From Frank Fabozzi's Bond Markets, Analysis and Strategies:
The price of any financial instrument is equal to the present value of the expected cash from the financial instrument.
Bond Price = sum(i=1 to T) CF(t)/(1+r)^t, where r is the required yield (or interest rate in this case). In the market, when the demand of the bond increase, the bond price increases. According to formula above, this means r has to decrease. Same reasoning applies when the demand of the bond decreases. (Note: the bond market usually uses the bond price and back out the yield rate)
Note that, when the demand of the bond increase, this usually means the market experience high uncertainty (slowdown in economy), and as a result, interest rate needs to decreases in order to stimulate the economy. This may also explain when interest rate falls, then the bond price increases.
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