# How does a bond's coupon rate differ from its market yield rate?

I'm trying to understand how bonds work and am confused by coupon rate vs market yield rate.

Suppose company Foo issues a bond with a face value of \$1 million, a 4% annual coupon rate payable semi-annually (so I believe a 2% semi-annual rate), a maturity of 5 years, and a market yield rate of 6% (which is 3% semiannual).

In that case, from company Foo's perspective, how much is it paying every 6 months - is that determined by the coupon rate? (and not the market rate)?

If so, what does the market rate mean? At first, I thought that the market rate was a constantly changing rate based on current interest rates, but the market rate in this context seems to remain constant throughout the lifetime of a bond. Does this relate to how the bond is initially issued at a discount?

Let's say I bought a bond with \$1000 face value that pays 5% a year for the next five years. And suddenly interest rates drop to say 4%. If you bought a \$1000 face value bond paying 4%, that would be less valuable than my 5% bond. So if you want to buy my 5% bond, I won't sell it for \$1,000, but say for \$1,050, because it will pay five times \$50 where a bond that you could buy for \$1,000 would only pay five times \$40.

So the coupon rate of my bond is 5%. The "market yield rate" calculates how much interest you get on what you paid (the \$1050), which will be only around 4%. (The numbers are just roughly calculated).

Every time the interest rate goes down, my bond becomes more valuable, and every time the interest rate goes up, my bond loses value. The coupon rate stays 5% forever, the market yield rate will be close to the current interest rate (plus speculation that the interest rate will change).

To simplify the discussion I'll use annual compounding rate, only one dividend/coupon/interest per year, and I'll change your number.

If I borrow \$1000 for 1 year at 5% from you, I'll return \$1000+\$50 to you after a year. That's the usual way we understand interest rate.

Using the same ratio I can also borrow \$1000/1.05~\$952.38 from you now, and return \$1000 to you after a year, that's also 5% rate.

Let's generalized a little bit,

For each 1 dollar I borrow from you, after t years I'll have to return 1.05^t dollars (compound annually).

For each 1 dollar I am going to pay back t years from now, I can borrow 1.05^(-t) dollars now.

Now consider the case I borrow some money from you, promise to return \$1000 face value in 5 years, and only pay you \$20 coupon every year, but at 5% interest rate, what does that mean?

I'll pay you back \$20 after 1st year, \$20 after 2nd year, \$20 after 3rd year, \$20 after 4th year, \$1000+\$20 after 5th year. Therefore, the amount you should lend me now, to match the 5% rate is,

20/1.05+20/(1.05)^2+20/(1.05)^3+20/(1.05)^4+1020/(1.05)^5~870.12

So now you lend me \$870.12, I pay you back according to the schedule I just described, That's the same as saying you have a bond with face value \$1000, 2% coupon rate, but 5% yield.

Bonds are marketable securities which means the market sets the value from day to day and minute to minute. If a bond is issued at a 7% yield the company issuing the bond always pays that 7% to whoever happens to own the bond. If the market adjusts to a new interest rate of, say, 6% that means the new holder of the bond just paid more for the bond than it cost when it was issued while a yield of 8% would mean the bond costs less on the market now than it did at issue.

The market effects on the price of the bond don't impact the coupon payments but the yield may be different than the stated amount on the bond depending on what the holder paid for it.