# How are sovereign bond interest rates determined?

They are a lot of questions about the price of a bond or its yield.

Mine is specifically about the interest rate (the coupon):
How does a government (or its central Bank) determine the price of the coupon when issuing a new bond?

Why would a country want to pay a 1% coupon if few month before they had a negative or 0% fees debt? (assuming a 0% or negative interest rate)

Is it related to the bank rate (EONIA/ESTER) or the "urgency" for the government to issue more debt at a given time?

Edit: the charts I linked was yield instead of coupon gross value, thank you @user253751 for noticing.

• The interest will make the bond price higher, making the interest rate more negative. Maybe they'd like to have some additional money between now and whenever the interest is paid May 31 '21 at 15:12
• But I think you're misreading bond yield. AFAIK, yield is not the same as coupon. Yield is the actual interest you get after accounting for the purchase price, par value, and coupons all together, and even inflation. May 31 '21 at 15:12

What you show on the picture is not the value of the coupon but the bond yield.

For example, for a zero coupon bond to maturity of a bond is calculated:

$$YTM = \left(\frac{FV}{ P}\right)^{1/t}-1$$

Where $$YTM$$ is yield to maturity, $$FV$$ is face value (value printed on the bond), $$P$$ is the bond price, $$t$$ is the number of time periods for the bond to reach maturity. Even though the above formula coupon would be zero the YTM will be non-zero.

For a non-zero coupon bond YTM would be given by solving the below for YTM (although unfortunately this have to be done numerically as there is no simple formula to do it analytically see discussion in Berk et al Fundamentals of Corporate finance 197):

$$P = C \frac{1}{YTM} \left(1 - \frac{1}{1+YTM}^t \right) + \frac{FV}{(1+YTM)^t}$$

where $$C$$ is the coupon payment.

The YTM is determined by market forces (i.e. supply and demand) as any price. You could ask equivalently, why in some store store clerk does not charge million euros for chewing gum, well because nobody would by it at that price.

When it comes to the coupon itself, that is usually set at a current market rate, but note it technically does not matter. The reason for that is that the bond price will always adjust in a way to make, all else equal, the YTM equal to equilibrium interest rate.

For example, if the equilibrium interest rate would be $$10\%$$ and government would try to sell 0 coupon 3 year bond with face value of 1000 then the market value of that bond would be:

$$P=\frac{1000}{1.1^3} \approx 751.32$$

People would simply not buy it for more as if the market interest rate is $$10\%$$ investing 751.31 dollars into some alternative project would yield 1000 dollars after 3 years, so why would you buy the bond (assuming all else e.g. risk etc is equal) for anything more than 751.31? Unless there is some special reason (e.g. you are forced to buy the bond etc) that not going to happen.

Typically you will see governments issuing bonds with coupon rate close to market interest rate (i.e. in such situation YTM will be close to coupon rate). There is no reason for it other than that means the bonds will sell approximately at par value (e.g. at their face value).