How an Inverted Yield Curve Implies Lower Forward Rates

# Inverted Yield Curve


# Inverted Yield Curve












In an article of FT today, Matthew Klein writes, "The yield curve represents the cost of borrowing over different amounts of time. Lenders generally prefer getting their money back sooner rather than later, so short-term debts tend to have lower interest rates than longer-term obligations. The curve is “inverted” when short-term interest rates are higher than long-term rates. This happens when traders believe short-term interest rates will be lower in the future than they are today."

I don't understand the final statement in the context of an inverted yield curve. If traders believed short term interest rates would be lower in the future wouldn't the current curve reflect that?

## Answer by phdstudent (score 4)

https://quant.stackexchange.com/a/38967

Usually when the yield curve is inverted that means that traders believe that interest rates are likely to decrease in the future.

To see that, notice that under no arbitrage, the following relationship must be true:

\begin{equation} (1+r_{t,t+2})^2 = (1+r_{t+1})(1+f_{t+1,t+2}) \end{equation}

where $r_{t,t+n}$ is the interest rate between today and $n$ years from now and $f_{t+1,t+2}$ is the forward rate between year 1 and 2.

If the yield curve is inverted then that means that:

\begin{equation} (1+r_{t,t+2}) < (1+r_{t+1}) \end{equation}

which immediately implies that:

\begin{equation} (1+f_{t+1,t+2}) < (1+r_{t+1}) \end{equation}

which mean that the forward rate is lower than today's interest rate. We also tend to think that forward rates are our best expectations of future interest rates (this is true under mild assumptions in many models), meaning that:

\begin{equation} E_t[r_{t+1,t+2}] = f_{t+1,t+2} < r_{t+1} \end{equation}

Therefore we expect interest rates tomorrow to be lower as the statement you read on the FT implies.

## Answer by NBF (score 2)

https://quant.stackexchange.com/a/38976

The first reasonable attempt to link the yield curve shape to economics and expectations was by Campbell Harvey in his PhD thesis in 1986. ("The Real Term Structure and Consumption Growth", JFE, #22 1988, p. 305-333 link) He used a Consumption CAPM framework over multiple horizons. In C-CAPM, one usually gets risk free rates (see Cochrane on Asset Pricing) are given by $$r(t)\propto a+b_1 \cdot g- b_2 \sigma^2(c)$$ where consumption is assumed to be lognormal, $c_t=c_0e^{g+\sigma\epsilon}$ for $\epsilon \sim N(0,1)$. Risk-free rates are proportional to growth, and will be decreased when consumption uncertainty is high (also called precautionary saving). Here all expectations are taken over the $P$-measure (the physical or subjective measure).

If we take this same result over multiple horizons, we get the result $$r(t_2)-r(t_1)\propto a+b_1 \cdot (g_2-g_1)-b_2 \cdot (\sigma^2_2-\sigma^2_1).$$ We ignore the vol terms and essentially, slopes are steep when growth expectations are higher in the future than they are now (i.e., during a recession), and slopes are flat when growth begins to falter (i.e., the peak of the cycle). Harvey's theoretical paper led to the later inclusion of the yield-curve slope into the set of leading economic indicators.

Less theoretically speaking, yield curves flatten when there are no further yield-buying opportunities, when all other risky assets have rallied and there is no more juice in them. Long dated yields are the last to rally. This will reduce the premium, but at the same time, long-dated yields usually do have a premium (i.e., future short rates will likely fall and long-dated bonds will still offer a premium over investing in short-rates and rolling).