Why CAPM Does Not Determine Stock Returns from Interest-Rate Shocks
Summary
The note examines an apparent contradiction in using the CAPM to infer how stock returns respond to a change in the risk-free rate. Setting a broad market’s beta to one seems to imply that its expected return does not change when the risk-free rate moves, which conflicts with the intuition that higher rates can make equities less attractive and with the market decline cited during 2022.
The explanation is that the calculation treats the expected market return as fixed while changing the risk-free rate. CAPM relates an asset’s expected return to the risk-free rate and the market risk premium; it does not specify how the market return itself responds to a rate shock. If the excess market return is held constant, the expected market return rises with the risk-free rate, and prices may fall as required returns increase. The discussion does not provide an empirical estimate or a causal model for that relationship. It notes that a fuller analysis could model the risk premium or include additional factors.
Key ideas
- CAPM relates expected asset returns to the market risk premium and the risk-free rate.
- The model alone does not specify how the expected market return changes when interest rates move.
- Holding the market return fixed while changing the risk-free rate creates the apparent contradiction.
- Assuming a constant market risk premium implies higher expected returns when the risk-free rate rises.
- Empirical rate and equity relationships require assumptions beyond the basic CAPM.
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# Relationship between stock returns and interest rates from CAPM model
# Relationship between stock returns and interest rates from CAPM model
I am trying to derive the relationship between market prices of various assets and changes in risk-free interest rates for a study project. For bonds this is straightforward, you can just start from the discounted cash flow model and get a relationship like $1/r^t$, which perfectly makes sense from an empirical point of view. The next step is to to extend this to stocks. The easiest model for stock prices is probably the CAPM. If I start with the equation: $$ R_0=r_0+b(R_m-r_0) $$ and assume an exogenous interest rate shock $r_0=\Delta r_0$ and calculate the ratio $\Delta R_0/R_0$, I obtain: $$ \frac{\Delta R_0}{R_0} = \frac{\Delta r_0 (1-b) + b R_m}{\ \ \ r_0 (1-b) + b R_m}. $$ First of all, I want to derive how a very broad market index such as the S&P 500 changes when interest rates change. To do this, you can simply set $b\rightarrow1$ and you get $\Delta R_0/R_0=1$. That would mean that the broad market doesn't react to interest rate changes, but only a portfolio with higher or lower risk than the market would react. This result doesn't make any sense to me. Because when interest rates rise, equities become less attractive than bonds, demand falls and so do prices. And it can also be clearly shown empirically that the S&P 500 also fell sharply when interest rates rose sharply in 2022, for example. Can someone help me to identify the flaw in my approach?
## Answer by D Stanley (score 1)
https://quant.stackexchange.com/a/82316
The main flaw is that you are assuming that $R_m$ is constant with respect to $r$. The point of the CAPM model is to assume that there is a "market risk premium" which is defined as the difference between the expected market return and the risk-free rate. The expected return of a stock is modeled as a multiple of that risk premium plus the risk-free rate.
Put another way, the CAPM model is not designed to define the relationship between the expected return of stocks (or the market) and risk-free rates, but the relationship between the expected return of stocks and the expected return of the market. A change in risk-free rates would also affect the expected return of the market, but that relationship is not defined by this model.
If you treat the excess market return ($R_m - r_0$) as constant, then you see that the expected return of the market (and stocks) rises (and hence stock prices fall) as interest rates rise.
From there, you could extend your model to include multiple factors (betas) and to question whether the market risk premium is, in fact, constant.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.