3.4 A Model with Floating Redemption Price
3.4.3 Stable NAV
As a response to the defects of secondary market trading, the shadow bank can intervene and eliminate the pecuniary externalities by participating in the trading.
Suppose the shadow bank offers a fixed price ϕ for equity share purchase and re- demption in period 1. Again, the result is that there will be zero trading volume in the secondary market.
When the true state is good, only the impatient investors will redeem their shares, while all patient investors will reinvest the dividend and purchase more. The shadow bank needs to pay out λϕ while receiving (1−λ)d1, which will cancel out with each
other if ϕis set at 1−λλd1. Each early consumer (impatient investor) will getcG1 = d1+ϕ in period 1, while the late consumers will havecG2 = 1−1λd2G = 1−d1
1−λYH.
When the true state is bad, the impatient investors and informed patient investors will sell their shares back to the fund atϕ, and uninformed patient investors will use the dividend to buy more shares. The consumption of early consumer (impatient investors and informed patient investors) will bec1B =d1+ϕ, while uninformed patient investors will use their dividend to buy d1
ϕ(1−θ)(1−λ) shares and have (1+ dϕ1)(1−θ)(1−λ) shares in total. Note that the number of newly purchased shares is smaller than the num- ber of redeemed shares λ+θ(1−λ) because the market price is fixed by the shadow bank.
The net inflow of shares is
λ+θ(1−λ)−d1
ϕ(1−θ)(1−λ) = λ+ [θ− d1
ϕ(1−θ)](1−λ)
The net redemption value is therefore
(λ+ [θ− d1
ϕ(1−θ)](1−λ))ϕ= [λ+θ(1−λ)]ϕ−(1−θ)(1−λ)d1
Given λ, θ, the shadow bank issuing uninsured equity shares would like to choose
d1 and ϕto minimize the net redemption value (i.e., net outflow of cash, which requires assets liquidation), which means a high first-period dividend d1 and a low priceϕ. The lowest feasible value ofϕis exactly one unit fiat money, since a price higher than one will not minimize the net redemption value, while a price lower than one will make house- holds unwilling to purchase shares in period 0.
Whenϕ∗ =1, the net redemption value will be minimized to zero when
d∗1 = λ+θ(1−λ) (1−θ)(1−λ)
A feasible d1 requires (λ1−+θθ)((11−−λλ)) < 1, or λ+θ(1−λ) < 0.5. In other words, at least half of the investors should be uninformed and patient to sustain the result. This parameter restriction corresponds to the retail money market fund, where investors are natural persons who seek prudent investments and largely overlap with the clientele of commercial banks.
The ϕ∗ and d∗1 achieve market-clearing in the primary market directly operated by the shadow bank. Without relying on liquidity support, the stable NAV helps the shadow bank to avoid liquidating assets, which is vital to surviving adverse economic conditions. Therefore in the bad state, the early consumers (impatient investors and informed patient investors) will consumec1B =d∗1+ϕ=, while the late consumers (uninformed pa- tient investors) will consumecB2 = ( 1
1−θ)(1−λ)d2B = (1−θ)(11−λ)(1−d1)YL =
1−2λ−2θ(1−λ) (1−θ)2(1−λ)2 YL.
Note that under the optimal dividend policy and stable NAV, the shadow bank needs to pay outλwhile receiving (1−1θ−)λθ+θ = λ+1−θθ > λ20 in the good state. The net value
θ
1−θ will be stored and redistributed to all remaining shareholders in period 2.
Hence in the good state, the consumption of early consumers (impatient investors) is
c1G = d1∗+ϕ∗ = (1−θ)(11−λ) in period 1, while the late consumers (patient investors) will havecG2 = 1−1λd2G+1−θθ = 1−1λ1−(2λ−2θ(1−λ)
1−θ)(1−λ) YH+1−θθ. The fixed purchase and redemption price avoid liquidation in the bad state, but doesn’t help with risk-sharing in the good state.
Let’s look back and check whether the uninformed investors will demand early re- demption under the fixed NAV through amortized cost method. On book, the total value of the remaining assets in period 2 is changed from (1−d1)YH to (1−d1)YL, probably
through writing down assets, since the state of the economy is know. But should liqui- 20A negative net redemption value, or a positive net new purchase value in period 1, is often not desirable either. This is because the projects have to be invested in period 0 to generate productive income. When more capital is contributed to the open-ended shadow bank, effectively all shares are diluted. This type of dividend policy might not have been approved by the shareholder in period 0, who were identical ex ante. But in this case, the probability of being patient is large enough (over 50%), and it is the patient investors, informed or uninformed, who are the only remaining shareholders of the shadow bank. Therefore the “dilution" is more like a stock split, which will not affect the actual value received.
dation occur, the market value will further drop to (1−d1)(1−δ)YL, while δ captures
the liquidation loss. With a fixed NAV other thanϕ∗, in order to pay for the net redemp- tion, the shadow bank, which has no liquidity backstop, has to resort to asset liquidation, which will lead to a downward spiral of asset values. Therefore a fixed NAV at ϕ∗ = 1 is the best available tool to reduce net redemption in bad state to zero and avoid asset liquidation.
The amortized-cost net asset value of the shadow bank share in period 1 will be
N AVAC = (1−d1)YL, which will be greater than or equal to one ifYL ≥ 1−1d1. The mark-
to-market net asset value of the shadow bank share after the liquidation in period 1 will be
N AVMM = [1−d1−Net redemption value
(1−δ)YL
](1−δ)YL
Clearly, in an adverse market environment where the shadow banks desire liquid- ity the most, N AVAC > N AVMM. With the shadow bank itself propping up the price, there will not be information extraction as long as the net asset value is maintained. The uninformed will not panic, and the optimistic will buy the shares at the constant price.
The main difference between traditional banks and MMFs at maintaining a stable price of the contracts lies in the source of liquidity to satisfy the unanticipated early with- drawal/redemption due to mood swings on uncertainties. Traditional banks rely on liq- uidity backstop provided by deposit insurance and federal reserve at the lender of last resort, which is more efficient and cost-effective in coping with aggregate uncertainties since the central bank controls the money supply. Shadow banks, on the other hand, have to depend on selling assets, often at a discount, to maintain a stable price.
As long as the fraction of patient households receiving negative signals (and there- fore requesting redemption) is not very large, the uncertainties regarding future payoff will have no effect on the share price. However, if the fraction is large, more early re- demption thanλwill either result in interrupting ongoing projects or induce the MMF to
hold more cash reserves. And since the signal is randomly drawn, it is hard to predict the fraction. The only way to make sure is to hold 100% cash reserve, which is the same as the storage and worse than the autarky state (where at least patient households enjoy R
units of goods.) Otherwise the large redemption will make it profitable for the rest of the patient household to run as well, making it impossible to sustain the direct redemption and fixed price/NAV regime. The solutions are (1) do not allow direct redemption and let the secondary market work, such as the ETFs; (2) allow direct redemption but with floating NAV to reflect the fair market price; and (3) allow direct redemption and stable NAV only if the liquidation loss is very little.