Friday, 14 June 2013

Modelling QE

This post outlines a theoretical model I've used for looking at the mechanical process by which QE impacts on the economy.  It is based on the UK economy and draws heavily on Tobin's portfolio approach to financial markets.*

The model is best understood by examining the balance sheet matrix on which it is based, a stylised version of the UK's financial balance sheet.  Although highly simplified, it still retains sufficient granularity to bring out the required mechanics.

PNFC securities


RoW securities







PNFC = private non-financial corporation
MFI = monetary financial institution (bank)
BoE = Bank of England
RoW = rest of world 

Asset prices and yields are driven by portfolio allocation functions for Households and RoW.  These are based on Tobin's, providing a demand for each asset based on the yields on all assets and an overall budget constraint.  The model then solves by bringing the market for each asset class into equilibrium, within the balancing constraints of the matrix. Three "price" variables need to be found: the price of private securities, the price of gilts and the exchange rate.  As each of these variables affects expected future returns, there is a high degree of inter-dependency.

The financial model sits within a simple Post-Keynsian style model of the real economy.  This is only intended to map out the long-run response path of the economy, which provides the feedback for expectation formation. Expectations place a crucial role in the model, and therefore issues such as whether the QE is expected to be temporary or permanent have a significant impact on the results.  Expectations are initially based on rule-of-thumb estimates, but the model includes an algorithm which enables yield expectations to be iteratively mapped to actual outcomes out to a distant future point.

Because the model is based on portfolio preferences, the spread between single period asset yields varies with the supply of those assets.  This means that traditional arbitrage model (whereby long term rates are a compound of expected short-term rates) does not in general hold, although there is a fair degree of correlation. Yields and prices can vary even in scenarios where the short-term rate is permanently fixed and even when the QE is only a temporary measure.

The structure of the model is explained in a little more detail below. 


All sectors of the actual economy are represented.  Financial institutions that are not MFIs are treated as part of the household sector.  The BoE is treated separately from private MFIs.  Public corporations are treated as part of Government.


Marketable securities are grouped into three classes: PNFC securities (equity and debt securities of UK PNFCs), gilts and RoW (rest of world) securities.  The latter includes all foreign equity and bonds, whether private or issued by foreign governments.

The value of each of these asset classes depends on some measure of price.  For PNFC securities and gilts, value is the number of securities multiplied by a price index.  For RoW securities, the value is the (exogenous) foreign currency denominated value divided by the exchange rate.

The last three asset classes: deposits, loans and reserves do not have a variable price.  The return on each is based on an interest rate which is assumed to be determined exogenously by central bank policy.  Deposits from RoW is intended to represent a net figure, i.e. sterling deposits from abroad less sterling loans / deposits placed abroad.

In the real world, UK MFIs have substantial non-sterling assets and liabilities (greater than their sterling positions) but these are assumed to be matched and therefore ignored.  Likewise, non-sterling liabilities of PNFCs are ignored.  PNFC cash is netted off against loans.

Portfolio Allocation Functions

Household demand for each asset is based on a portfolio allocation function, using single period expected rates of return.  RoW has as demand functions for PNFC securities and gilts based on single period expected returns in the three marketable security classes and trend GDP.  Demand for sterling deposits is based on trend GDP and the relation between the (exogenous) foreign interest rate and the expected equivalent return on sterling deposits after expected exchange rate movements.

The functions require expected next period values for: the distribution on PNFC securities, the price of PNFC securities, the price of gilts and the exchange rate.  The same exchange rate expectations are used for expected deposits returns and for expected return on RoW securities.


QE simulations involve an exogenous increase in gilts held by the BoE, matched by an increase in reserves. This increase in demand for gilts (or reduction in supply available to the rest of the market) increases the price and reduces the expected yield.  Portfolio re-balancing by households and RoW leads on to higher equity prices and a lower exchange rate.

Market prices respond immediately (even beforehand if expectations are allowed to reflect prior knowledge of the purchase).  The impact on the real economy is slower.  This involves (within the model) private sector spending response to increased wealth and reduced long-term rates, as well as exchange rate effects.  The increase in reserves is not modelled as having any direct impact on lending, as MFIs are not treated as reserve constrained.

* A General Equilibrium Approach to Monetary Theory, J. Tobin, Journal of Money, Credit and Banking (Feb, 1969)


  1. For trying to work out the effects of QE, I guess it is vital to know the "value" of holding safe zero yield bank deposits. I suppose zero yield short term T-bills are equivalent to safe zero yield bank deposits. For both I guess their value is all about them being an "option" to buy any other asset at any time. As such their value is totally dependent on the price volatility of all of the other assets. If any of the other assets is very volatile, then cash will be a great thing to hold because it will allow you to buy the volatile asset whenever it is cheap so that it can subsequently be sold when it is expensive. But of course it is hard to know what volatility will be. But that same issue is true for pricing any option I guess.

  2. I suppose it is not just the volatility of all other assets that sets the "value" of zero yield "cash" (safe bank deposits or short term treasury debt) as an asset class; it is also the extent to which the volatility of the other assets is syncronized between those asset classes. If one could be sure that say the FTSE100 and 50year gilts had price fluctuations that allways moved in opposite directions, then cash would not be such a sensible thing to hold because if say stocks fell in price, long term gilts could be sold to reballance and vice versa so why bother holding zero yield cash. The value of "cash" comes because such relationships are ficle. Cash provides a sure thing reserve for reballancing whenever everything else crashes in sync.

    I sometimes get the impression that these things work out a bit like the witches' prophacies in Macbeth. Basically perhaps zero yield "cash" has a value that is pegged by the real economy as being so much real goods and services. It also has a value as an asset class that is set as being its value as an "option". Both values must coincide so the volatility of other assets takes on a character that is consistent with the "option" value.

    Perhaps what we get from QE is syncronous bubbles and crashes across asset prices :)