In this post, I explain a simple but under-appreciated result about the dynamics of asset returns in response to changes in discount rates. The basic point is that increases in discount rates co-move negatively with current returns and positively with future returns. This implies that one change in the discount rate — a single increase or decrease — will induce a “reversing” or hump-shaped response in realized returns. Not only is this non-intuitive, but it complicates how we can use returns to measure discount rate shocks.
Suppose an asset operates at a steady state where it enjoys a constant expected return of $r_t = r$. That is, each period it achieves a return of, say, 5% that in a rational world reflects compensation for the risk it imposes on an investor’s portfolio. While this example speaks generally in terms of risky assets, the logic here will apply to interest rates on any asset — mortgage rates and house prices, interest rates and treasuries, etc.
Abstracting from idiosyncratic noise, the realized return of this asset (i.e. the actual change in prices) and expected return (i.e. another term for discount rate) are the same — 5% in our example. Since the price series must also grow at this rate, we achieve straightforward shapes for the three series:
Fig. 1 Expected Returns, prices, and realized returns with a constant discount rate of 5%. All dividends are paid past $t=20$.
Now suppose that one day, investors wake up and unanimously decide that the asset is more risky than they previously thought: they now demand 8%, rather than 5% return to compensate them for this risk. Importantly for this example, investors do not change their view on expected cashflows: that is, they expect the same dividends over time. How do expected returns, prices, and realized returns change?
Before looking at the immediate impact, let’s first consider the long run behavior. In the new steady state long run, the expected returns on the asset, by construction, are 8%: this is the compensation investors demand (and so must receive) to hold the asset. In other words, in the long run, the one-period realized returns will attain the same level as the new expected return, just as we saw in figure 1.
The interesting dynamics occur in the interim transition to this long run. At the time of the increase in discount rate, the price of the asset must drop. Intuitively, the asset is now less attractive since it is riskier, so demand falls. So the immediate response is a drop in price, i.e. negative realized returns. Combining these two effects, a single increase in discount rates causes negative short run returns and positive long run returns.
Fig 2. Expected returns, prices, and realized returns with a change in discount rates from 5% to 8% at $t=9$.
We can see this illustrated in Fig 2. The first panel, expected returns, restates the thought experiment: investors start out with discount rates of 5% that rise to 8% at $t=9$. The second panel shows the price response, which features a drop at $t=9$ when investors find the asset less attractive. Thereafter, returns must grow at the new level of 8% to compensate investors for returns. The rightmost panel of Fig. 2 illustrates the full dynamics of the return response, first turning negative then rising above the previous steady state.
To many economists, the dynamics described above should be obvious. So why write about it? The first answer is that this result, while straightforward, is perhaps non-intuitive. It’s strange to think that a one-time, uni-directional increase in a constant parameter — in this case increasing an asset’s discount rate — can induce a response that has different effects in the long and short run on the return of that asset. In simple models, we often think of single causes having effects in a single direction. Here, the reason for the observed response is that expected returns and future realized returns move in the same direction; expected returns and current realized returns move in the opposite direction. In fact, it is precisely because prices are low that future returns can compound at a greater rate. This is also the reason for the “non-intuitive” result that higher interest rates lead to lower returns on bond portfolios.