Why Bond Prices Fall When Interest Rates Rise
Summary
This note reconciles three explanations for the inverse relationship between bond prices and yields: the return implied by paying a price for a fixed payoff, the opportunity cost of holding a bond when other interest rates rise, and the discounting of future coupon and principal payments. It concludes that these are equivalent views of the same valuation relationship rather than competing explanations.
For a coupon bond, discounting each future payment at a higher rate lowers the present value. The opportunity-cost intuition describes the same repricing: less money invested today at the new rate can replicate the bond’s future cash flows, so the equivalent bond price is lower. The answer also points out that the simple discount-bond formula needs a maturity adjustment unless it describes a one-year bond. The explanation is introductory and assumes the bond’s promised cash flows are fixed; it does not address credit risk or changing cash flows.
Key ideas
- A bond’s price and yield move inversely when its promised cash flows are fixed.
- Discounting future coupons and principal at a higher rate lowers their present value.
- The opportunity-cost explanation and the present-value calculation describe the same repricing.
- A simple discount-bond return formula needs a maturity term except for a one-year bond.
Tags
Full text
# Interest Rate and Price of Assets
# Interest Rate and Price of Assets
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 by compilation-error (score 2, accepted)
https://quant.stackexchange.com/a/21707
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.