*An important question facing the FOMC today is how many fewer 25-basis-point hikes in the fund’s rate are appropriate given the withdrawal of accommodation from balance sheet normalization. We call this the “substitution effect,” how many fewer hikes would just offset the effect on aggregate demand of the projected rise in the term premium as a result of normalization.1 We attempt to answer this question using simulations with FRB/US. Jonathan Wright, the senior adviser to Monetary Policy Analytics, provided valuable input into our analysis. *

Funds Rate Equivalents

It is useful to be able to convert balance sheet policy into federal funds rate equivalents. For example, how many cuts in the fund’s rate did asset purchases substitute for when the fund’s rate was at its effective lower bound? Today, the question of interest to policymakers is how many 25-basis-point rate hikes to balance sheet normalization will substitute for. But a key question, apparently unsettled, is what “equivalent” should mean in this context.

The Fed’s Approach

Inside the Fed, there is a rule of thumb that is frequently used for translating balance sheet policy into funds rate equivalents. It is very simple—in our view, too simple. Yellen alluded to this rule of thumb earlier this year when she discussed how many 25-basis-point funds rate hikes should be forgone as a result of a projected rise in the term premium this year.2 She cited projections based on staff research that this effect would increase the term premium by about 15 basis points this year.3 That rule of thumb is based on a simple historical correlation between contemporaneous movements in short-term and longer-term rates: Short-term rates, on average, move three times as much as longer-term rates. We call this the three-to-one rule: A higher term premium can be offset by lowering the fund’s rate three times as much.4In this case, given the estimated 15-basis-point rise in the term premium from a diminished effect of the balance sheet, there should be about two fewer 25-basis-point hikes in the fund’s rate this year than would otherwise have been appropriate.

^{1}In this piece, when we say “the term premium” we are referring to the 10-year U.S. Treasury term premium. ^{2 }Chair Janet L. Yellen at the Stanford Institute for Economic Policy Research, Stanford University, Stanford, California, January 19, 2017: The Economic Outlook and the Conduct of Monetary Policy.

^{3 }Engen, Eric M., Thomas Laubach, and David Reifschneider (2015). “The Macroeconomic Effects of the Federal Reserve’s Unconventional Monetary Policies,” Finance and Economics Discussion Series 2015-005. Washington: Board of Governors of the Federal Reserve System, January, http://dx.doi.org/10.17016/FEDS.2015.005.

^{4 }By the way, the Board staff in the past has used a four-to-one rule of thumb. See, for example, Dave Reifschneider, John Roberts, and Jae Sim, “Incremental Balance Sheet Policies,” October 24, 2011, which was distributed to FOMC participants to inform their discussion of QE3.

Our Methodology: Ask FRB/US

In our view, a rule of thumb like this simply starts with the wrong question. It asks: What is the simple historical correlation between movements in short- and longer-term rates? But the right question is: What decline in the fund’s rate would just offset the effect of the projected increase in the term premium on aggregate demand? Answering this question requires the use of a macro model that provides the link between changes in the fund’s rate, changes in longer-term rates, and aggregate demand.5 We will use FRB/US, the large-scale structural model used by staff at the Fed.

The Effect of Balance Sheet Policy on the Term Premium

We have to start with an estimate of how balance sheet normalization is affecting the term premium. Here we use estimates from Fed staff research, which we suspect FOMC participants generally accept.6 The cumulative effect on the term premium gradually diminishes even before the size of the balance sheet declines as the duration of the portfolio declines, the size of the economy increases, and the expected time of the end of reinvestment and balance sheet normalization draws nearer. This diminishing effect continues until reserves

reach the minimum level consistent with the operating framework and the composition of the Fed’s asset holdings normalizes. The effect of asset purchases on the term premium—shown in Figure 1—is estimated to have been -100 basis points in 2016:Q4 and is projected to diminish to -50 basis points by 2020:Q4.

^{5}It might seem simpler to find the path of the fund’s rate in response to a rise in the term premium using a policy rule. But such prescriptions do not provide an estimate of the substitution effect. As we define it, the substitution effect is the decline in the fund’s rate that completely offsets the effect of a rise in the term premium on real GDP. Policy rules produce a decline in the fund’s rate that does not completely offset the shock. When we use a policy rule, we are, in effect, assuming that the FOMC does not know the source of or have an estimate of the size of the shock that lowers aggregate demand. It just responds to changes in economic variables such as the unemployment rate and inflation.

^{6 }Bonis, Brian, Jane Ihrig, and Min Wei (2017). “Projected Evolution of the SOMA Portfolio and the 10-year Treasury Term Premium Effect,” FEDS Notes. Washington: Board of Governors of the Federal Reserve System, September 22, 2017, https://doi.org/10.17016/2380-7172.2081. Here we use the estimates from the scenario with a higher level of longer-run reserves. The high-reserves scenario is consistent with our expectation that the Fed will opt to continue with its current operating framework of abundant reserves rather than return to a framework with scarce reserves. In any case, the estimated differences in term premium effects are fairly modest over the period of interest.

Simulation of Funds Rate Offsetting a Rise in the Term Premium

In our simulations with FRB/US, we assume the 10-year Treasury term premium rises by an amount consistent with the estimates in the high-reserves scenario in Bonis, Ihrig, and Wei (September 2017) of the diminishing negative effect of balance sheet policy. In FRB/US, there are also two other Treasury yields, the 5- and 30- year yields. For each of these points on the curve, we assume that the term premium increases by an amount proportional to the rise in the 10-year term premium, using ratios consistent with Board staff research.7 We employ VAR-based expectations for these simulations, rather than model-consistent expectations.^{8}

Table 1. Path of the Funds Rate Offsetting the Rise in the Term Premium

Impact | 2017:Q4 | 2018:Q4 | 2019:Q4 | 2020:Q4 |

Real GDP (%) | 0.0 | 0.0 | 0.0 | 0.0 |

Ten-year yield (bp) | 8 | 14 | 20 | 24 |

Term premium (bp) | 14 | 27 | 39 | 49 |

Funds rate (bp) | -21 | -34 | -44 | -52 |

Number of 25-bp cuts | 0.8 | 1.4 | 1.8 | 2.1 |

Note: Figures shown are the differences in the variables indicated relative to baseline. The difference in real GDP is expressed as a percentage change.

As shown in Table 1, the 10-year yield increases by less than the term premium because of a partially offsetting decline in the fund’s rate expectations component. To offset the rise in the term premium from the end of 2016 through 2020, there are two fewer 25-basis-point hikes, including one fewer in 2017. Since we’re already in the fourth quarter of 2017, this suggests one fewer hike (or a bit more) is warranted through the end of 2020 because of the effect of balance sheet normalization on the term premium.

With the higher path of the term premium, for the level of real GDP to be unchanged relative to baseline, the fund’s rate must be lowered by a slightly larger amount, with that amount varying over the simulation horizon. In 2017, a 25-basis-point lower funds rate is needed to offset a term premium that is about 15 basis points higher by the fourth quarter. By the end of 2020, the fund’s rate is just over 50 basis points lower, whereas the term premium is just under 50 basis points higher. This suggests the substitution effect depends on the horizon you’re looking at because it declines over time, from closer to two (twice as large a change in the fund’s rate to offset a given change in the term premium) to close to one after several years.

Bottom Line

There are many caveats, of course. As you know, we always take any model estimate with a grain of salt, maybe two grains in this case. But model-based empirical estimates can nevertheless be indispensable to inform our judgment, rather than to substitute for our judgment. These results are model-specific and indeed even specific to the choice of expectations formation we chose within FRB/US. Sorry, just the facts of life in the modeling world. Every step along the way is subject to considerable uncertainty, including the projected path of the term premium. But these results do strike us as reasonable and probably in line with what some

FOMC participants are thinking and assuming in their dots.

~~ ~~ ^{7}In particular, the ratios of the changes in the 5- and 30-year term premiums relative to the change in the 10-year term premium are 0.8 and 0.35, respectively.

^{8 }This means that, within the model, expectations of agents for various economic variables are based on projections from small-scale VAR models that will generally differ from the actual simulated values.

Appendix

An Analytical Expression for the Substitution Effect

Now we turn to try to quantify these factors that determine the size of the substitution effect. We start with a basic equation describing a situation in which the effect on aggregate demand of a change in longer-term rates is exactly offset by a change in the fund’s rate so that the level of real GDP is unchanged:

(1) m^{TP}dTP = −m^{r}dr ,

where MTP is the term premium multiplier (percent change in the level of real GDP for a given change in the term premium), DTP is the change in the term premium, MRIs the fund’s rate multiplier, and dr is the change in the fund’s rate.^{9}

Then we rearrange (1) to get the decline in the fund’s rate that just offsets the effect on aggregate demand of the projected increase in the term premium. The ratio of that change in the fund’s rate to the change in the term premium is what we call the substitution effect:

Multipliers for a Change in the Term Premium and a Change in the Funds Rate

Equation (2) shows that what determines the size of the substitution effect is the ratio of the multipliers for the term premium and the fund’s rate. The multiplier for the term premium, MTP, is fairly straightforward.10 The corresponding multiplier for a change in the fund’s rate, Mr, is more complicated because part of the effect of changes in the fund’s rate comes through the effect on longer-term rates. We define α to be the degree of passthrough of a change in the fund’s rate to longer-term rates—the ratio of a change in longer-term rates in response to a change in the fund’s rate to the change in the fund’s rate. The fund’s rate multiplier (Mr) depends on α, the multiplier for long-term rates (MTP), and mr̃, which we define to be the multiplier for the effect of the fund’s rate holding constant longer-term rates:

(3) m^{r }= αm^{TP }+ m^{r̃}.

Substituting (3) into (2), we get the expression for the substitution effect in terms of just two parameters, α and β:

β represents the relative importance of the fund’s rate relative to longer-term rates in the transmission mechanism. The implication is that, the more complete the passthrough from a change in the fund’s rate to longer-term yields (α), and the larger the relative importance of short-term rates in the transmission mechanism (β), the more smaller the size of the substitution effect.

Putting It All Together: Quantifying the Size of the Substitution Effect

In Table A.1 we show estimates of the substitution effect as well as the estimated multipliers and coefficients that determine those figures, as discussed previously. For simplicity, the expressions outlined in this paper are for a case where there’s one long-term rate and one short-term rate (the fund’s rate). In FRB/US, as we alluded previously, there are also 5- and 30-year Treasury yields, in addition to the 10-year yield. Multipliers for these three yields vary, as does the passthrough from the fund’s rate to each of these yields. Furthermore,

^{9 }For simplicity, the expressions outlined here are for a case where there’s one long-term rate and one short-term rate (the fund’s rate). 10 In FRB/US, the term premium and expectations components of a Treasury yield don’t have any direct impact on other variables in the model; their only impact is through the Treasury yield. So there aren’t separate multipliers for each of these variables.

the effects of balance sheet normalization on the term premium are also different for each point on the curve. Here, however, we stick to the simple case for illustrative purposes. As it turns out, the estimates of the substitution effect looking only at the 10-year yield are not far off from the more detailed case. All estimates are from simulations with FRB/US using VAR expectations.

Note: Figures for the term premium and funds rate correspond to the years 2017-2020. The substitution effect ratio is simply applied to the term premium estimates from Bonis, Ihrig, and Wei (September 2017) to get a translation into funds rate cuts.

To estimate β, we ran two simulations. The first was for the multiplier for the term premium, MTP, which is fairly straightforward. We simulated with the path of the 10-year Treasury yield 100 basis points above baseline, holding other points on the yield curve as well as the fund’s rate so that they are unchanged relative to baseline. This multiplier will reflect the full *transmission mechanism*, that is, the effects of a rise in longer

term Treasury yields on the financial conditions that directly affect aggregate demand, including yields on private securities, equity, and home prices, and the exchange rate. The second was an exogenous change in the fund’s rate, holding constant all longer-term rates in FRB/US; that gave us mr̃. In Table A.1, for α we show the change in the 10-year Treasury yield in response to an exogenous change in the fund’s rate divided by the change in the fund’s rate, with VAR expectations.