Yield Differentials and Liquidity Benefits

I recently had an interesting discussion with the ever thought-provoking JP Koning in the comments section of his recent post.  The issue concerns a comment made by Stephen Williamson on why the yield on Treasury bills might be below that on reserves.  Williamson suggests that this could be because reserves are actually less liquid than T-bills.  This is based on the fact that T-bills can be held more widely - anyone can hold T-bills, but reserves may only be held by certain banks.  I think Koning and I broadly agree on the key issues here.

We might write the total yield (Y) on an asset (a) as the net of three items, the expected cash return (R) plus the liquidity benefit (L) less some valuation of the credit risk (C).

Y_a = R_a + L_a - C_a

Then we might suppose that in equilibrium the return on each asset will be equal - otherwise people will sell assets with lower yields and buy ones with higher yields.  So for T-bills (t) and reserves (h), we have:

R_t + L_t - C_t = R_h + L_h - C_h

Now, we can assume that the credit risk on T-bills and reserves is the same (C_t = C_h).  So, we can deduce the difference in liquidity returns from the difference in monetary returns:

L_t - L_h = R_h - R_t

As R_h is greater than R_t, this suggests that the marginal liquidity benefit on T-bills is greater than that on reserves.  Could this be because T-bills can be held more widely held?  Although, that might sound right, the problem is that when we think about liquidity of an asset, what we are thinking about is how easily that asset can be converted into "money".  For banks, that means reserves.  And how can any asset be more convertible into reserves than reserves themselves?

To resolve this, we first need to note that it is only reserve banks that are in a position to bring about the equilibrium implied from our equation.  Such a bank will derive both a higher cash return and a higher liquidity benefit from holding reserves rather than holding T-bills.  So if the bank should find itself holding T-bills for liquidity purposes, it would make sense to sell them and hold the reserves instead.

However, it can only sell the T-bills it actually holds.  Once it has sold these, the equation changes.   In principal, the bank can short T-bills by borrowing and selling them, but it then has to collateralise the stock loan.  The reserves cannot be used for collateralisation, so it has to obtain the collateral from elsewhere at a cost.  So the no-arbitrage condition is different for the long position to the short position, and we may well find that we end up in the corner solution, where reserve banks do not hold T-bills for liquidity purposes (although they may hold them for other purposes, for example as trading stock).

In that case, the price of T-bills gets determined by the no-arbitrage condition of other investors.  As those investors cannot hold reserves, the returns on reserves are irrelevant.  We therefore need to make the comparison with bank deposits (d).

R_t + L_t - C_t = R_d + L_d - C_d

Now, however, the credit risk cannot be taken to be the same.  T-bills will generally be seen as a better credit risk and T-bill yields will therefore normally be below that of deposits.  Furthermore, the excess reserve position means that banks will tend to pay deposit rates below that of the rate on reserves.  The net effect is a T-bill yield below the cash return on reserves.

So, it is certainly the case that the yield differential is a result of the fact that only certain parties can hold reserves.  However, this does not mean that reserves are yielding lower marginal liquidity benefits than T-bills for anyone in a position to hold both.