Question

Why do bond prices converge to par as the time to maturity gets shorter?

Answer 1

Bonds are brought by the investors either at a discount (when coupon rate is less than yield to maturity) or premium (when coupon rate is greater than yield to maturity) or at par (when coupon rate is equal to yield to maturity).
The issuer pays the face value or par value of the bond at maturity to the investors irrespective of whether the investor brought the bond at a discount or premium.
The value of a premium bond moves down to par value and the value of a discount bond moves up to become equal to the par value at maturity.
Example: Suppose a one year bond (of par value $1000) is issued at $980. Over the one year period, the bond value increases gradually from $920 to $1000. This is called as accretion of a discount bond.
Similarly, when a one year bond (of par value $1000) is issued at a premium of say $1020. Over the one year period, the bond value decreases gradually from $1020 to $1000. This is called as amortization of a premium bond.