Final answer:
The relationship between annual returns and yield to maturity on a zero-coupon Treasury bond is based on the total returns from the discounted purchase to the face value at maturity. Rising interest rates result in lower prices for existing bonds, whereas falling rates cause them to appreciate. A zero-coupon bond’s effective yield reflects these dynamics over the bond's lifetime.
Step-by-step explanation:
When you buy a zero-coupon Treasury bond for $800 with a $1,000 face value, the yield to maturity (YTM) is a function of the discounted purchase price, the face value at maturity, and the time to maturity. The relationship between the annual returns and the yield to maturity lies in the concept that YTM encompasses total returns, including both capital gains and interest income (implied in the case of zero-coupon bonds since there are no periodic interest payments), over the bond's life.
Should interest rates rise, existing bonds with lower rates become less attractive, leading to a discount in their pricing. Conversely, if rates fall, bonds with higher coupon rates become more valuable, selling for a premium. In this case, the bond yields a 12% return due to the price difference from purchase to maturity. The coupon rate remaining at 8% means that if the bond had regular interest payments, it would pay this rate, but its true earnings are influenced by prevailing interest rates and time until maturity.