For the purposes of further discussion all crypto assets are divided into the following baskets:
 Tier 1.1 (core): BTC, ETH;
 Tier 1.2: 1st class stablecoins;
 Tier 2.1 (1st class DeFi): DeFiPulse Index constituents;
 Tier 2.2 (1st class layer 1 nonTier 1): DOT, Atom, XTZ;
 Tier 3: all the rest.
We start with the premise that FEI is a Central Bank for DeFi, as such is a provider of liquidity and potentially lender of last resort, while PCV is its Reserve Fund.
Hence primary PCV management objectives are:
(a) To preserve capital;
(b) To ensure daily operations, at all times, are fully backed by highly liquid assets;
(c) Subject to (a)â€“(b), to achieve an investment return that will help preserve the Reserve Fundâ€™s longterm purchasing power.
PCV is separated into the Backing Portfolio and the Investment Portfolio. The Backing Portfolio ensures daily operations are fully backed by Tier 1 highly liquid assets, while the Investment Portfolio is invested into Tier 12, potentially with some small proportion of hedging, like perpetual swaps, and insurance instruments with a view to preserving PCVâ€™s value and its longterm purchasing power. It also includes a countercyclical capital buffer.
Technically assets are injected into or transferred out from the Backing portfolio when the trigger point is reached: this is implemented on the Balancer V2 smart pool by means of Balancerâ€™s continuous pool rebalancing mechanism. This arrangement enables a higher investment return on excess assets while ensuring sufficient liquid assets in the Backing Portfolio.
So, when determining the conceptual framework of PCV management the first problem is related to determining the goals of holding and managing the Backing Portfolio based on the analysis of compliance of the Backing Portfolio available volume with the criteria of adequacy and optimality.
The nature of the Backing Portfolio management could vary significantly depending on its goals and purposes. Essentially, the first task of the Backing Portfolio management is to assess the amount of reserves needed to compensate for excess shortterm volatility in Fei market (based on the experience of past crises and stress tests). This portion of reserves should be invested in liquid and relatively reliable assets (including stablecoins), while amounts in excess of this amount may be invested in riskier assets in order to generate income and preserve capital.
Itâ€™s necessary to come up with criterion for adequacy of the Backing Portfolio. In the world Central banking practice there is no generally accepted criteria of this kind. The most common criteria for determining the lower limit of international reserves for a country are: import coverage ratio; monetary base coverage ratio/M2; GuidottiGreenspan rule. Thatâ€™s a place to start.
At present there is a revival of academic interest in the problem of determining the optimal level of international reserves for a particular country taking into account specific aspects of its economic situation. It has been estimated that maintaining international reserves costs developing countries about 1% of their GDP annually. These costs are thought to be the cost of having an insurance mechanism against external economic shocks â€“ a guarantor of stability and a source of liquidity in extreme situations. In determining the optimal level of reserves, the cost of reserves is compared to the benefits the country receives.
In our case the accumulation of reserve assets in the Backing Portfolio in excess of adequate and optimal levels leads to a decrease in investment resources in the Investment Portfolio preventing acceleration of Fei growth, curtailing development of its ecosystem.
The second part of the PCV management framework is related to the portfolio optimisation model. In accordance with the specific objectives and functions of PCV it should meet the following requirements:

minimising the risk of having to rebalance the portfolio due to changing market conditions;

ensuring high liquidity of investments;

providing high diversification of the portfolio to ensure safety of funds;

portfolio profitability is secondary to the goals of safety and liquidity.
The conceptual development of portfolio theory goes back to H. Markowitzâ€™s article "Portfolio Selection. The BlackLitterman model is an asset allocation model developed in 1990 by F. Black and R. Litterman. It is a combination of ideas from the Capital Asset Pricing Model and the Markowitz Mean Dispersion Optimization Model and serves as a tool for investors to compute the optimal weights for assets in their portfolios according to given parameters.
The BlackLitterman portfolio allocation model largely solves the problems of insufficient diversification and high sensitivity of the portfolio structure to the quality of incoming data associated with the application of the classical Markowitz approach. The model is a method for constructing an effective portfolio of assets that allows the investor to take into account their personal forecast regarding the correlation between the returns on specific assets and their equilibrium market returns.
The presence of the market portfolio as a point of reference in the BlackLitterman model causes a higher stability of resulting weights in the optimal portfolio. If an investor has no predictions about future returns, the BlackLitterman model suggests holding a portfolio containing financial instruments proportional to their market capitalisation.
The introduction of the modelâ€™s own forecast of returns on assets allows, subject to competent forecast, to improve the resulting portfolioâ€™s returns, while slightly increasing the variability of its structure. The degree of increase in the variability of the optimal weights is largely due to the variability of the investorâ€™s predictions of future returns.
The BlackLitterman model uses equilibrium returns on assets as a neutral starting position, which are obtained from the assumption that the market is efficient at the moment. A Bayesian approach is used to combine investorâ€™s subjective views about expected returns on one or more assets with a vector of equilibrium market returns to form a new mixed assessment of expected returns, which allows the investor to input his personal macroeconomic development forecast and the behaviour of the analysed assets into the calculations. For the Markowitz model it can be quite difficult to obtain reasonable data on expected asset returns. Black and Litterman have overcome this difficulty by not requiring the exact values of expected returns. The portfolios built with the BlackLitterman model are characterized by higher stability of asset weights in the portfolio, which allows for less frequent portfolio restructuring and saves on transactions. The disadvantages of the BlackLitterman model are rather complicated calculations accompanying the construction of this model, as well as higher requirements to the stability of the covariance matrix of returns on the assets in the portfolio.
Let us recall the essence of the BlackLitterman model. The expected return of the asset portfolio in the framework of this model is determined by the following equation:
E[R]=[(\tau\Sigma)^{1}+P^T\Omega^{1}P]^{1}[(\tau\Sigma)^{1}\Pi+P^T\Omega^{1}Q], \space \space\space\space\space\space\space (1)
where
E[R] â€“ a new combined vector of returns (NĂ—1, column vector);
\tau â€“ scaling factor;
\Sigma â€“ covariance matrix of asset returns (NĂ—N matrix);
P â€“ matrix identifying the assets involved in the calculations (KĂ—N matrix);
\Omega â€“ diagonal matrix of covariance of standard errors of forecasts, describing the uncertainty of each forecast (KĂ—K matrix);
\Pi â€“ vector of expected equilibrium returns (NĂ—1), element values of which are defined by equation (2);
Q â€“ forecast vector (KĂ—1);
K â€“ number of investor forecasts;
N â€“ number of assets in the portfolio.
In the BlackLitterman model the vector of supposed equilibrium profitability is a set of expected returns of assets under the supposition that the stock market exhibits efficiency properties and all investors have identical forecasts concerning future levels of portfolio asset returns. The vector of expected equilibrium returns can be calculated by inverse Sharpeâ€™s optimisation based on the available data on the market portfolio structure using the following formula:
\Pi=\lambda\Sigma w_{mkt}, \space \space\space\space\space\space\space (2),
where
\Pi â€“ vector of implied equilibrium returns (NĂ—1, column vector);
\lambda â€“ investorâ€™s risk appetite coefficient;
\Sigma â€“ covariance matrix of asset returns;
w_{mkt} â€“ share of assets included in the portfolio of the total market volume (NĂ—1, column vector)
The equation (2) contains investorâ€™s risk appetite coefficient, which characterises for a given investor acceptable proportion between a reduction in portfolio risk and a reduction in the return on that portfolio. This coefficient is estimated by the following formula:
\lambda=\frac{E(r)r_f}{\sigma^2}, \space \space\space\space\space\space\space (3)
where
E(r) â€“ expected market (benchmark) return;
r_f â€“ riskfree interest rate;
\sigma^2 â€“ variance of the market (benchmark) portfolio, which can be calculated by the formula w^T_{mkt}\Sigma w_{mkt}.
In the absence of forecasts, investors should form a portfolio whose structure corresponds to that of the market portfolio. However, BlackLitterman model allows investors to take into consideration their subjective forecasts about future changes in the expected returns on individual assets in the portfolio, which may differ from the expected equilibrium value. Investor forecasts can be entered into the model in absolute or relative form. Vector of forecasts Q is usually a column vector KĂ—1. The uncertainty of the forecasts leads to the existence of a random, independent, normally distributed error of the forecast vector, which we denote by vector (\varepsilon). Letâ€™s assume that the expectation of this uncertainty is 0 and the variance is constant. In addition, letâ€™s denote the covariance matrix between the values of prediction errors as \Omega. The elements of this matrix will be the values of cov(\varepsilon_i; \varepsilon_j). Thus investorâ€™s predictions can be represented as a sum of the deterministic part of predictions and the random part: Q +\varepsilon, or in the expanded form:
Q +\varepsilon=\begin{bmatrix} Q_1\\ ...\\ Q_k \end{bmatrix}+ \begin{bmatrix} \varepsilon_1\\ ...\\ \varepsilon_k \end{bmatrix}.\space \space\space\space\space\space\space (4)
As can be seen from the equation (4), the error vector (\varepsilon) indicates the uncertainty of investor predictions. When an investor is 100% sure of his predictions, the error vector (\varepsilon) has zero value. However such situation seldom occurs in reality. The difficulty of evaluating the expected return on assets in a portfolio always leads to the presence of uncertainty in investorsâ€™ forecasts. Therefore, as a rule, the error vector (\varepsilon) has a nonzero value. The greater the variance of the error vector (\varepsilon), the greater the forecast uncertainty. However, as can be seen from formula (1) of the BlackLitterman model, the error vector is not included in this equation. Instead, the BlackLitterman model includes a matrix of individual error vector variances \varepsilon, denoted by \Omega. This matrix is diagonal, with offdiagonal elements equal to 0, because this model assumes independence of predictions:
\Omega=\begin{bmatrix} \omega_1 & 0 & 0\\ 0 & \omega_2 & 0\\ 0 & 0 & \omega_3 \end{bmatrix}.\space \space\space\space\space\space\space (5)
One of the main difficulties of the BlackLitterman model implementation is to come up with the \Omega matrix. This forces an investor to determine the probability density function for each representation. The second difficulty in the model implementation is the choice of parameter \tau in the equation (1), i.e. the scaling factor. There is an ongoing debate regarding its value. Black and Litterman believe that this parameter should be close to 0, others believe that it should be equal to 1, and others suggest that it should be calculated as 1 divided by the number of observations.
G. He and R. Litterman published a paper in which they calibrated individual variance values \omega_i of the matrix \Omega such that the ratio \frac{\omega}{\tau} is equal to the variance of the forecast portfolio p_k\Sigma p^T_k, i.e., \omega_k=\tau p_k \Sigma p^T_k. From this point of view \Omega takes the following form:
\Omega=\begin{bmatrix} p_1\Sigma p^T_1 & 0 & 0\\ 0 & p_2\Sigma p^T_2 & 0\\ ... & ... & ...\\ 0 & 0 & p_k\Sigma p^T_k \end{bmatrix}.\space \space\space\space\space\space\space (6)
Investor forecasts, which are written in the matrix \Omega, are assigned to specific assets with matrix P, which has dimensions KĂ—N:
P=\begin{bmatrix} p_{1, 1} & 0 & p_{1, N}\\ ... & ... & ... &\\ p_{K, 1} & 0 & p_{K, N} \end{bmatrix}.\space \space\space\space\space\space\space (7)
When investorâ€™s forecast is absolute, the intersection of the row corresponding to this forecast and the column corresponding to the selected asset has 1, and the rest of the row should be filled with 0. If the forecast is relative, then the sum of all row values should be equal to 0: highyielding assets in the matrix P have positive values, and lowyielding ones â€“ negative.
It was shown that the BlackLitterman model, when tested in comparison to other models, produced most stable results in terms of returns and variability of weights, while offering highly diversified portfolios. At the same time, the BlackLitterman model has a number of drawbacks that must be considered when forming a portfolio. For example, in the case of highly correlated returns on individual assets, forecasting the expected return on one of the assets sometimes entails a substantial revision of the weights of all other assets in the portfolio. As a result, there is an increased risk of instability of the resulting weights of the investment portfolio, their high variability depending on the formulation of the forecast and its accuracy. It was also shown by the researchers of the Royal Bank of Scotland that in cases when there is no investorâ€™s forecast concerning the return of all assets and there is a forecast â€śreturn of some asset will coincide with its equilibrium returnâ€ť the model gives out two different portfolio structures, which at first sight contradicts common sense. It is therefore important for the model user to express his/her opinion about the return of each of the assets (to use the â€śfullâ€ť, square matrix of forecasts).
Thus, we can conclude that the BlackLitterman model can serve as a starting point for PCV portfolios management. However, a substantial refinement of the model taken as a basis is required to take into account all specific objectives and constraints as much as possible.
For the riskreturn portfolio optimization in combination with the BlackLitterman model we recommend Expected Shortfall (CVaR), a coherent spectral risk measure, vs VaR risk measure. As the Basel Committee on Banking Supervision noted: â€śa number of weaknesses have been identified with using VaR including its inability to capture tail riskâ€ť. But thatâ€™s a topic for further discussion.
A few final notes. FEI is a Central Bank, as such is a provider of liquidity and lender of last resort for DeFi. PCV is its Reserve Fund. The most lucrative business is printing money. The protocol can mint Fei and lend it out against collateral nominated in Tier 12 assets at differentiating rates to platforms like Compound, while they could already lend it to users. Minted Fei will be on PCVâ€™s balance, thus collateralization ratio will grow. Maybe it can be also open to retail borrowers (maybe not to all, but with some preferential access). Thank you @cozeno for the discussion on this matter.
In the short run to survive the coming bear market a basket of some stablecoins, like USDC, is perfect (I feel that USDT could collapse any moment, DAI is too dependent on ETH, many have too limited liquidity for PCVâ€™s purposes).
However in the long run, for FEI to bring significant value a separated, diversified, relatively conservatively managed PCV is needed.