Interest rates going down is a bad thing for a bond portfolio if you haven’t bought your bonds yet, because now you can’t get as much return for your investment (a bit like the stock price going up is a bad thing if you haven’t bought your stocks yet).
So that means that bond yields going down is a good thing for your bond portfolio if you already have bonds in them, because the price goes up and your portfolio is worth more. It is a bad thing for any bonds you buy in the future.
At some level it may not matter. The bonds you have have gone up in value, but you can’t sell them and buy higher yielding stuff (unless you take on more risk). So if you think that rates have to go up soon, you might sell your bonds, hold cash until rates climb and then buy them again, but of course timing the bond market is unlikely to be any easier than timing the stock market, plus your investment mandate may not let you do something like that.
Let’s suppose your bond is worth 100 at a 5% yield and rates go down to 3%. Maybe your bond went up in value to 110 because of duration. You have more money today, but you’re effectively getting 3% on 110 rather than 5% at 100. So you get both a gain in present value but a reduction in future yield. Often it balances out, but if you are trying to time interest rates, or you have specific liabilities to neutralize and the durations don’t match, then you may want to trade them.
Ultimately, nothing has really changed, you’ll still get your payments as scheduled.
Stocks are not all that different, but stocks trade on price, whereas bonds tend to trade on yield. Plus the future cash flows from stocks are subject to much more uncertainty, which is why they have a required return. When a stock price goes up faster than its required return, if nothing has changed in the underlying business, it means that the future returns will now be lower, so if you have the stock in your portfolio, you have an unrealized gain, but a lower expected future return. With stocks, that’s often a sign to sell, because you figure that there might be some other stock that is undervalued and might do the same thing, and you can try to swich horses. It’s harder in the bond market because the differences tend to be on the order of basis points, but one of the advantages of the bond market is that it is substantially larger than the stock market (and can include non-public companies).