Interest Rate and Price of Assets

Question

I have a very basic question about finance. I know that for an asset, the price is inversly related to the yield to maturity, or the interest rate. However, I have three ways of thinking about this relationship.

First, think of a discount bond. The interest rate is given as: $$ i=\frac{F-P}{P} $$ Holding the face-value fixed, $\frac{\partial i}{\partial P}<0$ . This is quite obvious- holding the face value fixed, the more you have to pay now, the lesser interest you make off the bond.

The second way I think about this is that if the interest rates rise, then in equilibrium, prices of bonds will have to adjust downwards because the opportunity cost of holding that specific bond increases (\emph{i.e. }one can make more off other assets and as a result, the bond in question will have to reduce in price to make it relatively attractive).

The last way I think about this is that if interest rates rise, then we discount future payments more. Think of a coupon bond: $$ P=C+\frac{C}{1+i}+...+\frac{F}{(1+i)^{n}} $$

The maturity is at time period $n.$ Here, we discount payments more because $i$ increases.

Now, which one of these is the correct way to think about the relationship? The first and third one are mathematical definitions. The second is an intuitive one. Are these explanations even mutually exclusive?

Answer 1

As @Alex C mentioned, they are all equivalent.

Specifically, 1 and 3 are the exact same thing. (1 is missing an n - unless it's a one year bond).

2 is the intuitively the equivalent of putting a smaller amount of money today in the bank (whatever rf inst. guarantees the $i$ rate of interest) to have same payments as the bond in the future. Since this amount becomes lesser as the interest rate $i$ gets higher, the price of the equivalent bond goes low in the same measure. 1 and 3 are just saying this exact same thing in formulas.