This statement is true only if the investor holds the bond until maturity. Reinvestment risk means that a bond investor risks having to reinvest bond cash flows (both coupon and principal) at a rate lower than the promised yield. Reinvestment risk increases with longer maturities and higher coupons, and decreases for shorter maturities and lower coupons. While a bond investor can eliminate price risk by holding a bond until maturity, he usually cannot eliminate bond reinvestment risk. One exception is zero-coupon bonds, since these bonds deliver payments in one lump sum at maturity. There are no payments over the life to reinvest.
The statement, "Long-term bonds should be purchased if the investor anticipates higher reinvestment rates," should read, "Short-term bonds...".If an investor expects interest rates to rise, he would want a bond with a shorter maturity so that he received his cash flows sooner and could reinvest at the higher rate. Also, there is less prepayment risk with shorter maturities.
The statement that begins, "Unless the reinvestment rate...," is partially true. However, the holding period return (covered in a later LOS) could be less or greater than the original yield to maturity (YTM). Over the investor's holding period, interest rates are likely to fluctuate both up and down; at some points the investor will reinvest at a higher rate than the original YTM and sometimes he will reinvest at a lower rate.