How Coupon Rates and Yields Determine Bond Prices

# Bond interest rate, the relationship between a bond's interest rate and its present value, and discount rate


# Bond interest rate, the relationship between a bond's interest rate and its present value, and discount rate












Consider this equation for calculating the Present Value of Bond that pays a coupon and its face value at maturity:

C is the coupon, r is the interest rate on the bond, m is the number of times it is compounded per year, n is the number of years, and F is the face value of the bond paid at maturity.

Now consider this table used to calculate the Present Value of the bond which pays a Coupon of 50 semi-annually and has $1000 face value that will be paid at maturity, where m is equal to 2, for different values of n and r:

First, what is the interest rate on a bond supposed to represent, is it simply discounting, taking away, from the Coupon payments and the payment of the face value at maturity, or is it adding value? Although its proper term is the yield-to-maturity, I seem unable to grasp what it means, i.e. what the interest is actually doing in this case.

For the interest rates of 0% and 2.5%, why does the present value of the bond increase when the bond is held for more years? Why does the present value of the bond stay constant at 5% regardless of the number of years it is held. And lastly, why for the higher interest rates of 6.5% and onwards (to the right in the table), does the present value of the bond decrease when it is held for longer?

## Answer by nbbo2 (score 1, accepted)

https://quant.stackexchange.com/a/53240

The bond issuer (company or government) decides the Coupon Rate once and for all (though it is less important than beginners think). But the Promised Yield to Maturity (PYTM) is going to be set in the marketplace evrey day depending on economic conditions, bond risk and PYTM available on comparable bonds. The PYTM is what investors care about.

In your examples in every case the Coupon Rate is 5%. When the marketplace wishes to set the PYTM of the bond at 5% the bond is said to "trade at par". In this case the price of the bond is always 1000 and the investor gets his return in the form of coupons only. You have verified this experimentally by showing that the price in column 3 is always 1000, but this can be shown analytically (try it).

When the market decides that this bond must earn a PYTM greater than the coupon, no investor will touch the bond at the price of 1000, the price will sink to a value < 1000 ("below par") before anyone buys it. Columns 4,5,6 and 7 illustrate this case. For example if the bond has maturity of 10 years and the market has decided that it must be priced at 6.5%, the bond price will be 891. If you buy it at this price, you will get your Total Return of 6.5% in the form of two components: 50 dollar coupons (the Coupon Income) and Price Appreciation (since you buy the bond for 891 but it is worth 1000 at redemption, you get a capital gain). Again you should verify analytically that the two components add up to 6.5% Total Return over any holding horizon.

When the market place is satisfied with a low PYTM (such as nowadays, with interest rates very low), the 5% coupon bond we are discussing will sell for > 1000 ("above par"). This is the case in columns 1 and 2. The investor receives an exorbitant 50 coupon but since he bought at price > 1000 he suffers a price depreciation (capital loss). The net result is that he again gets exactly the market rate of interest in total. For example if market interest rates are 0% (first column) and he buys a 5 year bond at 1250, he will suffer a -250 capital loss at maturity (1000-1250) but he also receives 5 coupons of 50 each. In total he makes 0 profit, which is exactly what we meant when we said that (for this example) "market interest rates are 0%".

## Answer by JazKaz (score 2)

https://quant.stackexchange.com/a/53251

The interest rate or rather the Yield to Maturity, Like @noob2 said, is based on market beliefs, economic conditions and comparable bond rates. If the bond price formula was altered it’s possible to get the present value annuities formula plus the face value part on the end. This shows that the YTM acts similarly as a discount rate

Based on memory, there are four key relationships in bonds:

One is that if the YTM > coupon rate, the bond is called a discount bond. it is worth much less that the face value. A premium bond would be where YTM < coupon rate

Another is that the price of the bond and the yield have an inverse relationship

As time increases to maturity the risk increases of default and economic conditions and the price often is more variable

As a bond reaches towards maturity it will begin to equal the face value