate over finite lives and that land does not, our approach is similar to that taken in the analysis of roads in the previous section. The duration of a particular class of property, hospitals for example, is computed as a value-weighted average of the durations of the land and buildings comprising the asset. As with roads, the weights are based on replacement costs.

Real Discount Rate for Real Property

For roads it was assumed that the range of real discount rates was 4 to 6 percent. Given the specific risks in property markets, the real discount rate for real property is likely to be higher. If much of the land and buildings could be managed as in the private sector or, indeed, sold to the private sector, it would seem reasonable to add a market risk premium to the government general purpose borrowing rate in order to reflect the rate in the private sector that would be appropriate for valuating the property. Assuming a market risk premium of 2-4 percent (based on our real case), we may employ real discount rates from 6 to 10 percent.

Real Growth Rate for the Stream of Services from Real Property

The growth rate in the stream of services provided by real property can be estimated to be the growth rate of the activities for which the property is used. For example, the Ministry of Education may project an average annual growth rate ranging from .376% (the assumed annual growth rate for population) to 1.25% (based on an aggregate policy for the expansion of education programs) for the school system. A similar approach applies to hospitals, justice and other services made available on government real property. For justice and police buildings, it is reasonable to assume that the stream of services will grow at the same rate as the population. This annual population growth is assumed to be .376 percent in this paper.However, as in the case of highways, future growth rates of the stream of services for hospitals and many other services are likely to reflect changes in pricing policies or government expansion policies. Accordingly, we consider three possible real growth rates for use in our paper: zero percent---the pessimistic case of government restraint, 0.376 percent--the middle ground representing growth at the same rate as the population, and 1.33 percent ---the highest value reflecting an increased utilization of services.

Asset Lives for Real Property

The third parameter needed to estimate real property durations is asset life. The value of real property is the sum of buildings which depreciate over a finite life and land which is non-depreciable. Our approach is to estimate the lives of the depreciating assets. Later we will calculate property duration as the weighted average of the durations of buildings and land.

Assuming that buildings are depreciated over 50 years, and that they are built, maintained and replaced continuously at a constant rate, the average age would be around 25 years. Given other complications, asset lives may vary from this norm.

One guide to the likely variation in the lives of buildings is accumulated depreciation shown on the balance sheet. For example, if buildings are only a few years old, and so have remaining lives greater than half their design lives, accumulated depreciation will be low relative to the initial total cost. Following this approach, we calculate the remaining life as:

Life = 50 - [(Accumulated Depreciation) / Cost] x 50

To illustrate, suppose the summary of a country’s Police Land and Buildings at the end of a particular year is:

Cost $50,261,390

Accumulated Depreciation $ 3,060,843

Net Book Value $47,200,547

Based on our equation, the remaining average life of these buildings is about 47 years:

Life = 50 - [3061/50,261] x 50 = 47 years.

Alternatively, for some categories of real property we may obtain breakdowns of the remaining lives for individual properties by regions, allowing us to calculate average life and thus the average remaining life. We assume that the average remaining life for hospital buildings is 37 years. For schools, we assume the economic life to be 25 years. Although this average asset life is arbitrary, its use gives us some idea of the duration variability as a function of building life.

Durations for Real Property

We illustrate our methodology by calculating the real durations for police buildings, schools and hospitals.

Police Real Property

Summarizing the parameters for police buildings, we have a range of real risk adjusted discount rates of 6 to 10 percent and a growth rate in the stream of services of 0.376 percent. The buildings have a life of 47 years and the land has an infinite life. To obtain estimates for economic lives, we assume that police property consists of about 33 percent land and of 67 percent buildings in terms of total property values at replacement cost.

Table 3 calculates the duration of police real property in three steps. First, we compute a range of durations for police buildings shown in the column labelled DUR Bldg. Next, using the formula for infinite lived assets, we calculate the duration of land for zero real growth (DUR Land g=0). The third step weights the durations of buildings and land to find Weighted Overall Duration. (One added column in this and following tables, labelled "DUR Eqn.8" will be discussed below.)

The approach for schools is similar to that for police property. We assume a range of growth rates from 0.376 percent to 1.25 percent, discount rates run from 6 to 10 percent, an asset life of 25 years and the weightings at replacement value of12 percent for land and 88 percent for buildings.

The duration output takes the same form as for police buildings and appears in Table 4. The results suggest that, for schools, duration is somewhat lower than for police property mainly due to their shorter economic life, and because of the greater concentration of the real property in buildings. The variability of the real duration for schools stems largely from the variability of the discount rates and the growth rate. Assuming a 1 percent growth rate in building services, 0 percent growth rate in land services, and a 25 year life of school buildings, the real school duration varies from 9.12 years to 11.4 years as the discount rate ranges from 6 percent to 10 percent. A mean real duration for schools is 10.25 years.

Growth rates in services range from zero to 1.33 percent over an asset life of 37 years for hospital buildings. The discount rate remains in the range of 6 to 10 percent. The resulting duration calculations are in Table 5. The durations for hospital real property are closer to those for police property and highways than to schools mainly because of longer asset lives for hospital buildings. For schools the overall duration is calculated by the weightings of 12% for land and 88 percent for buildings and this can account for some of the differences.

Concluding Comments on Duration for Real Property

This section shows how our stream of services model applies to real duration for real property. Applications have included police property, schools and hospitals. The methodology developed in this section for the determination of the real duration of land and buildings is clearly feasible. The ranges of duration estimates for various properties could undoubtedly be narrowed down with more precise information on growth rates and especially on the viable economic lives of economic activities undertaken jointly with land and buildings. We have considered a very broad range of discount rates, 6-10 percent. A more precise estimation of appropriate discount rates would also diminish considerably the range within which the real duration estimates have fallen.

Conceptual Issues

There are two main models that can be used for quantifying the real durations of the government's equity investments in state operated enterprises (SOEs): the asset/liability model and the dividend-growth model. It is assumed in both models that the government owns 100% of the equity. In an asset/liability model, the real durations of the SOE's assets and liabilities are estimated using appropriate service growth rates, asset lives, and real discount rates. The duration of net worth or equity is then computed as the difference in real asset and liability durations adjusted for the capital structure (the debt/equity ratio). In the dividend-growth model, the expected dividends to the government constitute the expected cash flows generated by the government's equity investment in the SOE. These expected dividends represent expected residual receipts after accounting for net asset inflows, net liability outflows, and the reinvestment of retained earnings. As an example of how these techniques may be utilized, in this section we derive estimates for the real durations of the government's equity investments in an electric company.

Duration for an Electric Company

Using the Asset-Liability Model: The duration of equity, using the asset-liability model, is very similar to the DGAP model of equation (4). Equity as a function of the rates of return on assets and liabilities can be expressed as

E(r_E) = A(r_A) - D(r_D) (10)

where E is equity, A is the level of assets, D is the level of the liabilities or debt, r_E is the real rate of return on equity, r_A is the real rate of return on assets, and where r_D is the real rate of return on debt. Of course, the real rate of return on equity, r_E, is dependent on the real rate of return on assets, debt, as well as the levels of assets and debt. One may regard r_E as the real cost of capital. Taking the derivative of (10) with respect to r_E, assuming all rates change by the same amount, the duration of equity can be written as

D_E = D_A()() - D_D()() (11)

where D_E is the duration of equity, D_A is the duration of assets, and D_D is the duration of the liabilities.

For electricity assets we assume a range of 4 to 7 percent for real discount rates. This relatively wide range most certainly includes the governments’ real borrowing rates as well as the weighted average cost of capital for an electric company. We assume real growth rates of 1.5 to 2% in the use of electricity.

The asset life calculations for electricity assets (Table 6) are based on correspondence and conversations with those in charge of a specific power company and their auditors. The lives represent "typical assets". The figures for remaining average lives for electricity assets differ dramatically among asset classes. The overall asset duration will depend critically on the weights attached to each type of asset. The weights may be obtained from the data presented in the latest annual report of power companies. However, usually the information is more accurate for the lives than for the weights. Although the simple average life was calculated to be 32.5 years, we assume a weighted average of 30 years for our calculations.

Table 6 here Table 6 shows duration calculations for the electricity assets using a weighted asset life of 30 years, growth rates of 1.5 to 2 percent, real discount rates ranging from 4 to 7 percent, and a zero terminal value. The results show that the duration of electricity assets ranges from around 12 to 14 years. It is fairly insensitive to small shifts in the discount rate. Our empirical estimates of asset duration for electricity assets are very close to these numbers.

We present an example to show how equation (9) can be used. Suppose that D_A = 13.36 years---as obtained from Table 6 with r_A = .05, g = .02, and with an asset life of 30 years. If we then suppose that the debt equity ratio, D/E , is equal to one, a risk adjusted rate of return on equity is r_E = .07, and r_D = .055, then the duration of equity is D_E = 21.4 years. The equity rate of .07 is not an arbitrary rate, though it is a reasonable value; it is derived below using the CAPM model to account for risk.

Using the Dividend-Growth Model: Equity duration measures the interest rate sensitivity of flows to shareholders. The measure we use is based on the well-known dividend valuation model in which P = D₁/(r-g) where P is the equity value, g is the perpetual growth rate, r is the discount rate, and D₁ is the following year’s expected dividend. As noted earlier in equation (9), the duration measure for a security with a perpetual series of growing cash flows can be stated as:

D = (1+r) / (r-g). (12)

This duration for assets having infinite lives is very sensitive to small changes in the growth rate of dividends, ceteris paribus.

Equity duration considers only dividends as flows to the government . These flows are essentially residual cash flows that are derived by deducting the liability cash flows from the asset cash flows. The resulting flows to equity may consequently be more interest rate sensitive than the assets and may thus have a duration that is larger than the duration for the assets.

The interest rate, r, utilized in equation (12) should be a risk-adjusted discount rate. This rate can be obtained by use of the Capital Asset Pricing Model (CAPM). Based on several studies on equity market and electric utilities, we assume that the equity beta for a typical electric company is about 0.5 and that the long-run measure of the market risk premium is around 5 percent given real borrowing rates of 4.5 percent. This produces a discount rate of 7%.

R_j = 4.5% + 0.50 [ 5.0% ] = 7.0%,

Further useful information on the return on equity comes from discussions with officials of power companies who mostly believe the firm's cost of equity capital is around 12 percent. Since this is a nominal rate, an inflation adjustment of around 2 percent is in order producing an estimate of 10 percent for the real return on equity. We may also use the accounting return on equity (ROE) as a check on our measure of the real discount rate. Drawing on all three approaches, our sensitivity analysis varies the real discount rate from 7 to10 percent.

Earlier analysis of asset duration for power companies resulted in projecting the annual growth in real electricity output to be in the range of 1.5 to 2 percent. Here, we consider whether dividends will also grow at that real rate. In general, two factors inhibit dividend growth for a utility: failure to plowback sufficient earnings into the company and heavy capital requirements for financing assets. We use two measures of financially feasible growth and find that the projected real growth rate in dividends falls between them.

The simplest measure of feasible financial growth is given by

g = f x ROE,

where the parameter, f, represents the plowback ratio and ROE represents the real return on equity. Based on this formula, assuming a 20% plowback ratio (consistent with our real case), and considering the 7-10 percent range of discount rates, we obtain .25 x 7% = 1.75% to .25 x 10% = 2.5% .

Although this is a useful beginning, calculating growth as the product of the reinvestment and plowback rates ignores the contribution of debt financing to growth. The sustainable growth formula (as presented below) improves on our estimate by recognizing the capital structure.

g = ROE x f / [1 - ROE x f] (13)

Applying this formula results in a sustainable growth rate ranging from 1.78 to 2.56 percent. In this case, the formula makes a modest upward revision in the range of growth rates. However, the sustainable growth rate is the maximum possible rate given the plowback ratio and the return on equity. In this paper, we calculate equity duration using a range of growth rate from 1.75 to 2 percent.

Combining the estimated range for real growth and the range of risk adjusted real discount rates, and applying equation (13) gives an estimated range of 13.3 to 21.4 years for the equity duration as shown in Table 7.

We can use either an asset/liability model or the dividend growth model to quantify the real durations of the government's equity investments in SOEs. Given the quality of available data, one may argue that the dividend-growth model provides greater precision in estimating real duration for SOE's than does the asset/liability approach, because the dividend-growth model requires fewer inputs and thus involves fewer assumptions. However, as the owner of an SOE, the government may wish to monitor the structure of the SOE's balance sheet. By using asset/liability methods, financial officers of the government can observe how closely the SOE is matching the durations of its assets and liabilities.

Methodology

The final step in macrohedging government’s balance sheet is to arrive at a narrow range for an estimate of the real duration number that represents the total assets on the balance sheet. Our approach is to choose a representative duration for each asset category and to calculate a weighted average across categories, adjusted for interest rate relatives. This adjustment is essential unless all assets in the category have the same rate of return. The duration formula for asset category, P, is defined as follows.

(14)

The aggregate duration could be determined by reapplying Equation (12) over all categories as follows:

(15)

Here T_A is the total value of the assets, r_A is the rate of return (WACC—the weighted average cost of capital) on total assets, and A_P, r_P and D_P are as defined above.

Inputs

Because detailed information is not available for every asset in the government’s balance sheet, in each category of assets, representative subcategories may be selected to calculate the duration for the category. The range for the aggregate portfolio duration may be obtained from Equation (13). This range is shown in Table 8 along with individual durations for various asset categories on the government's balance sheet. In the calculation of the aggregate duration in Table 8, we considered only long-term (Non-Current) assets because, in principle, current assets should be financed by current liabilities and by and large are not subject to long-term planning.

To arrive at an aggregate number, we must first calculate the percentage weight of each category in relation to total long-term assets. These weights may be calculated from the figures in the Balance Sheet. In most cases, these figures are based on replacement costs, and may be very different from market value weights. These weights are reported in Column 2 of Table 8. We ignore the intangible assets category to simplify the analysis. Columns 3 and 4 show the duration (or duration range) and the discount rate for each asset category is in column 5. As we explained above, these durations are calculated by applying Equation (13), but, for illustration purposes, we could only apply this equation to one or two representative assets in each category. However, usually, representative assets form a significant portion of each category's value. For example, state highways, land, and buildings often form about 80% of the value of physical assets. In the following paragraphs we explain the details of the calculations for each part.

Investments

This asset category is usually contained in both the current and non-current portions of the Statement of Financial Position. We may obtain a list of such investments and their characteristics (such as duration and yield to maturity). Since we are only interested in long-term assets, we may pick only those assets which had more than one year to maturity. Column 3 of Table 8 contains an example of the duration for this asset category. This duration is the present-value weighted average duration of the investments on our selected list. Similarly, the discount rate for this asset category reported in Column 5 is obtained by calculating the present-value weighted-average yield to maturity of the assets on our selected list.

Receivables

Receivables and advances are usually short term. Here we have ignored this category in the calculation of our aggregate duration, realizing that this would result in slightly overestimating our aggregate number since it implies that they have the same duration as the typical asset in the long-term portion of the statement.

State-Owned Enterprises (SOE) and Government Entities

In this category we have only considered durations for electric companies and the government-owned forestry corporation (ignored in this paper because duration calculations involved the same process we developed for equity duration of electric companies.) We assume this asset category forms 33% of total non-current assets. Table 9 shows the detailed computations for this category.

As previously mentioned, we assume that about 80 percent of the value of physical assets relates to state highways, land, and buildings. Therefore, these three items together are a good representation of the physical asset category.

Earlier, we calculated duration for the highway system assuming a 4-6 percent real discount rate along with various growth rates and asset lives. In order to narrow the range of durations for the purpose of calculating an aggregate duration for non-current assets, we focus here on the range of 4.5-5.5 percent for the real discount rate. Assuming a 9 year life and a real growth rate of 2 percent, this results in a range of 13.2 -15.6 for the duration of highways as shown in Table 10.

Buildings form 43.59 percent of the value of the physical asset category. Calculating duration for buildings also requires estimates for asset life, the appropriate discount rate, and the growth rate for the stream of services. Above we calculated building durations for different uses. Here, we present only one category for all buildings. We assume that the useful life for a typical government building is considered to be 50 years. This implies a 25 year remaining average life if we assume an even age distribution among all buildings. However, as shown above, for some buildings such as police property and hospitals, the life estimates are much longer. Therefore, our estimate of asset life for the building category is adjusted from 25 to 30 years. Our earlier work also shows that the growth rate of the stream of services for most real estate properties is quite low. We assume a range of zero to one percent. For the rate of return we simply chose 8%, the midpoint of the 6-10 percent range which we used for our calculations in the real estate section. These estimates resulted in a range of 10.1 to 10.7 years for the duration of buildings as shown in Table 10.

Land is the smallest of the three categories in Table 10 with a weight of 11.54 percent. To compute the duration for land, we set the real discount rate at 8% following the work for buildings. Land has a perpetual life and produces a stream of services that exhibits zero real growth. These parameters result in a duration of 13.5 years for land as shown in Table 10.

Combining the estimates for buildings, land and highways, we calculate the duration for the physical asset category as a weighted average (adjusted for risk) of these three asset categories. Table 10 shows a range of real physical asset durations from 11.86 to 13.21 years. Combining the durations for individual categories showing that Aggregate Duration for non-current assets ranges from 11.71 to 12.51 years, as indicated in Table 8.

Having determined a set of durations for the assets and liabilities under the government’s control, how should the government proceed to utilize this information in an appropriate manner? Should it undertake hedging strategies as though it were a private firm? Although the government, acting in the public interest, is not always attempting to maximize its profits or the value of its assets, is it appropriate for the government to hedge the value of its interest-rate sensitive assets? Given the extent of modern information and the availability of hedging strategies in the private sector, can the government undertake strategies comparable to those used in the private sector? Should one regard the government’s control of interest rate risk as simply another aspect of its general control of risks and should it approach the management of this risk in the same way that large multi-product private enterprises approach it? In this section we explore some of the issues and implications of macro-hedging by government organizations.

Issues of Measurement Value

As noted earlier, to compute the duration of a group of assets or liabilities in the same risk class, it is necessary to know or estimate the proportion of the total market value represented by a particular asset or liability. Except for new investments, the market values of the various components are not commonly available from accounting statements where historical cost bases with simple arbitrary depreciation rules are in effect. Market value accounting is not a standard practice in modern accounting. As a consequence, the market value proportions for any particular asset or class of assets must be estimated. The estimation process may vary considerably. In some cases, one may utilize readily available costs of replacement and in other cases, one may be able to derive estimates by use of comparable information on similar assets or liabilities in the private sector. In many cases, the meaning of "market value" itself is dubious. The method of computing the market value of a highway system, for example, is by no means an obvious undertaking. In much of this paper, especially in the sections on real estate and the highway system, we substituted replacement costs or we used authoritative estimates of the proportions. Duration measurements undertaken for securities having observable market values are very simple but they become more difficult in the circumstances envisioned in this paper.

Many banks and other depository institutions in the U.S. and Canada have developed their own concepts of GAPs, and they are not necessarily DGAPs. To undertake a similar exercise for the interest returns and costs on government assets and liabilities need not be ruled out, although most certainly the traditional duration estimates can best be developed for cases involving actual cash flows rather than non-cash benefits. Although the procedures outlined in this paper may not provide accurate substitutes for a market-based duration analysis, the processes may be feasible and they can be designed as a supplementary aspect of the government’s internal cash flow analyses. Non-zero GAPs for the government treasury would not have exactly the same meaning as for a depository institution. The government treasury, unlike a depository institution, is not exclusively in the business of borrowing and lending so as to make a profit on the spread. Yet, in the sense of the economics of benefit/cost analysis, one may view the government treasury as an investing intermediary entrusted with the capacity to borrow on behalf of the public.

Given the real duration gaps as opposed to GAPs used by depository institutions, one can attempt to construct the cash flows and maturity structures of the assets and liabilities so as to achieve zero DGAPs. One may also attempt an extremization of the DGAPs (making them as close to zero as possible) subject to various constraints on the assets and liabilities. Moreover, it is also possible to utilize off-balance sheet activities (swaps, options, and futures) so as to affect the values of the DGAPs.

Hedging What?

The government’s assets and liabilities can be roughly categorized as either marketable or non-marketable or as promising cash flow benefits or non-cash flow benefits. Such a scheme would place the assets and liabilities into the following four cell arrangement:

Cash Flows Non-Cash Flows

Marketable 1 2

Non-Marketable 3 4

Most financial securities would be classified into cell number 1. Most such securities have a market, however thin, in which one may obtain information on market values. Highway systems or networks would undoubtedly fit into cell number 4. The benefits accrue to the many commercial and non-commercial users of the highway system. By and large such assets are non-marketable because of the difficulty and costs of conversion to an asset marketable to the private sector. Many such assets are part of the infrastructure and make the operation of an efficient economy possible. To some extent, we may wish to classify parts of a highway system into cell number 3, given that one can connect various user-specific tax receipts with the asset. Many assets, like parks and recreation areas, may generate cash flows, however modest, but are non-marketable or cannot be sold into the private sector without incurring considerable costs of conversion. Such assets are in cell number 3. Finally, the government has many assets that are readily marketable -- government office buildings and military equipment --- but which generate little or no cash flows. These assets belong to cell number 2.

Hedging positions in the private sector are most easily undertaken with respect to the financial securities in cell number 1. For example, a short futures position implies that an increase in interest rates would result in cash flowing in from the futures market to offset the decline in the value of the financial securities. Other types of hedges would work similarly. Because the securities are readily marketable, the changes in value resulting from interest rate changes can be accurately calibrated and corresponding futures positions for the hedging activities can be fairly precisely specified. Hedging activities, here, are common and well understood.

Some cash flows from non-marketable assets or liabilities may be sensitive to interest rate changes. Insofar as these flows are predictable, they can be hedged even though the assets or liabilities corresponding to them are nonmarketable. The hedges can consist of futures positions or interest rate put options. If consistent with other objectives, interest rate swaps that transform these flows into fixed cash flows are also possible.

Interest rate changes may affect the value of non-marketable assets and liabilities, whether in cell 3 or cell 4. To achieve marketability, there may be considerable and perhaps prohibitive conversion costs, but these assets or liabilities may have considerable value after conversion and be sensitive to interest rate changes. It is difficult to imagine, however, that the functional efficiencies of non-marketable physical assets would be impaired by changes in interest rates, per se. Unless the government is planning to sell non-marketable assets to the private sector, the only criterion of consideration should involve the functional efficiency of the assets in place.

Most likely, interest rate changes would affect the costs of maintaining these physical assets. For example, higher interest rates may increase the costs of acquiring bulldozers, road graders, and other equipment used to maintain or rehabilitate such assets periodically. Here, however, the interest rate changes are affecting cash inflows and outflows associated with such assets and their cash flows can be hedged separately from any hedge on the asset itself. The value of such assets may change with interest rates because the costs of reproduction or the costs of new assets of a similar nature may change. Although the value of the asset, after conversion costs have been incurred, may decrease as rates rise, this increase in rates is of no consequence unless the government is anticipating conversion and sale in the imminent future. The values of such sales and of new investments can be and are hedged in the private sector. The Treasury could do the same.

New Investment Applications

Rather than applying duration techniques to existing assets, one could apply them only to new investments. In this way, as part of the usual analysis of costs and benefits attending the decision to undertake an investment, one can estimate the duration. At this stage in the decision process, considerable effort can be put into the duration estimation and considerable accuracy can be expected. These data can then be maintained indefinitely into the future and be used for determining the optimal debt characteristics that fund the investments where applicable.

Investment Priorities and the Real Rate of Interest

Benefit-cost analyses often precede public investments. If the present value of the benefits, often derived by discounting estimated service flows, exceeds the costs of the investment, then the public investment is regarded as worthwhile. At any time, one might visualize a list of possible public investments being considered for activation. If these investments are ranked by the amount that the present value of benefits exceeds their costs in percentage terms for a given real interest rate that is applicable to public investments, it may be possible to determine a marginal investment project such that investments ranked better than this investment have positive percentage benefits over costs and investments ranked worse have negative percentage benefits over cost. When real interest rates increase, the decline in present value (in percentage terms) is greater for the long duration projects than for the short-duration projects. As a consequence, the increase in real rates can drive many long-duration projects below the marginal investment. In this way, increased real rates can reduce the duration on average of public investment projects that remain economically viable. Moreover, many projects may be redesigned so as to reduce their durations, and many long-duration projects may be postponed. Given that many assets comprising infrastructure are long-duration projects, periods of high real interest rates may be periods in which there is considerable deterioration in the infrastructure. On the other hand, as the urgency for repairs and rehabilitation is enhanced so also are the conceivable benefits. Rising real rates may thus bring about a mixture of increasing urgencies and delayed investments.

When an economy is undergoing inflation, a confusion may easily develop because real rates are more difficult to discern. As a consequence, many public investments may be inappropriately undertaken if the real rate is estimated to be less than it really is and many investment projects may be unnecessarily postponed if the real rate is estimated to be greater than it really is.

This section discussed a number of related implementation issues. A common theme is how a government might adapt duration hedging techniques employed by financial institutions in the private sector. Although many of the techniques discussed may appropriately be carried over from private sector applications, it is important to note several unique features in potential public sector applications. Unlike financial institutions for which hedging is centralized in the treasury function, governments may wish to decentralize hedging. Decentralization may be particularly efficient if the impact of hedging activities in one department does not significantly affect the impact of hedging in other departments. Further, hedging by the government must take cognizance of the distinction between cash and non-cash benefits. Third, the overwhelming predominance of real assets on the government's balance sheet means that particular attention must be paid to evolving real interest rate trends.

In this paper we have argued that the techniques of macrohedging can be extended from financial institutions to non-financial corporations and to governments. However, we have also noted that the transfer of the duration methocology to the public sector is not a straight forward exercise. It consists of techniques that must be integrated into traditional cost/benefit analyses and into a careful analysys of the impact of changing real interest rates. In summary, the following two points are very a propos.

1. Before meaningful hedging strategies can be designed, macro-hedging techniques must be modified and adapted to the specific nature of government organizations and their decision-making processes.

2. The policy implications of prospective governmental macro-hedging strategies should be carefully reviewed and analyzed before implementations are attempted. In some situations for some assets and liabilities, macro-hedging may be appropriate and work very well, but in other situations, macrohedging may be counter-productive, expensive, or not worth while.

Some may argue that athe variation in equity value in the government’s balance sheet may not be as strongly linked to the degree of duration matching as it is in the case of financial institutions. Then why would oneduration estimates for the assets and property held by the government? This argument arises because, for most government assets, the values are more strongly linked to variables other than interest rate movements. Nevertheless, although this means that macrohedging is less important for governments than it is for financial institutions, it does not mean that it is completely irrelevant for the government. If preserving equity value is a concern for the taxpayers, attempts should be made to immunize the equity value against interest rate movements. Even though this may not be as important as protecting the government’s equity against other factors in efficient use of government assets, it would not be completely irrelevant.

Many of the concerns of private firms are not also concerns or problems for government organizations. Bankruptcy is a case in point. For governments, bankruptcy concerns may be irrelevant for at least two reasons. First, taxpayers may be more concerned with the flow of services (cash or non-cash) from the government’s assets than in the government’s equity position. This is because the government’s equity value cannot be as easily observed because their assets are non-tradable or they will never be sold. Nevertheless, even if the taxpayers concern is with variability of the flow of services generated by these assets rather than their values, immunization is not meaningless. So long as these flows of services are affected by interest rate movements, hedging strategies could serve to reduce their variability. However, in this case, it may make more sense, to design hedging strategies that are similar to gap management (as opposed to duration gap management) strategies conducted by commercial banks. Second, unlike private firms, governments have power to tax. When cash flows are adversely affected by economic conditions such as interest rate movements, governments may be able to offset part of the loss by imposing taxes. However, the government’s power to tax is not unlimited. Changing tax rates requires following certain political process which may take time. Further, there is no guarantee that increasing tax rates increases government’s income to the extent that the reduced cash flow is compensated. If tax rates become too high, it may even reduce tax revenues. Moreover, hedging strategies are proactive and aim to prevent adverse effects on values (or flow of services). On the contrary, imposition of taxes to mitigate adverse economics conditions is a reaction to events.

Applying the principles of macro-hedging to large scale multi-product enterprises such as represented by a government organization may require extraordinary innovation. It is clearly a challenge to modern methods of risk management. A government organization that is contemplating macro-hedging on a large scale might do well to study the risk management methods already in use by large enterprises in the private sector. For example, the Royal Bank of Canada, one of the largest banks in North America, has attempted to centralize its control of risk management in its Treasury which operates as a central controlling unit as well as a clearinghouse for all risk-reduction and risk-transferring activities. The notion that one should simply consider the balance sheet of an enterprise and then undertake an immunization strategy is much over-simiplified. For many government activities it may simply be very difficult to identify a balance sheet. Government debt often consists of general debt obligations; this debt, unlike municipal revenue bonds in the U.S., cannot be closely identified with specific projects or enterprises. Developing balance sheets for diverse government activities may be very arbitrary. Even large private firms have found it difficult to develop separate balance sheets for its various related on-going enterprises. Various models of risk control for large scale enterprises, are available, however, and government organizations can consider molding some of these approaches into an effective risk management process for itself.

Given the difficulty of accounting accurately for the diverse assets of government organizations, as well as the difficulty in establishing a risk management authority comparable to that of large scale enterprises in the private sector, it may be much wiser (and cheaper) to focus exclusively on the maturity structure of a government’s debt and to undertake immunizing activities with respect to it only. To this end, the availability of various derivative securities provides government treasuries with a much less expensive alternative to macrohedging than may be required for the "on balance-sheet" techniques noted above or to the establishment of a central risk management authority. Also, the government can pair individual assets with individual liabilities without being concerned about macro-hedging. It may substitute a variety of micro-hedging activities for an over-all macro strategy. In other words, there is much the government can do to insulate its assets and liabilities as well as its cash flows and non-cash flow benefits from the ravages of interest rate fluctuations as well as other risks.

The main purpose of this paper is to demonstrate the applicability of duration as a risk management tool for government organizations. Drawing on a real case, we present methodologies for quantifying (1) the durations of real assets on a government’s balance sheet, and (2) the durations of the financial assets represented by shares in State Owned Enterprises (SOEs). In the area of real physical assets on the balance sheet we focused on the highway system and on real estate owned by the government. The methodology for measuring durations of SOEs focused primarily on an electrical utility .

Our main conclusion is that it is feasible to derive excellent practical measures of the real durations of physical assets on a government’s balance sheet. This paper has demonstrated that these measures are far from academic curiosities--they can be estimated with a fair degree of accuracy in practice.

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