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When we try to find the present value of a bond I understand we should discount the coupon payments and the face value by the ''current interest rates''.

My question is what are these interest rates? From what I knew is that bonds are a debt security meaning that the bondholder receives interest rates as coupon as payments and does not have to pay interest themselves.

So I was wondering whether the discounting comes from the time value of money (i.e the potential money I could make investing somewhere else, and those interest rates could be estimated just using the CAPM model)?

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    When you own a bond, the interest is received by you, not paid. – JoeTaxpayer Apr 26 '15 at 18:55
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Just to confirm, you don't pay interest when holding a bond, the issuer of the bond pays you interest.

The idea of calculating "present value" is as you suggest. You discount future payments using an appropriate rate. These future payments include both the coupon payments you receive through the life of the bond and the principal repayment at the maturity of the bond - each should be discounted from the due date of the payment to today's date.

A typical rate to use would be the interest you yourself could earn by investing elsewhere (this gives you some idea of how much it would cost to get those payments another way), or perhaps some standard rate, for example the interbank rates such as LIBOR or FEDFUNDS.

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