I. Introduction

The phenomenon of financial contagion has achieved considerable attention in both academic and policy circles in recent years. The tequila crisis of 1994–95, the Asian crisis of 1997, the Russian default and the collapse of Long Term Capital Management in 1998, the boom and bust related to the Internet bubble in the late 1990s, the response of international markets in the immediate aftermath of September 11, and the run-up to the Argentine debt default in late 2001, all were accompanied by the transmission of financial market volatility across borders. In the case of emerging markets, the prices of assets of countries which were not related through direct macroeconomic links (e.g. trade channels, linked exchange rates, or vulnerability to similar commodity prices) showed comovements in excess of what could be explained through traditional macroeconomic linkages.

The literature on contagion can broadly be classified into its empirical and theoretical strands. The empirical strand has focused on definitions of contagion to account, for instance, for simultaneous increases in volatility which show up as increased correlations, or the impact of common external factors. The theoretical strand has tried to identify the possible channels of contagion, including the herding behavior of investors, the transmission of panic, and automated risk management procedures. This paper will focus on contagion as a transmission of a negative or positive shock to another country where financial markets are not linked by economic fundamentals but affected by the behavior of different types of fund managers to parameters such as index weights, volatility of the returns of the assets, the level of risk aversion, and trading strategies.

Policy approaches to contagion have relied mainly on the argument that informational asymmetry drive excess comovements of prices as investors watch each others’ actions and often tend to reinforce each others actions. This has prompted calls for greater disclosure both of market positioning and key financial and economic data from countries vulnerable to contagion. In a related vein, policymakers and researchers have also focused on the role of particular investor groups in driving market prices. Particularly after the Asian crisis, hedge funds and other highly leveraged institutions were the subject of much debate on the causes of contagion and policies were targeted at reducing the sources of contagion from such sources.

A key assumption in much of the literature is that the main financial markets are efficient and that contagion is a deviation from the norm. This could be driven by information asymmetry, market manipulation through size, or the destabilizing effects of leverage. This paper aims to analyze the phenomenon of contagion by showing that the institutional structure of markets can play a significant role in creating market architectures that may lead to contagion.

In particular, the incentives fund managers face can lead to contagion even in a market with no asymmetric information dominated by certain classes of institutional investors—a key feature both of emerging debt markets as well as major equity markets. The different compensation mechanisms of different classes of fund managers, themselves an outcome of optimal principal-agent relationships between fund managers and their clients, are a root cause of deviations of asset prices from what may be the efficient market outcome. This also suggests that asset prices may continue to significantly deviate from underlying “fundamentals” and the behavior of fund managers is optimally guided not just by the fundamentals, but by their expected compensations for taking on risky positions. The paper finds that given the domination of markets by distinct types of portfolio managers, who are distinguished by their mandates and compensation mechanisms, the optimal responses of these investor classes to the same information set and market conditions vary considerably. While groups of investors behave in well-defined ways in response to shocks, the paper finds that the impact on equilibrium market prices and fund managers’ rebalancing of their portfolio weights is based on the type of shock and the relative sizes of the two fund manager classes, and the initial conditions in the market.

While this model was motivated by emerging markets, this framework can also be used to analyze comovements in prices in different assets within the borders of the same country, for example between stocks and the bond market. A key conclusion that emerges from this paper is that managerial compensation systems are a key source of distortions in financial markets, and may be the source for long-term deviations of prices from the so-called fundamentals. This also leads to the conclusion that the opportunity to arbitrage away such deviations may be limited for long periods of time, and markets may be over- or undervalued and be perceived as such for extended periods.

The paper considers two types of fund managers—dedicated and opportunistic—in the model. Dedicated managers are compensated based on deviations from an emerging market index and are not allowed to borrow cash or short any asset. Opportunistic managers are compensated based on the absolute return on their portfolio and are allowed to short any asset and borrow cash. First, the optimal weights for each asset for each type of investor are derived. The paper finds that dedicated investors tend to rebalance their portfolios towards the index when asset volatility or their risk aversion increases. It also finds that opportunistic managers decrease the amount of leverage in response to increased asset volatility or increase in risk aversion. Second, the paper derives equilibrium expected asset returns and prices. The paper finds that a demand shock in one asset affects the expected price of the other asset. Specifically, the relative contribution of one type of trader to contagion depends on underlying market conditions.

The paper is structured as follows. The next section provides a brief overview of the literature. Section III presents the basic framework of the paper and discuss features of the demand functions of three types of fund managers. In Section IV, equilibrium prices are calculated and the impacts of changes in parameter values are investigated. Section V offers some concluding thoughts.

II. A Review of the Literature

This paper best fits in the theoretical literature about contagion where the reallocation of assets by investors is not necessarily based on market fundamentals.² Calvo and Reinhart (1996) distinguish between fundamentals-based contagion and “true” contagion where channels of potential interconnection are not present (also see Kaminsky and Reinhart, 2000). Contagion is defined as the propagation of a shock to another country’s asset when there are no fundamental linkages between the country hit by the shock and the other countries, and the comovement of asset prices across borders is based on the behavior of global investors.

Calvo and Mendoza (2000) suggest that information regarding investments in a portfolio may be expensive and investors may choose to “optimally” mimic market portfolios. They find that financial globalization in an environment of imperfect information may increase contagion where investors face high costs to gather information on market fundamentals and rely on the actions of other investors. Kyle and Xiong (2001) construct a continuous-time model with two risky assets and three types of traders—noise traders, long-term investors, and convergence traders. When convergence traders suffer large capital losses in one market, they liquidate positions in both markets. The liquidation of the portfolio amplifies and transmits the shock from one asset to another. Contagion in their model is generated through the wealth-effect of convergence traders. Kodres and Pritsker (2002) construct a multiple asset model to study contagion through cross-market rebalancing when one country faces an idiosyncratic shock. Countries may be weakly linked in terms of macroeconomic risks. They also find that asymmetric information increases a country’s vulnerability to contagion. Schinasi and Smith (1999) suggest an alternative view to contagion from those based on market imperfections such as asymmetry of information. They construct a partial equilibrium framework to study different portfolio management rules and rebalancing events and their effects on contagion. They find that a shock to an asset in one country may have effects on risky assets in other countries because of the underlying portfolio management rules and the parameterization of the joint distribution of asset returns. Furthermore, they find that rebalancing may be affected by whether or not the investor is leveraged. Leveraged investors will reduce their exposure to risky assets if the return on the leveraged portfolio is less than the cost of funding.

This paper extends the literature by considering the case where investors optimally rebalance their portfolios based on an idiosyncratic shock to one market in terms of increased volatility and a demand shock to an emerging market asset potentially resulting in contagion. Unlike the previous literature, the focus is on the managerial incentives of fund managers and their role in dampening or exacerbating contagion. Fund managers are often restricted in the amount that they can invest in emerging markets. In addition, they may also be compensated on the relative return on the portfolio to the emerging market index. The paper considers two types of international fund managers, dedicated and opportunistic fund managers, which are discussed in detail below.

III. The Model

This paper considers a simple discrete time model with two risky emerging market assets (A and B) a mature market asset (Z), and cash (M). The emerging market and mature market assets can be viewed as long-term bonds. There are three types of traders: dedicated emerging market fund managers (investing in only emerging market assets and cash), global opportunistic fund managers (investing in emerging markets and mature markets), and noise traders (local investors). Risk averse managers will attempt to maximize their risk-adjusted compensation.

Local Investors trade in asset A or asset B, and do so based on conditions in other asset markets in that country. They do not invest outside of their respective country, and hence only choose between asset A (or B). For purposes of this model, noise traders add a random element to the demand of assets A and B.

Dedicated fund managers allocate their capital between two risky assets A and B and a risk-free asset (cash), and can only invest in these assets (their mandate does not allow investing in the mature market asset Z). The compensation of dedicated fund managers is tied to the performance of the funds under their management relative to the benchmark index for emerging market assets.³

Opportunistic fund managers are allowed to invest in all three assets A, B and Z. While their main investment universe is defined as mature market assets, they have the opportunity to invest in the emerging market asset class to enhance their overall returns. Thus, their decision is whether to invest a small amount of their portfolio in emerging market assets or mature market assets. Opportunistic mangers may either increase or reduce their exposure to assets A, B and Z depending on the relative returns/volatilities of mature and emerging market assets.⁴ Asset Z can be interpreted as a risk-free asset such as U.S. Treasuries with fluctuations in secondary market prices. Unlike dedicated managers, opportunistic managers may sell assets short to finance long positions in other assets.⁵

Since comprehensive data on the composition of the investor base is difficult to compile, one has to rely on the evidence presented by international banks who are the main market makers for emerging market debt, in gauging the relative size of investor classes. The total sovereign emerging bond market universe investible by international investors is estimated at some $225 billion. While the size of outstanding bond market capitalization is somewhat larger, the above estimates exclude smaller and illiquid sovereign bond issuances, and emerging market corporate issuances of about $100 billion, and others not meeting the criteria for inclusion in the major market indices. Of this pool of available assets, between 40–50 percent is thought to be held by dedicated investors including both emerging market mutual funds as well as emerging market funds managed independently but belonging to a larger family of funds. Hedge funds typically comprise between 10 and 20 percent of the investor base. The remainder is dominated by global investors who either invest in the whole emerging markets index or who selectively and opportunistically “cross over” into emerging markets. Direct retail investors do not form a significant proportion of overall emerging market investor bases.

A. The Investment Horizon

For the purposes of modeling portfolio managers’ behavior, this paper considers a time horizon consisting of three periods as below (see figure 1):

The Timeline

Citation: IMF Working Papers 2004, 199; 10.5089/9781451860146.001.A001

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The Timeline

Citation: IMF Working Papers 2004, 199; 10.5089/9781451860146.001.A001

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The Timeline

Citation: IMF Working Papers 2004, 199; 10.5089/9781451860146.001.A001

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$t=\{\begin{array}{l}0\\ T\\ T+1.& \end{array}$

Period 0 is the initial period, where fund managers begin with a certain portfolio allocation, and a certain knowledge of prices and returns, which is an outcome of the previous period’s portfolio decisions and shocks. They then update their information set I₀ in this period based on which they form their expectations of the future demand of local investors for each asset, and the variance (distribution) of all assets. Based on that, in a rational expectations framework, they make a decision on their new optimal portfolio, based on expectations of the variables.

Period T is when portfolio managers, based on I₀ and their initial conditions, put in place their new portfolios, and when the realization of the random variable takes place. The actual outcome of equilibrium prices and returns in period T will be the result of the realization of the random variable on the new portfolio positions. These equilibrium prices have to be compared against the prices under alternative scenarios to analyze the dynamics of contagion. Since expected returns are the inverse of prices, as will be shown, the allocation of a proportion of a portfolio to an asset will help determine its price, and hence expected return.⁶

Period T+1 is a terminal condition on prices. The terminal condition is significantly beyond the time period focused on in the model. The terminal price is based on asset and economy-wide fundamentals is fixed and known. These assets may be viewed as long-term bonds where the terminal payout is known but the price in secondary markets fluctuates.

The model will be based on the rates of return of various assets, which is the inverse of their prices. The model will determine the total demand for each asset, and set that against a fixed supply of each asset to determine equilibrium prices. Note that the rates of return will be computed as the difference between the equilibrium prices determined in the model and the terminal prices.

D. Dedicated Fund Manager’s Compensation Structure

Let r^D denote the net return on the portfolio held by the index investors from period T to period (T + 1), where λ is the proportion of their wealth invested in asset A and τ is the proportion of their wealth invested in asset B, with (1 − λ − τ) being the proportion invested in cash. Then, the net return on the dedicated manager’s portfolio is:

$r^D=\lambda (r^A)+\tau (r^B)+(1-\lambda -\tau )(r^M),$

where:

$\begin{array}{c}\frac{P_{T+1}^A+P_T^A}{P_T^A}\equiv r^A,\\ \frac{P_{T+1}^B+P_T^B}{P_T^A}\equiv r^B,\\ \frac{P_{T+1}^M+P_T^M}{P_T^A}\equiv r^M.\end{array}$

Let r^D − r^I denote the total excess return of the dedicated fund manager’s portfolio at time (t +1). The excess return is defined as the return of the managed portfolio over a portfolio which simply tracks the market index. The fund manager’s compensation is a fixed proportion k of the excess return she earns for the portfolio, and her utility is increasing in his expected income and decreasing in the variability of his income (with (a) denoting the coefficient of constant absolute risk aversion). Assuming that each fund manager’s initial portfolio value is normalized to one, the dedicated fund manager’s optimization problem is as follows:

$\underset{\lambda ,\tau }{{max}}\left\{kE[U(r^D-r^I)]\right\},$

where:

$U(r^D-r^I)=-e^{(-a(r^D-r^I}))$

and

$E[U(r^D-r^I)]=-e^{(-a[E[r^D-r^I)-\frac{1}{2}aVar[r^D-r^I])}.$

The excess return of the portfolio is given by:

$r^D-r^I=(\lambda -\alpha )r^A+(\tau -1+\alpha )r^B+(1-\lambda -\tau )r^M.$

Then,

$E(r^D-r^I)=(\lambda -\alpha )E(r^A)+(\tau -1+\alpha )E(r^B)+(1-\lambda -\tau )r^M$

and

$Var(r^D-r^I)=((\lambda -\alpha {)}^2{\sigma }_A^2+(\tau -1+\alpha {)}^2{\sigma }_B^2).$

The return on cash is a known constant r^M. To isolate the effects of index-linked investing on comovement of asset prices, it is assumed that Cov(r^A, r^B) = 0, i.e. it is assumed there is nothing inherent in asset prices of A and B that already has contagion incorporated in it. Maximizing the expected utility of wealth (since the fund manager gets a fixed percentage k of the excess returns on the portfolio, he will maximize his utility by maximizing the excess returns on the portfolio) is equivalent to maximizing:

$E(r^D-r^I)-\frac{1}{2}aVar(r^D-r^I).$

The following function is maximized with respect to λ and τ:

$\begin{array}{ccc}\underset{\lambda ,\tau }{{max}}\left\{\begin{array}{c}(\lambda -\alpha )(E(r^A))+(\tau -1+\alpha )(E(r^B))+(1-\lambda -\tau )r^M\\ -\frac{a}{2}[{(\lambda -\alpha )}^2{\sigma }_A^2+{(\tau -1+\alpha )}^2{\sigma }_B^2]\end{array}\right\}& & (5.2)\\ \text{subject} \text{to}:\{\begin{array}{c}\lambda \ge 0,\\ \tau \ge 0,\\ \lambda +\tau \le 1.\end{array}& \end{array}$

Note that dedicated managers are not allowed to short either asset A or B, or borrow cash.

The dedicated fund managers’ demand space for assets A and B is diagrammed in Figure 2. When cash holdings are zero, the manager is on the diagonal line. When cash holdings are positive, the manager is below the diagonal line. Because dedicated managers are not allowed to short either asset or borrow cash, their allocations are bounded from below by the λ and τ axes. If the manager is underweight asset A but overweight asset B, then she will be in the triangle labeled I. If the manager is overweight asset A and underweight asset B, she will be in the triangle labeled III. If she is underweight both assets she will be in rectangle II. Even if the manager knows an asset is likely to bring negative returns, the compensation and indexation structure results in her holding some amount of the asset under certain conditions as elaborated below.

Dedicated Investor Demand Space

Citation: IMF Working Papers 2004, 199; 10.5089/9781451860146.001.A001

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Dedicated Investor Demand Space

Citation: IMF Working Papers 2004, 199; 10.5089/9781451860146.001.A001

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Proposition 1

The solution of the dedicated fund manager’s optimization problem (5.2) is as follows: ¹⁰

For the region of parameter values where $\frac{E\left(r^A\right)-r^M}{a{\sigma }_A^2}\ge 0$ and $\frac{E\left(r^B\right)-r^M}{a{\sigma }_B^2}\ge 0$, the optimal portfolio weights (λ*,τ*) are:

${\lambda }^{*}=\frac{E\left(r^A\right)-E\left(r^B\right)}{a({\sigma }_A^2+{\sigma }_B^2)}+\alpha \mathit{\text{and}}{\tau }^{*}=\frac{E\left(r^B\right)-E\left(r^A\right)}{a({\sigma }_A^2+{\sigma }_B^2)}+(1-\alpha )$

For these parameter values, cash holdings are zero. The investor will be along the “no-cash” line in Figure 2 above.

For the region of parameter values where $\frac{E(r^A)-r^M}{a{\sigma }_A^2}<0$ and/or $\frac{E(r^B)-r^M}{a{\sigma }_B^2}<0$, the optimal portfolio weights (λ**,τ**) are:

${\lambda }^{**}=\{\begin{array}{c}\frac{E(r^A)-r^M}{a{\sigma }_A^2}+\alpha ,\text{whenever}\frac{E(r^A)-r^M}{a{\sigma }_A^2}+\alpha >0\\ 0,\text{whenever}\frac{E(r^A)-r^M}{a{\sigma }_A^2}+\alpha \le 0\end{array}$

and

${\tau }^{**}=\{\begin{array}{c}\frac{E(r^B)-r^M}{a{\sigma }_B^2}+(1-\alpha ),\text{whenever}\frac{E(r^B)-r^M}{a{\sigma }_B^2}+(1-\alpha )\ge 0\\ 0,\text{whenever}\frac{E(r^B)-r^M}{a{\sigma }_B^2}+(1-\alpha )<0\end{array}$

For these parameter values, cash holdings are:

$(1-\lambda -\delta {)}^{**}=\{\begin{array}{l}\frac{r^M-E(r^A)}{a{\sigma }_A^2}+\frac{r^M-E(r^B)}{a{\sigma }_B^2},\text{whenever}0<{\lambda }^{**}<1\text{and}0<{\tau }^{**}<1,\\ 1-\frac{E(r^A)-r^M}{a{\sigma }_A^2}-\alpha ,\text{whenever}0<{\lambda }^{**}<1\text{and}{\tau }^{**}=0,\\ 1-\frac{E(r^B)-r^M}{a{\sigma }_B^2}-(1-\alpha ),\text{whenever}{\lambda }^{**}=0\text{and}0<{\tau }^{**}<1,\\ 1,\text{whenever}{\lambda }^{**}=0\text{and}{\tau }^{**}=0.\end{array}$

Proposition 1 demonstrates that the index weights α and 1 − α are key determinants of a dedicated managers portfolio allocation towards an asset. Other things equal, a country with a greater weight in the index will automatically get a greater allocation of funds in an optimal behavioral framework. Note also that the deviation of the allocation from the index weight is independent of that weight.

Proposition 2 describes the behavior of dedicated managers when one or both emerging market assets underperform cash.

Proposition 2:

Suppose that the risk-adjusted excess return of an emerging market asset underperforms cash:

a) If E(r^A) > r^Mand E(r^B) < r^Mor E(r^B) > r^Mand E(r^A) < r^M, the manager will go overweight asset that outperforms cash. Conversely, if E(r^A) < r^Mor E(r^B) < r^Mor both, the manager will be underweight asset A (λ<α) and/or asset B ((1 − λ) < (1 − α)), but will not necessarily hold zero of either asset.

b) As the weight of asset A in the benchmark index α rises, a manager who is overweight the asset will increase her exposure further by maintaining the overweight. A manager who is underweight asset A will also increase her exposure, but maintain the underweight.

c) As the risk aversion coefficient (a) rises, the demand for asset A or B falls, if the manager is overweight the asset. If the manager is underweight the asset, an increase in (a) results in her reducing her underweight position. In other words, a higher degree of risk aversion causes “hugging of the index.”

d) As${\sigma }_A^2$ or ${\sigma }_B^2$ rises, the demand for asset A or B falls if the manager is overweight the asset. If the manager is underweight the asset, as ${\sigma }_A^2$ or ${\sigma }_B^2$increases, the manager reduces her underweight. In other words, an increase in ${\sigma }_A^2$ or ${\sigma }_B^2$ results in greater “hugging of the index.”

e) An increase in (a), ${\sigma }_A^2$ or ${\sigma }_B^2$, may reduce the demand for cash and greater hugging of the index.

Proposition 2 states that dedicated managers may hold positive values of an emerging market asset even when it underperforms cash. Intuitively, it is easy to see that while lower weights to an asset with lower returns than cash would increase utility, the low weight relative to the benchmark increases the risk of underperforming the index and hence lowering utility. For some ranges, the return element dominates and hence a zero allocation may be optimal, but in other ranges, the risk element dominates leading to a positive allocation.

This result can be easily generalized to more than two emerging market assets. When the dedicated manager rebalances her portfolio weights closer to the index, the demand for all assets where she was underweight will increase and the demand for all the assets where she was overweight will decrease. Thus, the behavioral characteristics of the dedicated investor results in linkages between otherwise unrelated markets based on whether the portfolio weight is greater or less than the market index.

Proposition 2 also states that dedicated managers tend to hug the index more closely when volatility of returns on emerging market assets and risk aversion increase. If the manager is underweight an asset and the volatility of that asset increases, she will increase her holdings of that asset. Interestingly, dedicated managers reduce their cash holdings when volatility and risk aversion increase.

The model next considers the case when both emerging market assets outperform cash.

Proposition 3:

Considering the case when λ + τ = 1.

a) The dedicated manager is overweight the asset with the higher expected return and is underweight the asset with the lower expected return.

b) An increase in risk aversion coefficient (a) would result in “hugging of the index” or allocations closer to the index. If the manager is underweight an asset, an increase in (a) would result in the dedicated manager increasing her exposure of that asset and decreasing her exposure of the other asset. Similarly, if the dedicated manager is overweight an asset an increase in (a) would result in a decrease in exposure of that asset and an increase in exposure of the other asset.

c) An increase in ${\sigma }_A^2$ or ${\sigma }_B^2$ reduces the size of the overweight/underweight positions as well, forcing the dedicated manager to move closer to the benchmark index.

When dedicated managers do not hold cash, they increase their holdings of an underweight asset when its volatility increases and decrease their holdings of the other emerging market asset. In other words, an increase in the volatility of an underweight asset results in a decrease in the demand for the other emerging market asset when there are only two assets. If there are more than two assets, the demands for all the underweight assets vis-à-vis the index increase while the demands for all the overweight assets decrease. In this sense, an increase in the volatility of one asset spills over into the demand for the other asset.

Propositions 2 and 3 state that changes in the expected returns, level of risk aversion, and variance of the emerging market assets may lead to changes in the demand for the underlying assets. It is also found that increases in ${\sigma }_A^2,{\sigma }_B^2$ or a would result in managers choosing allocations closer to the index.

IV. The Equilibrium

The previous sections derived the optimal behavior of two main classes of fund managers in emerging market bond markets, namely dedicated emerging market managers and global opportunistic managers. Now the equilibrium returns (and implicitly prices) that are derived from the interaction of these two classes of managers are computed.

The supply of asset A, (S^A), and asset B, (S^B), are known and fixed. D^D,A and D^D,B denote the dedicated mangers’ demand for assets A and B, respectively. Similarly, D^O,A and D^O,B, denote the opportunistic managers’ demand for assets A and B, respectively, and D^L,A and D^L,B denote the local investors’ demand for assets A and B, respectively.

For dedicated managers, their compensation mechanism is linked to the performance of their portfolio relative to a benchmark portfolio. Most dedicated investors are benchmarked to either the EMBI+ or the EMBI Global index. In equity markets, they are typically benchmarked to Morgan Stanley Capital International Emerging Markets Free index.

Hedge funds and the proprietary desks of commercial and investment banks act like the global opportunistic managers described above. They essentially are focused on the absolute risk-adjusted returns of their portfolios, and have access to both emerging and mature market assets, and can go long or short assets, thereby allowing significant expansions of their balance sheets. What the model shows is that such managers look at the relative risk-adjusted returns for all assets. The main determining factor for their positioning, including whether to go long or short any asset, is their expected excess return over other assets they can invest in, for given levels of volatilities. Therefore, whether they will treat two emerging market assets similarly or differently will depend on how the returns compare with that of the mature market asset in a three-asset case.

Defining contagion as a comovement of asset prices (and hence returns) in the same direction, and reverse contagion as the offsetting movements (in the opposite direction) of two asset prices, contagion can be analyzed by comparing the returns on the two assets when subject to a shock. The shocks of particular interest are when investor expectations of local traders in a particular country changes and its effect on the expected return on the other emerging market’s asset via the trading strategies of cross-border managers.

The impact on emerging market bond prices from the interaction of dedicated and opportunistic managers can be seen from the computation of equilibrium prices. For this, the total demand of assets A and B from two types of managers is set equal to their respective supplies and compute equilibrium prices. Suppose that there are n number of dedicated investors and q number of global investors. When dedicated and opportunistic managers along with local investors are present, the market clearing conditions are:

B. Dedicated Manager (Zero Cash Holdings) and Opportunistic Manager

This section examines the equilibrium expected prices when dedicated managers do not hold cash. Substituting the optimal portfolio allocations to each asset for each type of investor and plugging into (6.1) and (6.2) yields:

and

Solving for the expected returns for assets A and B yields: