Convertible Bond

If interest rates are not expected to change then the straight value of the bond will not change (ignoring the change in value resulting from the passage of time). I f the straight value does not change, then downside risk is indeed limited to the difference between the price paid for the bond and the straight value. If, however, interest rates rise as the price of the common stock falls, the conversion value will fall and the straight value will fall, exposing the holder of the convertible bond to more downside risk.

I thought the minimum value of convertible bond = max (straight value, convertion value). Why they mention the downside risk is the difference between the price and hte straight value?

Thank you!