If the treasury yield curve is downward sloping, how should the yield to maturity on a 10-year treasury coupon bond compare to that on a 1-year t-bill?

The yield to maturity of the 10-year Treasury coupon bond would be lower.

Explanation:

To get an idea of how the yield to maturity would differ between both the mentioned instruments, we need to understand what the treasury yield curve is, and what inferences we can draw from the slope of this yield curve.

The yield curve is simply a line on a graph with the y-axis depicting interest rates (yield on the instrument) and the x-axis depicting the time to maturity. The graph is plotted by marking interest rates applicable on instruments of different maturities. The slope of the resultant curve allows people to have an idea of any economic upswings and downturns in the offing, and on the general trend of interest rates in a given economy.

Now, the yield curve can theoretically take any 3 shapes: upward sloping, downward sloping, or flat. In the question at hand, the yield curve is downward sloping. This means that the yield (the interest rate) on a lower maturity instrument (a plot on the curve that is higher on the y-axis and lower on the x-xis) is higher than the yield on a longer term instrument (a plot on the curve that is lower on the y-axis and higher on the x-axis). Therefore, if the curve is downward sloping, the yield to maturity on a 10-year bond would be lower than the yield on a 1-year T-bill.

Note that the curve is generally upward sloping since the risk of a longer maturity instrument should generally be higher and therefore the yield should correspondingly be higher to compensate for the additional risk. The downward sloping curve would indicate that there is an economic recession that is expected which is driving down the yield on longer term bonds. In this case, investors would purchase longer term bonds early on to cash in on the higher yield before it goes down in the future. This would of course raise the price of long term bonds with respect to shorter term instruments but would have a drive down the yields correspondingly.  

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