## Thursday, 9 January 2014

### 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).

Ya = Ra + La - Ca

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:

Rt + Lt - Ct = Rh + Lh - Ch

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

Lt - Lh = Rh - Rt

As Rh is greater than Rt, 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).

Rt + Lt - Ct = Rd + Ld - Cd

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.

1. Good post Nick, I agree with most of it.

"T-bills will generally be seen as a better credit risk and T-bill yields will therefore normally be below that of deposits."

I agree on the point about credit risk. But what sort of deposits are we talking about here? A savings deposit will normally yield more than a t-bill, but a chequing deposit will normally yield less than both the equivalent term deposit and t-bill. I'd attribute this to Cd (credit risk on a chequing deposit) being higher than Ct but Ld being higher than Lt.

1. Thanks, and I appreciated our exchange on this on your blog.

We don't want to compare with term deposits or those with notice periods, because that then obviously has implications for the liquidity. If we are comparing with people that might be significant holders of T-bills, then maybe Fed Funds deposits are the best comparison. And yes, I'd say Cd > Ct and Ld > Lt.

2. Just brainstorming a bit... So the t-bills:reserves margin is in a disequilibrium of sorts because t-bills cannot be shorted in sufficient quantity and the t-bills:deposits margin is in equilibrium, but what about the reserves:deposits margin? If the total return (Y) on t-bills is less than that on reserves but equal to the return on deposits, by transitivity doesn't that mean that deposits yield less than reserves? If so, shouldn't the reserves:deposits margin be arbitrageable by the banks that participate in that market?

3. First of all, I'm not sure that we can say "the total return (Y) on T-bills is less than that on reserves but equal to the return on deposits. The "total return on bills" is not the same in each case because it's different people holding reserves to those holding deposits. Clearly the cash return on bills is the same to everyone, but I don't think we can say anything about the marginal liquidity benefit or the credit cost. (It's like the idea that we can say that marginal utility over price is equal over all goods for the same household, but not generally equal between households.) So I don't think the transitivity applies.

That said, it would seem to be the case that the liquidity yield to a bank will be less on a deposit with another bank than it will on reserves and the credit disutility will be higher, so the deposit rate would need to be above the reserve rate for reserve banks to want to be prepared to place deposits with other banks. As it is not (in the current situation), we should conclude that banks will not want to deposit with other banks.

If the deposit rate is significantly below the reserve rate, then banks would want to take in more deposits and hold more reserves, which will drive the deposit rate up. As there is no liquidity cost and little credit risk in this arbitrage, we would expect deposit rates to get quite close to the reserve rates. There are still likely to be some frictions though which would keep a small spread.

Does that sound right to you? I'm not 100% on this, but that seems to me how it break down, more or less.

2. " Clearly the cash return on bills is the same to everyone, but I don't think we can say anything about the marginal liquidity benefit or the credit cost...So I don't think the transitivity applies."

Although I completely agree that the marginal liquidity benefit thrown off by an asset is not the same to everyone, as I mentioned earlier on my blog I still think the market arrives at a single price that it is willing to pay for a certain asset's liquidity services, and therefore there exists a single liquidity return. It's on this basis that people can compare liquidities across assets and transitivity may hold. But I'm brainstorming here. I recall you were skeptical of the ability to strip out the price of liquidity.

I agree with your point that the deposit rate would need to be above the reserve rate, and if not, that rate would be driven back up towards the reserve rate. I do find it difficult to think about this since we're discussing an asset that has a limited ownership base, and that seems to be throwing a wrench in my brain gears.

1. I'll have to look at the earlier posts you mentioned. Yes, my sense is that it's not possible to complete isolate the liquidity benefits, particularly when the marginal benefits are small and more frictional costs come into play, but it's just a gut feel, rather than anything I can prove.

The fact that reserves have limited ownership is critical here, as is the fact that credit risk is asymetric - a cost to the lender, but not a benefit to the borrower. I can't see how anyone can explain the actual breakdown of current rates without factoring it in.

3. My understanding is that this is due to the GSEs (Fannie, Freddie). They cannot hold reserves, but have massive cash balances. Everybody knows this, and prices are adjusted accordingly. (I was not involved in US money markets, so my knowledge is hazy.) Nobody is really big enough to arb this away.

1. That sounds about right. I also have no direct experience in US money markets, so I'm just saying what I'd expect to see, rather than what I know for sure is happening.