# Estimating the global price of USD inflation risk

Executive summary for the mildly-interested reader: Although the average holding return for 1-year maturity US Treasuries during 2014-2018 was a *negative* .50% inside the US, the estimated return to such USD inflation risk in the global capital market was a *positive* .70% … a 1.2% difference. The basic lesson is that the price of a capital asset (and its underlying risks) within a single country is generally not the price that prevails in the global capital market.

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*Valuation professionals are fundamentally in the business of (i) forecasting expected cash flows from capital assets, (ii) determining the risks factors to which the cash flows are sensitive, and then (iii) estimating the stable market price of the risk factors. Perhaps the most fundamental risk factor influencing the value of capital assets is inflation risk; i.e., the risk that the future value of money–and therefore future cash flows–will differ from expectation. This suggests that, especially in emerging economies, a fundamental problem facing valuation professionals is estimating the stable market price of inflation risk.*

*It turns out that although valuation professionals understand the importance of inflation risk, they often avoid the problem by paradoxically defining “risk-free rates” as observable rates on governmental debt securities that carry only inflation risk (… what?!). The problem is, of course, that–with the possible exception of currency–there are no risk-free assets (see Aswath Damodaran’s excellent article on this titled “**Into the Abyss: What If Nothing is Risk Free**“).* *Professor Damodaran’s article presents a number of methods for estimating quasi-risk-free rates that exclude sovereign debt default risk premiums but, unfortunately, still include inflation risk premiums. From one perspective this is not a problem, however, because of the existence of **inflation-indexed debt securities** issued by sovereign governments, which allow us to observe expected risk-free yields to maturity (YTM) for a particular country / currency. Such securities are, however, not available for all countries / currencies.*

*From a valuation theory and methods perspective, this article is the continuation of a sequence of articles on biological asset valuation and risk pricing, where it was shown that (i) CAPM-based methods are not useful when estimates of the market prices of specific risks are required, and (ii) a method based on arbitrage pricing theory (APT) has existed for over 40 years that allows us to directly estimate the international market prices of specific risks. (As I mentioned in previous articles, I wrote a book on these topics for technically-oriented valuation professionals.) But more specifically, this article uses the exact same asset pricing theory and related no-arbitrage method to develop direct estimates of the global market price of US Dollar inflation risk over a 1 year horizon. Please read on …*

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### 1. Estimated USD inflation risk sensitivities

US Dollar inflation risk. We usually think of inflation and inflation risk as being measured relative to changes in price indexes published by governmental agencies. But there are several problems with using such indexes to measure inflation and inflation risk: (i) it is not clear which consumer price indexes, producer price indexes, or GDP deflator are the most accurate indicators of monetary value; (ii) it is not clear whether methods used to create the price indexes induce short-term errors in inflation risk measurements; and (iii) it is not clear whether such price indexes capture changes in the international value of a currency (see, e.g., discussions of Eurodollar time deposits). For these reasons, I will define US Dollar inflation risk over a 1 year horizon in terms of the risk that the aggregate principal and interest payments to be received on US Treasury securities with a 1-year maturity will have *actual values* that differ from their *expected values*.

It is common to think actual values of government debt principal and interest payments always equal their expected values (at least in the no-default case) because the payments are contractually specified; and, so, we intuitively think of such government debt securities as risk-free. But this intuition is inadequate: We’re not interested in *nominal values* of the payments; we’re interested in their *real economic values*; i.e., the quantity of goods and services the money can purchase. Anyone who has lived through mass- and hyper-inflation conditions in emerging economies has this distinction firmly in mind on an almost daily basis (cf. Brazil had inflation / currency devaluation of over 30,000% in 1990). Those of us who did not have this learning experience can develop the proper intuition by simply recognizing that a debt security is a promise to deliver a *nominal quantity of money* in the future, which potentially will have a *different real value* depending on a variety of economic conditions.

In this connection, it is reasonable to assume changes in the current price of US Treasury securities depend primarily on changes in the expected real value of the US Dollar at their future maturity date. With the common assumption that capital market traders and investors have rational expectations on average across time and economic conditions, it follows that–on average–changes in US Treasury market prices reflect differences in the real value of future principal and interest payments; justifying the use of changes in 1-year maturity US Treasury security prices (UST1) as a proxy for US Dollar inflation risk over a 1 year horizon.

Risk sensitivity estimates. Using the same APT-based approach used to estimate the market price of raw sugar risk in my previous article, the sensitivities of 1-year maturity German government bonds (DB1) and iShares MSCI All Country World Index ETF (ACWI) with respect to 1-year horizon US Dollar inflation risk were estimated using a least squares method:

The inflation risk factor (denoted *X* with UST1 subscript) defined as the monthly percentage change in the implied price of 1-year US Treasury securities (UST1) for the 60 months ended March 2019 was constructed from historical US Department of the Treasury yield curve data. Similarly, monthly market returns (*R*) over the same period were derived from publicly-available market data on DB1 and ACWI.

DB1 and ACWI were selected to price 1-year horizon US Dollar inflation risk because (i) German government bonds are highly-liquid, low risk debt securities that are acquired by international investors to achieve the same basic investing objectives as are US Treasury securities, and (ii) the ACWI exchange traded fund is designed to track the entire global equity portfolio (excluding frontier markets). So, both DB1 and ACWI were selected because they represent liquid securities traded in the global capital market that are, therefore, likely to price US Dollar inflation risk.

### 2. Optimal risk-minimizing asset portfolio

The fundamental assumption of APT. It is reasonable to assume that investors prefer increased return for increased risk and, therefore, are willing to accept decreased return for accepting decreased risk. It follows from this assumption that investors structure their capital asset portfolios–and therefore price capital assets–in a way that minimizes the aggregate risk in the portfolio *in expectation*.

Optimal risk-minimizing portfolio. If one is willing to study the following system of two simultaneous equations, it can be seen that it is possible to structure a portfolio comprised of DB1 and ACWI in a way that makes the aggregate sensitivity of the portfolio to 1-year horizon US Dollar inflation risk zero in expectation:

It can be shown that, given the estimated sensitivities of DB1 and ACWI to UST1 risk, the risk-minimizing optimal portfolio asset weights are a long position of 1.017 for DB1 and a short position of -.017 for ACWI. Because these two portfolio positions can be held feasibly (e.g., there is an active market for ACWI put and call options), the risk-minimizing portfolio is feasible. So, in combination with the fundamental APT assumption, it follows from this observation that it is likely that aggregate sensitivity to 1-year horizon US Dollar inflation risk in the market portfolio of DB1 and ACWI is zero in expectation. It turns out that if this fundamental APT assumption holds, then the global market price of US Dollar inflation risk can be extracted from the observable market returns of DB1 and ACWI, as shown next.

### 3. No-arbitrage risk pricing

If over the 60 month sample period investors essentially constructed a 1-year USD inflation risk-minimizing portfolio as shown above, and in so doing set DB1 and ACWI relative asset prices in a way consistent with the optimal portfolio weights, then it follows from APT that both the (historical) expected monthly return on the optimal portfolio and the global market price of 1-year USD inflation risk can be extracted from the historical returns on DB1 and ACWI:

I say *global market price of risk* because it is the price (risk premium) set simultaneously in the German government bond market and the global equity market; i.e., the market price of 1-year USD inflation risk does not exist in a single observable market such as the NYSE, NASDAQ, etc. but rather exists implicitly in the global capital asset portfolio.

Substituting the estimated risk sensitivities and estimated historical asset return expectations for DB1 and ACWI into the above expression, the resulting optimal portfolio risk price and 1-year USD inflation risk price are …

More accurately stated, these are estimates of monthly expected returns to optimal DB1-AWCI portfolio risk independent of 1-year USD inflation risk and to an asset with a sensitivity of 1 to such 1-year USD inflation risk:

At this point we are essentially done, having estimated the price of 1-year USD inflation risk from global capital market data. But I will finish the example by constructing the risk-adjusted expected market rate of return for an asset with a sensitivity of 1 to 1-year USD inflation risk next.

### 4. Risk-adjusted expected market rate of return

So, what would be the appropriate risk-adjusted expected market rate of return–i.e., risk-adjusted discount rate–be for an asset with a sensitivity of 1 to 1-year USD inflation risk? Similar to the article on biological asset risk pricing, I will use the observable 12 month yield-to-maturity on US Treasury inflation-indexed securities at 29 March 2018 (termed “TIPS”) of 0.525% as the risk-free component of the discount rate and transform the monthly 1-year USD inflation risk premium to an annualized rate, resulting in …

Contrary to the intuition provided by CAPM-based methods, it’s important to recognize that even risk premiums which our intuition would suggest are easy to observe–such as that for USD inflation risk–are in fact not. To see this consider the (non-)relationship between the no-arbitrage estimate of the 1-year USD inflation risk premium and a range of historical mean monthly returns from holding US Treasury securities with a 1 year maturity (amean = *arithmetic mean*; gmean = *geometric mean*; and median are different measurement of the historical average monthly returns):

To be clear, the historical averages of monthly returns from holding US Treasury securities with a 1-year maturity for the 60 month sample period ended March 2019 are *negative*, while the estimated no-arbitrage method monthly risk premium is *positive*.

Why so? Most succinctly, because in modern global capital markets, investment portfolios are constructed through pricing, hedging, and diversification such that specific risk prices are implicit in global asset portfolios; they are not necessarily explicitly priced within specific assets. Actually, it can be shown that …

**Capital markets do not actually price assets per se; ****most fundamentally, capital markets price risk.**

To develop intuition regarding why USD inflation risk pricing is generally not feasibly estimated within the US capital markets–and why estimating such risk pricing requires data from the global capital market–it is only necessary to pose (and answer) the question, How can USD inflation risk be hedged in US capital markets that essentially price risk in terms of US Dollars? Although this is not a good formal answer, the intuitive answer is that such hedging is most easily or perhaps best accomplished in capital global capital markets where returns are denominated in other currencies. And, so, if USD inflation risk is best hedged in the global capital market, then the price of the risk is best estimated from global capital market data.

### 5. A CAPM-based estimate and conclusions

As in my previous article, it is again worthwhile to stop and consider how an asset with a sensitivity of 1 to 1-year USD inflation risk would be priced under the CAPM method. Using the same risk-free rate as above, after estimating the sensitivity of 1-year maturity US Treasury security returns to AWCI returns (where AWCI is an arguably close proxy for the global capital asset market portfolio) and estimating the 60 month historical average return on AWCI of .06355, the CAPM-based estimate would be …

So, is this CAPM-based estimate a better estimate of a risk-adjusted expected market rate of return than the APT-based estimate? The answer ultimately depends on whether the CAPM- or the APT-based estimation method better explains historical returns and predicts future returns from holding 1-year maturity US Treasury securities. In my view, however, the simpler answer hinges on which method represents a *direct estimate based on minimum of assumptions* about investors knowledge, beliefs, and market behavior.

This is a somewhat complex topic that I address in my book, but the simple reality is that CAPM is based on theoretical assumptions that few people believe actually correspond to reality and it can be shown that the empirical explanations and predictions of CAPM do no hold in real-world capital market data. In contrast, it has been shown that the APT-based method can be used to construct optimal, risk-minimizing capital asset portfolios and–at the same time–provide no-arbitrage risk pricing estimates that demonstrably represent the opportunity cost of specific risks. This is *not* true of CAPM-based estimates … .

One of the more stunning comments I’ve heard from finance and valuation professionals in my career is that the APT-method of capital asset risk pricing, shown above, is “too complex to use in practice.” Hmmm, really? I’ve directly observed what is taught in undergraduate and graduate programs in finance since about 1995, and APT is almost always included in the curriculum. It seems that if we are truly finance professionals, we should understand both of the leading single period asset pricing theories: CAPM and APT; not just one of them. And if we actually understand both theories, surely the method presented in Sections 1-4 above is *definitely not* “too complex to use in practice,” is it?

MMc

São Paulo

**Caveats.** Please note: (i) views presented above are my own and do not reflect those of others; (ii) like anyone, I’m not infallible and am responsible for any errors; (iii) I greatly appreciate being informed of any significant errors in facts, logic, or inferences and am happy to give credit to anyone doing so; (iv) the above article is subject to revision and correction; and, (v) the article cannot be construed as investment or financial advice and is intended merely for educational purposes. MMc