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I understand that for US treasury bonds, that the price, its current yield and annual coupon is related by

price = coupon/yield.

But the prices and yields when I look them up online are always denoted for fixed periods such as 1 month, 3 month, 6 month, 12 month, 1 year, 2 year, 5 year, 10 year, 30 year and so on.

How can I find out (calculate or otherwise determine) the price on at a time after the bond was issued when falls in between any such period? Say I have a bond that matures in 3 years, 2 months, 2 weeks, 3 days, 3 hours 2 minutes, 4 seconds? How would I find out the current market price from first principles?

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The prices are never denoted for a fixed period. It's simply what the market trades the bond at at the moment. It seems you don't look at any specific bonds but computed yield curves. Trying to price a single bond with these is convoluted and will never get you the correct price (quoted price). An explanation can be found in this answer.

Yield (yield to maturity) itself is just the interest rate needed to discount the future cashflows to reach the current market price. You can see a detailed example matching Bloomberg here.

Your formula also only works if you look at current yield (which is computed as the bonds coupon rate / price). However, no one really looks at that and your quoted yield curves are definitely in terms of YTM.

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There isn't a simple formula. The selling price depends on what interest rate new bonds are offering or are likely to offer, compared with the rate on this bond and the remaining time on it. So it becomes a matter of the mood of the market as much as one of facts. The bond hasn't changed, but the market you're selling or buying it in may have.

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