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[Source:] Bonds fall in value when rates rise, and this will be reflected in the unit price of bond funds. When the rising rate cycle gives way to a decline in borrowing costs, bond funds will rise in price.

give way to = Allow oneself to be overcome by, or to succumb to (an emotion or impulse):

What does the bolded mean? Substitute the definition of give way to:

When the rising rate cycle succumbs to a decline in borrowing costs, bond funds will rise in price.

But then why does a decline in borrowing costs imply that the grey?
Does a decline in borrowing costs somehow relate to an increase in interest rates?

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Although people anthropomorphize the financial markets, they do not really have emotions. Use definition 2 instead: Be replaced or superseded by: Also 'borrowing costs' is usually just interest, which (for any given principal) is higher when the interest rate is higher. Thus:

  • When market interest rate goes up, bond values go down. Since bond funds hold bonds as assets, those fund values also go down.

  • When interest rate goes down, bond/fund values go up.

  • When interest rate was going up and changes to going down, values start going up.

Their point is that a fund (traditional or ETF) always trades at valuation (NAV or market price arbitraged toward NAV) which varies in this way -- plus or minus transaction costs, but they are ignored here -- while an individual bond varies if traded now but if held to (near) maturity it converges to its face value (unless defaulted).

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