Incorporating Currency in Portfolio Choice

Introduction

The standard one-period portfolio choice problem has the following elements.² There are M (i = 1, …, M) economic agents who prefer strictly more to less — the agents have strictly increasing concave utility functions U_i defined over random wealth W̃ⁱ. There are N (j = 1, …, N) risky securities with random returns r̃_j and one risk-free security with deterministic return r_f. Agent i starts with wealth W₀ⁱ, invests a_jⁱ dollars in each of the risky securities and the remainder in the risk-free security, and maximizes E[U_i(W̃ⁱ)], where

Under specific assumptions, such as multivariate normally distributed returns or quadratic utility functions, the well-known Capital Asset Pricing Model (CAPM) relation can be derived.

In the standard problem, the strategy of making no investment at all does not exist.³ Setting every a_jⁱ to zero does not produce the null trading strategy: it produces a risk-free position — a loan to a borrower at the rate r_f. Currency is omitted from the model. The standard defense of the omission is dominance: the risk-free security pays r_f > 0 and dominates cash which pays zero. A conclusion that follows is the participation theorem: when at least one risky security has a strictly positive risk premium, that is, E(r̃_j − r_f) > 0, every economic agent will make some risky-security investment.⁴

Currency and the Null Trading Strategy

The existence of a risk-free security is not the issue: the zero-beta CAPM replaces the risk-free security with a zero-beta portfolio.⁵ The issue is the absence of cash: currency — the best medium of exchange and the gold standard of liquidity — is missing. Currency allows for the trading strategy of making no investment at all, and a model that excludes the possibility of not taking a position in any security whatsoever — risky or risk-free — is flawed. The possibility of holding wealth in cash should be allowed.

Within the standard problem, the initial wealth of agent i, W₀ⁱ, is assumed to be held in securities. In effect, agent i merely rebalances his portfolio of securities. Agent i cannot start with currency and invest some of his wealth in securities, and agent i cannot convert any portion of his portfolio of securities into currency. Since securities are meant to transfer consumption intertemporally, the standard problem assumes that consumption occurs without the exchange of currency for goods and services.

In the case of wealth, preferring more to less is reasonable. The difficulty is not monotonicity; rather, it is the reference point. When economic agents are allowed to keep their money in their mattresses, cash earning 0% represents the trading strategy of making no investment at all. The dominance defense of the omission of the null trading strategy is merely an empirical claim — a claim that has failed. It is incorrect to assume that cash is dominated by risk-free securities. In nominal terms, cash returns zero, and there have been extended periods during which nominally risk-free securities returned less than zero: between 2014 and 2021, this was the case across the euro area, Denmark, Japan, Sweden, and Switzerland. In real terms, cash returns −r_I/(1 + r_I), where r_I is the inflation rate; therefore, whenever the nominal return of a risk-free security is negative, the real return of cash exceeds the real return of the risk-free security. A model that contains a strictly dominated asset is defective in a stronger sense than one that admits arbitrage: dominance is the more severe violation.

Suppose that currency carries a storage cost of δ ≥ 0 so that its net nominal return is −δ. Then, a nominally risk-free security is held only if r_f ≥ −δ. In what follows, without loss of generality, δ = 0 is assumed.

Maximizing the Expected Utility of Quasi-NPV

The NPV rule states the following: an investment should be accepted if it has positive NPV and rejected if it has negative NPV. However, it is well established that negative NPV projects can be accepted⁶ and that positive NPV projects can be rejected⁷.

The NPV concept is related to the definition of wealth: wealth is created by making positive-NPV investments, and wealth is destroyed by making negative-NPV investments. When the inflation rate is positive, the null trading strategy is a negative-NPV strategy: $1 tomorrow is worth less than $1 today.

Let there be L (j = 1, …, L) securities with random nominal returns r̃_j, and let there be K (i = 1, …, K) economic agents with strictly increasing, strictly concave, twice differentiable utility functions U_i defined over the quasi-NPV random variables NPṼⁱ. Every payoff — of every security and of the currency — is discounted by the same deflator: the deterministic inflation rate, r_I. Let

and

Agent i invests a_jⁱ dollars in each security and holds the remainder in currency. Currency holdings are nonnegative because private agents cannot issue currency. Borrowing occurs by short selling securities. Defining

The choice of deflator is key in the model. No assumption is made that any security is priced correctly: expected quasi-NPVs may be positive, zero, or negative. Defining W̃ⁱ = W₀ⁱ + NPṼⁱ,

so maximizing the expected utility of quasi-NPV is equivalent to maximizing the expected utility of real terminal wealth.

The optimization problem for agent i is to maximize E[U_i(NPṼⁱ)] over the set of a_jⁱ. By strict concavity, the first-order conditions that are both necessary and sufficient:

Define the risk premium of security j against currency as

The risk premium is positive when the expected nominal return is positive. Total abstention — with all wealth in currency — is optimal only when π_j ≤ 0 for all j.⁸ The risk-free security is not special: the case r̃_j = r_f represents a nominally risk-free Treasury bill (T-bill) with a premium vis-à-vis currency of r_f/(1 + r_I). When r_f > 0, currency is dominated, the nonnegativity constraint binds at zero, and the standard model of the Introduction re-emerges. When r_f < 0, the T-bill is dominated and unheld: there is a flight to cash.

Notably, inflation increases the cost of total abstention. The null trading strategy produces

with certainty and is strictly decreasing in inflation. The certain cost of total abstention rises with inflation: standard specifications — mean–variance, constant absolute risk aversion, and constant relative risk aversion — imply rising demand for securities. This is the Mundell–Tobin effect, manifesting here as a partial statement about the cost of total abstention with the distribution of nominal returns held fixed.⁹ In general equilibrium, nominal returns respond to expected inflation: for equities, the estimated relation is negative.¹⁰

A CAPM-Type Relation

Assume, in addition, that returns are multivariate normally distributed. From the first-order conditions and Cov(X, Y) = E(XY) − E(X) E(Y),

and, applying Stein’s Lemma,

Dividing by E[U_i″(NPṼⁱ)], defining

and summing over i,

Defining

and

it follows that

Let W₀ = ΣKi=1W₀ⁱ, and define the gross real growth of the market as

where w_j are the aggregate portfolio weights and w_I is the currency weight. Then,

so

Multiplying by w_j and summing over securities and currency, with currency contributing neither premium nor covariance,

Dividing the last two equations, provided that some risky security is in positive net supply so that Var(G̃_M) > 0, produces the CAPM-type relation:

In real terms, the benchmark R_I carries the inflation rate: the macroeconomic variable r_I sits inside the pricing relation. In nominal terms, the deflator cancels and the relation reads

with

where 1 + g̃_M = (1 + r_I) G̃_M is the gross nominal growth of the market. The security market line passes through the origin. Currency is the zero-beta asset, and its return is zero. The nominal form contains no inflation term.

The relation above is derived at an interior currency position (w_I > 0). When a T-bill with r_f > 0 is present, the nonnegativity constraint on currency binds, and the T-bill takes over the reference margin. To repeat the argument with the T-bill, let

so that

which is the Sharpe–Lintner CAPM. With currency dominated (r_f > 0), the standard relation governs. With the T-bill dominated or absent (r_f < 0), the relation of this paper governs. Each model is the other’s corner, and the economy selects the regime through the sign of the risk-free rate.

Remark on Mispricing

Nothing above assumes that market prices are fair, and nothing forbids an agent from believing that they are not. A view about the price of a security is a view about the distribution of r̃_j. Security-specific discount rates, as instruments of active selection in the lineage of Treynor and Black,¹¹ require an account of their origin. The proposed model does not require such an account.

Summary

The standard one-period portfolio choice problem contains no currency, defending the omission by a dominance claim that has failed empirically, and treats the risk-free security as special, defining risk premia against it. The proposed model restores currency and discounts every payoff by the price level. The null trading strategy is incorporated. The risk-free security becomes a security like any other, held only when it dominates currency, with the risk-free rate bounded below by the return on cash. Risk premia are defined against inflation. Inflation encourages investment in the demand-side sense of Mundell–Tobin. A CAPM-type relation holds, in which currency is the zero-beta asset and the security market line passes through the origin in nominal terms. The standard model is the corner of the proposed one, where the risk-free security dominates the currency. No efficiency assumption is made. The macroeconomic variable r_I can be observed by all economic agents.

Notes

1. The author can be contacted at nkkolev@gmail.com. Draft of August 21, 2026. ↩
2. Huang, Chi-fu, and Robert H. Litzenberger. 1988. Foundations for Financial Economics. New York: North-Holland. ↩
3. Pliska, Stanley R. 1997. Introduction to Mathematical Finance: Discrete Time Models. Malden: Blackwell Publishers. Pliska uses the null trading strategy — the strategy which starts with zero money and does no investment at all — in his comparison of trading strategies. ↩
4. No positive investment in risky securities is made only if none of the risky securities has a strictly positive risk premium. ↩
5. Black, Fischer. 1972. “Capital Market Equilibrium with Restricted Borrowing.” Journal of Business 45 (3): 444–455. Black derives the CAPM without a risk-free security, leaving the zero-beta rate to be estimated. The proposed model is a complement: the zero-beta asset is the one always present in the economy, and its nominal return of zero fixes the intercept. For the CAPM under uncertain inflation, see Friend, Irwin, Yoram Landskroner, and Etienne Losq. 1976. “The Demand for Risky Assets under Uncertain Inflation.” Journal of Finance 31 (5): 1287–1297. The proposed model is different because the deflator is deterministic and currency itself is the reference asset. ↩
6. Jensen, Michael C. 1986. “Agency Costs of Free Cash Flow, Corporate Finance, and Takeovers.” The American Economic Review 76 (2): 323–329. ↩
7. Myers, Stewart C., and Nicholas S. Majluf. 1984. “Corporate Financing and Investment Decisions When Firms Have Information That Investors Do Not Have.” Journal of Financial Economics 13 (2): 187–221. ↩
8. At the all-currency point, NPV is deterministic, so the marginal value of a position in security j is proportional to π_j. Any nonzero premium induces a position. ↩
9. Mundell, Robert A. 1963. “Inflation and Real Interest.” Journal of Political Economy 71 (3): 280–283. Tobin, James. 1965. “Money and Economic Growth.” Econometrica 33 (4): 671–684. ↩
10. Fama, Eugene F., and G. William Schwert. 1977. “Asset Returns and Inflation.” Journal of Financial Economics 5 (2): 115–146. ↩
11. Treynor, Jack L., and Fischer Black. 1973. “How to Use Security Analysis to Improve Portfolio Selection.” Journal of Business 46 (1): 66–86. ↩

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