It is an article of faith on Wall Street, and in much of the press, that when the Fed lowers interest rates, it sparks a “reach for yield,” a “bubble,” a decline in the risk premium of other assets including stocks and houses. Not so, says a recent excellent paper,

The correlation between bond and equity risk premia was negative during the Nagel & Xu sample from 1995 to 2022. The consensus of the existing literature is that it was positive in the pre-1995 sample. (Of course, the Fed only began providing after-meeting decision statements in 1994.) Also, we know that the results of the Campbell-Shiller decomposition are very sensitive to the predictor variables included in the VAR. Are the predictors the same as in Bernanke-Kuttner? And why don’t Nagel & Xu use the orthogonalized monetary policy surprises from Bauer & Swanson? You should be able to do this starting in 1988. (See Table 3 here: https://www.michaeldbauer.com/files/mps.pdf. The R2 is about 30%.)

Finally, Cieslak and McMahon have a recent paper that resolves some of inconsistencies you mentioned by looking at forward guidance and the effect on risk premia in intermeeting periods: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4560220.

Reading the paper, I was very confused by something. Yes, an interest rate shock changes p and p_B by roughly the same amount on average. However, as far as I could tell, changes in p aren't highly correlated with changes in p_B.

If changes in p_B are driving most of the movement in stock prices on FOMC days, shouldn't these be highly correlated? That is, I want to see that variation in p_B accounts for most of the variation in p on FOMC days.

This is why I love classical economics. It turns wives tales on their head. I have read through this post three times to try and grasp exactly what is going on.

One thing I experienced, but haven't seen studied was how risk preferences radically changed with 0% interest rates persisting for years. (It works in practice, but does it work in theory?)

This study looks at a short term shock and frankly, it's fascinating to someone who traded for a living. I was trading spreads in Eurodollar futures back in 2001. My experience from 2009-2021 was that with zero to very low interest rates, valuations of riskier assets increased significantly faster. For example, startup valuations. Intangible assets like cryptocurrencies saw big run ups in valuation. Also, riskier stocks appreciated seemingly faster than less risky stocks. Tesla vs GM. In liquid stocks, zero interest rates cannot account for all of the demand since future expectations are also priced in. When rates started going higher, the air came out of a lot of balloons.

The price of a perpetuity -- a bond that pays the same coupon forever, no principal -- is p = 1/r. The 1/ is nonlinear. So when interest rates are very low, the same bp change in interest rate has a larger effect on the asset price and rate of return than it does when rates are high. This observation strikes me as about long-duration assets in general, not about a risk premium in stocks vs long term bonds.

Just to add to JC, the Tesla v GM example is also exactly what you expect if Tesla is riskier and therefore was discounted more to begin with.

To take an exceedingly simple example, image two companies that pay no dividends now but starting in 5 years they will both pay $1 per period forever. However, there is a probability p(x) that the dividend will not materialize, at the end of the 5 years when the first dividend is meant to be paid all uncertainty is resolved, if the first payment is made then all future payments are. Company A is riskier than company B if p(A)>p(B).

The uncertainty can be accounted for in the price by simply saying the expected cashflow is lower and discounting at the risk-free rate or, completely equivalently, we can use evaluate expectations with respect to the risk neutral measure and account for the extra risk entirely by a higher discount rate. I stress the approaches are equivalent but the second approach makes the point easier to see.

In the risk neutral measure the initial price of company is A is lower because it is discounted at a higher rate. After say 3 years the price of A has appreciated more than B simply because instead of 5 years of discounting the same expected cashflow at a higher rate we now only discount at a higher rate for 2 years. The extra return is the compensation for risk and comes from starting out at a lower price.

Actually that doesn't quite work. Instead it should say that p(A) and p(B) are the same unconditionally but not conditional on something like aggregate consumption growth. A is riskier than B if p(A) has higher correlation with consumption growth. Then the last paragraph goes unchanged.

In the present context (i.e., the Nagel & Xu paper vs. the Bernanke & Kuttner paper) where the securities are not riskless securities, we should consider what the variation of a risky perpetuity would be when the risk-free rate is perturbed by a small increment. In the case of a risky perpetuity, the discount rate is given by 𝑟ₚ > risk-free rate 𝑟 ᶠ.

If the perpetuity price Pₜ = 1/𝑟ₚ , as it would be if the perpetuity is not callable, then

a small variation in the risk-free rate would give

Let the perpetuity be a risky security, non-callable as stated, then 𝑟ₚ = 𝑟 ᶠ + 𝛽∙(𝘳 ᵐ – 𝑟 ᶠ ) where 𝘳 ᵐ is the rate of return on the market portfolio, 𝑟 ᶠ is the risk-free rate, and

Separating the PDE into its constituent parts: ∂Pₜ/∂𝑟 ᶠ = – [Pₜ /𝑟ₚ] – [Pₜ∙ 𝛽∙(∂𝑟ₘ/∂𝑟 ᶠ – 1)/ 𝑟ₚ].

The first term on the RHS of the equals sign is the answer that would be par for a riskless non-callable security. The second term on the RHS is the added effect when the non-callable security is not riskless, i.e., 𝛽≠0.

Nagel & Xu conclude that ∂Pₜ/∂𝑟 ᶠ = – [Pₜ /𝑟ₚ], finding in their regression analyses no support for inclusion of the term – [Pₜ∙ 𝛽∙(∂𝑟ₘ/∂𝑟 ᶠ – 1)/ 𝑟ₚ]. Bernanke & Kuttner find the second term to be more determinative. The difference between the results of the two papers is likely to be found in the construction of Nagel & Xu’s Equations (1), (2), and (3). Specifically, in (3) where the term Gₙₜ is a function of Bₙₜ , i.e., Gₙₜ = 𝑓(Bₙₜ) ∝ 1/Bₙₜ.

For those who might have an interest in ‘resolving’ the controversy between postulate of Nagel & Xu and that of Bernanke & Kuttner, this might be a fruitful avenue of investigation.

I like posts like these.

The correlation between bond and equity risk premia was negative during the Nagel & Xu sample from 1995 to 2022. The consensus of the existing literature is that it was positive in the pre-1995 sample. (Of course, the Fed only began providing after-meeting decision statements in 1994.) Also, we know that the results of the Campbell-Shiller decomposition are very sensitive to the predictor variables included in the VAR. Are the predictors the same as in Bernanke-Kuttner? And why don’t Nagel & Xu use the orthogonalized monetary policy surprises from Bauer & Swanson? You should be able to do this starting in 1988. (See Table 3 here: https://www.michaeldbauer.com/files/mps.pdf. The R2 is about 30%.)

Finally, Cieslak and McMahon have a recent paper that resolves some of inconsistencies you mentioned by looking at forward guidance and the effect on risk premia in intermeeting periods: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4560220.

Great links. Thank you!

Excellent, and I love the decomposition analysis (of anything really).

Reading the paper, I was very confused by something. Yes, an interest rate shock changes p and p_B by roughly the same amount on average. However, as far as I could tell, changes in p aren't highly correlated with changes in p_B.

If changes in p_B are driving most of the movement in stock prices on FOMC days, shouldn't these be highly correlated? That is, I want to see that variation in p_B accounts for most of the variation in p on FOMC days.

This is a very good point. Wish I'd thought of it!

This is why I love classical economics. It turns wives tales on their head. I have read through this post three times to try and grasp exactly what is going on.

One thing I experienced, but haven't seen studied was how risk preferences radically changed with 0% interest rates persisting for years. (It works in practice, but does it work in theory?)

This study looks at a short term shock and frankly, it's fascinating to someone who traded for a living. I was trading spreads in Eurodollar futures back in 2001. My experience from 2009-2021 was that with zero to very low interest rates, valuations of riskier assets increased significantly faster. For example, startup valuations. Intangible assets like cryptocurrencies saw big run ups in valuation. Also, riskier stocks appreciated seemingly faster than less risky stocks. Tesla vs GM. In liquid stocks, zero interest rates cannot account for all of the demand since future expectations are also priced in. When rates started going higher, the air came out of a lot of balloons.

The price of a perpetuity -- a bond that pays the same coupon forever, no principal -- is p = 1/r. The 1/ is nonlinear. So when interest rates are very low, the same bp change in interest rate has a larger effect on the asset price and rate of return than it does when rates are high. This observation strikes me as about long-duration assets in general, not about a risk premium in stocks vs long term bonds.

Just to add to JC, the Tesla v GM example is also exactly what you expect if Tesla is riskier and therefore was discounted more to begin with.

To take an exceedingly simple example, image two companies that pay no dividends now but starting in 5 years they will both pay $1 per period forever. However, there is a probability p(x) that the dividend will not materialize, at the end of the 5 years when the first dividend is meant to be paid all uncertainty is resolved, if the first payment is made then all future payments are. Company A is riskier than company B if p(A)>p(B).

The uncertainty can be accounted for in the price by simply saying the expected cashflow is lower and discounting at the risk-free rate or, completely equivalently, we can use evaluate expectations with respect to the risk neutral measure and account for the extra risk entirely by a higher discount rate. I stress the approaches are equivalent but the second approach makes the point easier to see.

In the risk neutral measure the initial price of company is A is lower because it is discounted at a higher rate. After say 3 years the price of A has appreciated more than B simply because instead of 5 years of discounting the same expected cashflow at a higher rate we now only discount at a higher rate for 2 years. The extra return is the compensation for risk and comes from starting out at a lower price.

Actually that doesn't quite work. Instead it should say that p(A) and p(B) are the same unconditionally but not conditional on something like aggregate consumption growth. A is riskier than B if p(A) has higher correlation with consumption growth. Then the last paragraph goes unchanged.

In the present context (i.e., the Nagel & Xu paper vs. the Bernanke & Kuttner paper) where the securities are not riskless securities, we should consider what the variation of a risky perpetuity would be when the risk-free rate is perturbed by a small increment. In the case of a risky perpetuity, the discount rate is given by 𝑟ₚ > risk-free rate 𝑟 ᶠ.

If the perpetuity price Pₜ = 1/𝑟ₚ , as it would be if the perpetuity is not callable, then

a small variation in the risk-free rate would give

∂Pₜ/∂𝑟 ᶠ = – (∂𝑟ₚ/∂𝑟 ᶠ )/𝑟ₚ² = – Pₜ∙(∂𝑟ₚ/∂𝑟 ᶠ )/𝑟ₚ .

Let the perpetuity be a risky security, non-callable as stated, then 𝑟ₚ = 𝑟 ᶠ + 𝛽∙(𝘳 ᵐ – 𝑟 ᶠ ) where 𝘳 ᵐ is the rate of return on the market portfolio, 𝑟 ᶠ is the risk-free rate, and

𝛽 = ℂovₜ[Pₜ , Pₘ]/√𝕍arₜ(Pₜ)∙𝕍arₜ(Pₘ).

Hence, ∂𝑟ₚ/∂𝑟 ᶠ = 1 + 𝛽∙(∂𝑟ₘ/∂𝑟 ᶠ – 1), giving ∂Pₜ/∂𝑟 ᶠ = – Pₜ/𝑟ₚ∙[1 + 𝛽∙(∂𝑟ₘ/∂𝑟 ᶠ – 1)].

Separating the PDE into its constituent parts: ∂Pₜ/∂𝑟 ᶠ = – [Pₜ /𝑟ₚ] – [Pₜ∙ 𝛽∙(∂𝑟ₘ/∂𝑟 ᶠ – 1)/ 𝑟ₚ].

The first term on the RHS of the equals sign is the answer that would be par for a riskless non-callable security. The second term on the RHS is the added effect when the non-callable security is not riskless, i.e., 𝛽≠0.

Nagel & Xu conclude that ∂Pₜ/∂𝑟 ᶠ = – [Pₜ /𝑟ₚ], finding in their regression analyses no support for inclusion of the term – [Pₜ∙ 𝛽∙(∂𝑟ₘ/∂𝑟 ᶠ – 1)/ 𝑟ₚ]. Bernanke & Kuttner find the second term to be more determinative. The difference between the results of the two papers is likely to be found in the construction of Nagel & Xu’s Equations (1), (2), and (3). Specifically, in (3) where the term Gₙₜ is a function of Bₙₜ , i.e., Gₙₜ = 𝑓(Bₙₜ) ∝ 1/Bₙₜ.

For those who might have an interest in ‘resolving’ the controversy between postulate of Nagel & Xu and that of Bernanke & Kuttner, this might be a fruitful avenue of investigation.