Why would an investor want the minimum yield? I am puzzled by a problem related to bonds.
When a bond is callable, the purchase price (present value of the bond) can fluctuate and I also understand the difference when the bond is purchased at a discount or premium.
However, I am still a bit confused by the fact that the investor wants to find the minimum yield rate.  Would not one want the maximum?  Because the higher the yield the more money you receive later... can someone explain to me with the following example?

A 5% bond with face value 100 has callable dates when t = 10 to t = 15.  It is purchased by 80.  What is the minimum yield rate?

 A: The objective of the investor is to receive a return. The expected return is indicated by the yield rate.
If the bond were not callable, the investor will receive back the face value of $100$ in $15$ years along with $5\%$ coupon payments on earlier dates. 
If the bond is callable for $ 10 \leqslant t  \leqslant 15$ at $80$, then the issuer has the option to buy back the bond from the investor for $80$ during that period.  It is in the interest of the issuer to do so if the future price of the bond (discounted future remaining cash flows) is greater than or equal to $80$. When the bond is repurchased, the investor will receive no further coupon payments and will not receive a lump sum payment of $100$ at t = $15$.  Instead, the investor will receive a payment of $80$ at the time the bond is called.
Nobody knows when or if the the bond will be called in the future. So the mimum yield is calculated under the worst case assumption that the bond is called at an eligible time between $10$ and $15$ years.  You calculate various yields assuming the final payment of $80$ is received at each of those times and determine the minimum.
That is yield-to-worst -- giving the investor a worst-case lower bound on the expected return on the investment.
Note that initially the investor has sold a call option to the issuer and this is reflected in the purchase price of the bond.  Otherwise, there would be no point in buying such an instrument.
