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Can someone please explain the below two statements with their logic?
Statement 1: If the spot curve is upward sloping, par rates will be near—but below—spot rates, particularly at the long end of the curve.
Statement 2: When spot rates are negative while the spot rate curve is upward sloping. In such instances, par rates lie above spot rates.
Statement 1: “If the spot curve is upward sloping, par rates will be near—but below—spot rates, particularly at the long end of the curve.”
Spot Curve: The spot curve represents the yields of zero-coupon bonds of different maturities plotted against their respective maturities. An upward sloping spot curve means longer-term yields are higher than shorter-term yields.
Par Rates: Par rates refer to the interest rates at which the present value of a bond’s cash flows equals its par value. In other words, par rates represent the yields at which bonds are issued at par (their face value).
Logic: In an upward-sloping spot curve, longer-term yields are higher. However, par rates are determined based on the average of the spot rates over the term of the bond. Since par rates are calculated based on longer-term yields, they will be near but slightly below the spot rates, especially at the longer end of the curve. This is because par rates average out the higher long-term yields with the lower short-term yields, resulting in rates that are slightly lower than the longest-term spot rates.
Statement 2: “When spot rates are negative while the spot rate curve is upward sloping. In such instances, par rates lie above spot rates.”
Negative Spot Rates: Negative spot rates occur when the yields of zero-coupon bonds are negative. This means investors are willing to accept a guaranteed loss in exchange for the safety of certain assets.
Upward Sloping Spot Curve: As mentioned earlier, an upward sloping spot curve means longer-term yields are higher than shorter-term yields.
Logic: In instances where spot rates are negative but the spot rate curve is upward sloping, it implies that shorter-term rates are even more negative than longer-term rates. Par rates, on the other hand, are calculated based on the average of these rates over the bond’s term. Since par rates consider the average of these negative rates, they will be higher than the negative spot rates. This is because the averaging process would dilute the impact of the most negative short-term rates, resulting in par rates lying above the spot rates.