Do governments issue debt at low r to pay debt with high r? I have a question. I know governments do debt rollover from time to time. But, to me it seems logical that governments do this when the interest rate of bonds they have issued in the past is higher than the one they could get now. This way, they pay more expensive debt with cheaper debt. To me this seems like a no-brainer. But is this what they do?

With current QE and low interest rates, are governments doing this? I think they are not. Why?
The phrase governments "do debt rollover from time to time" is an understatement. The entire stock of Treasury bills is rolled over continuously, as well as the coupon debt. For example, a 3-month Treasury bill needs to be rolled over four times a year.

As for the current situation, it is somewhat strange.


Government fiscal agencies (Treasury, Ministry of Finance) have tended to increase the average maturity of their debt issued. (The answer by Facet_Velvet discusses this, but in order to keep this answer self-contained, I will provide the same link: link to OECD report.)
Many (but not all) developed central banks have been buying government bonds, replacing them with short-maturity governmental liabilities (deposits at the central bank).


These two actions tend to cancel each other out. The only defense of the policy that I can offer in this context is that the central bank is the party taking the interest rate risk. (The central bank has nominally independent finances.) As a disclaimer, I think Quantitative Easing as a policy makes no sense, so I cannot come up with a good rationale for this policy mix.

However, the answer to your question is that we cannot think of the central government as being like a household or business. As the currency issuer, it has a very different perspective on finances, and minimising interest costs is actually not an important policy objective.

For example, it is extremely easy for the government to minimise interest costs: it just needs to set interest rates at 0% by legislative fiat. (Some may object that this is impossible; the experience with interest rate pegs during and after World War II show that the policy is feasible.) However, just because we can minimise interest costs does not mean that it is a good idea. In order for monetary policy to work, the government needs to have bonds that set the rate of interest at a positive level, which can then be varied by the central bank to help control the business cycle.

The primary interests of the government in issuing bonds are:


Create a yield curve that act as a benchmark for private sector borrowing. When Australia was running large surpluses, I believe that the Australian banks pleaded for continued government bond issuance to keep the yield curve liquid. (I do not have a reference.)
Have a liquid Treasury bill market that gives risk-averse investors a place to park cash. In particular, central bank foreign exchange reserve managers are not willing to take large amounts of interest rate or credit risk, and so you need a liquid Treasury bill market if you want foreigners to hold your currency as foreign reserve assets.
They need to meet the demands of institutional investors for long-dated credit risk free bonds.


These competing demands imply a continued juggling act to balance out issuance at various maturity points.

Finally, governmental debt managers do not want to lock in their portfolio of issued bonds as long-dated instruments, as they can do nothing about them. Unlike mortgages, almost all government bonds cannot be repaid early. (The U.S. issued some 30-year callable bonds in the early 1980s.) If the debt is termed out, and the the government has a fiscal surplus (which happened for a brief period in the late 1990s), the government would end up with no Treasury bills outstanding and a surplus of liquidity. It would have to go into the private markets and lend to private sector entities with that surplus, which the government does not want to do. Hence, they need to have a sizeable amount of short-dated paper that can be run off in such a situation.