Monday, 14 May 2012

Explaining the real cost of funding government borrowing (and why David Smith is very confused..)

[Update. David has responded here. His post simply ignores the second-to-last para below. which addresses the substantive point.  At this point, I really don't know whether it is because he's dug himself into a hole and doesn't want to admit it, or because he genuinely doesn't understand the government's intertemporal budget constraint - a standard identity, taught in any decent graduate macro or public finance course, if not before.].  

My post yesterday claimed that

"if the government were, as I suggest, to fund a £30 billion (2% of GDP) investment programme, and fund it by borrowing through issuing long-term index-linked gilts, the cost to taxpayers - the interest on those gilts - would be something like £150 million a year."  

David Smith, however, tweeted repeatedly that this was wrong, arguing (for example)
"No, again, £150 million of revenue doesn't give £30 billion of funding. £1 billion - real yield plus inflation - might."
Why is David wrong?  Well, as the chart in the post showed, real interest rates for very long term index linked gilts have been hovering around the 0.5% mark for the last couple of years. Indeed, they are currently lower, and recently the government has actually auctioned index-linked gilts at real rates of basically zero, but lets take 0.5% as the government's current real borrowing rate.  David hasn't argued with that. 

What does that mean? It means that if the government borrowed £30 billion at 0.5% real, then it would have to pay interest of £150 million per year, uprated for inflation for the term of the debt. See here for an explanation.   So, in year 2, it would have to pay £150 million uprated for inflation between year 1 and year 2, and so on. So David's tweet is simply wrong - £150 million per year of revenue (uprated for inflation) perfectly well could yield £30 billion of funding at current market real interest rates, consistent with the results of recent government bond auctions.  David may think the markets are somehow "wrong", but those are the rates at which they are currently lending to government. 

What about when the debt comes due?  At the end of the term, it would have to repay the £30 billion, uprated for inflation (ie an amount worth £30 billion in today's prices). But alternatively, and more likely, it could refinance the debt. Assuming real interest rates are the same then as now (and the yield curve is actually downward sloping at the far end, implying that as far ahead as the markets can see they will be falling not rising), then it could be refinanced again, with annual interest payments again being £150 million at today's prices, uprated for inflation. Of course market prices could change and real interest rates could rise - but again, this is what the markets are saying now. 

In other words, £150 million per year, uprated for inflation, and payable for ever, is sufficient to fund borrowing and spending now of £30 billion.  Is this surprising?  Not at all, given current market real interest rates; it's just arithmetic. Indeed, this is an absolutely standard textbook result from the basic macroeconomics of public finance, generally known as the intertemporal government budget constraint, and summed up in this equation:

See here for a formal derivation.  Again, it may look a bit complicated, but it's really just arithmetic.  It just says that b (the value of debt at time 0) must be financed by future primary surpluses s, discounted at the real rate of interest r.  What does this mean for my example?  Letting n take the limit to infinity, and plugging b=30 billion, r= 0.5% and solving for s will give you s=150 million (don't take my word for it - go ahead). In other words, saying it again, £150 million per year, uprated for inflation, is sufficient to fund borrowing and spending now of £30 billion. 

What about the pasty tax?  Well, David is right (and my original post admits) that the revenue estimate of £150 million or so relates to closing VAT loopholes, of which pasties were merely the largest. So I admit to some poetic license in the title of my post. However, the estimated revenues grow consistently faster than inflation over the forecast period, reaching £190 million by 2016-17; and normally one would assume that in the longer term revenues from a specific tax grow in line with GDP, that is faster than inflation.  And they will continue as long as we continue eating pasties (or other hot takeaway food), which I think we can safely assume is forever.  So there's no problem there.  

So there is absolutely no doubt that, from a textbook economic perspective, the best estimates we have now from current market prices is that the revenue from the pasty tax (and other minor loopholes) could finance current borrowing of £30 billion or so.  So what then was David actually talking about, and where did he get his £1 billion? I'm still not quite sure. But he seems to be referring to the interest payments, and ultimate repayment of principal, on nominal (non-index linked) debt.  This is silly and economically irrelevant. The value of nominal debt is eroded over time by inflation; nominal interest payments compensate for that as well as paying the actual carrying cost of the debt.    So over time, the real value of the debt reduces, and eventually is completely wiped out. But the pasty tax revenues go on for ever, and, as noted above, increase in line with inflation at least, and quite probably more. He is not even trying to compare like with like. 

I have spent some time on this, because it's an important point.  David, and others, seem determined, against all the evidence, to deny the fact that government can at the moment borrow at unprecedentedly cheap rates. As Martin Wolf and I have long argued, this means that even substantial borrowing would have only a marginal impact on long-run fiscal sustainability.  This in turn constitutes a very strong argument for doing just that.    


  1. "The value of nominal debt is eroded over time by inflation;"

    But this doesn't happen with index linked gilts. The nominal value of the gilt is adjusted for the rate of RPI. So that the redemption amount of the bond is the original nominal x the RPI over the duration of bond.

    I don't disagree with you that borrowing is currently extremely cheap for the government and that a shot of infrastructure and housing spending makes economic and moral sense.

    I just don't think it is quite as cheap as you argue.

    Nor that infrastructure spending always adds to long term economic productivity. Japan's bridges to nowhere provided a short term boost to local employment but not much else.

  2. On the redemption amount, I am well aware of how it works with indexed gilts, and that's exactly what I said in the post:

    "At the end of the term, it would have to repay the £30 billion, uprated for inflation (ie an amount worth £30 billion in today's prices)."

    So it's baked into the arithmetic set out in the post (and in every graduate macro/public finance course). So it is exactly "as cheap" as I argue. Unlike David, you seem actually interested in the facts, so if you're going to disagree, you do have to set out an actual argument as to why my arithmetic is wrong

    As for the productivity of infrastructure spending, of course you're right but that's a separate and much more complex discussion, for another post.

  3. Suppose the government were to increase borrowing by a large amount, say, £300 billion. Undoubtedly interest rates would rise. This higher rate would then apply to all borrowing, not just the £300 billion. So the additional interest payable as a result of the increase in borrowing is the interest on the £300 billion plus the additional interest that is paid on the rest of the debt as rates go up.

    I know that you are only suggesting £30 billion but surely the same effect applies only at a smaller scale.

    So the additional borrowing may not be as cheap as you suggest. Your main point still looks pretty valid though.

    1. Bill, You say
      "Suppose the government were to increase borrowing by a large amount...Undoubtedly interest rates would rise." Undoubtedly? Govt borrowing over the last few years has gone up, and interest rates it pays have gone down. See also Japan. I have (slightly unfairly) taken out your figure of £300 billion - at some figure your point might be right, but so far we haven't reached it.

      "This higher rate would then apply to all borrowing, not just the £300 billion." Again, is that really right? The prices of bonds might fall and/ or the rate on new issues might rise (though not so far). But that does not affect the amount the govt actually has to pay on all outstanding debt. It's just when it has to redeem or refinance it. A lot of govt debt does not have to be redeemed/refinanced for decades. When it does fall to be re-financed, rates may be higher - indeed that expectation is priced into long gilts. But if rates are higher, that will be because we're out of the current mess.

      Am I missing something?

    2. Luke, yes you are right this will not apply to all debt immediately. But it will, as you state, apply to debt that has to be re-financed. And if the funds borrowed are invested wisely to increase GDP the overall result might actually be a reduction in rates.

      It's certainly true that other factors will also influence interest rates when the debt is refinanced. But, all things being equal, an extra £300 billion in debt will likely produce higher cost of borrowing. Although, as you state, this is by no means certain.

  4. So in 20 years we need to borrow more to pay off what we borrow now plus borrow more to continue spending at the same rate?

    The interest rate being offered is based on the amount of risk perceived in each sale; so if the government suddenly announced an increase in desired borrowing the interest rate would jump i.e. the rate at which it was possible to borrow the money would not be the market rate currently in operation.

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  6. I shouldn't waste more time on this but....I have read DS's piece (I had no idea he was econ editor of the ST - I thought you were humouring some interested amateur like me.)

    What gets me is the idea that as the debt is increasing in line with inflation, it is somehow increasing in real terms (if that's not what he means, then what does he mean?). If he has some IL gilts or National Savings, he must be a very happy man as he sees his wealth increasing every year.

    Then he makes the astounding discovery that borrowing by means of IL gilts costs much the same as borrowing in standard gilts. You mean bond investors aren't idiots? Who would have thought it?

    1. Indeed: I don't know whether to be depressed or encouraged that "interested amateurs" like you and Russell below understand the basic issues under discussion here, while the economics editor or the ST doesn't.

  7. Agree with your thoughts that index-linked borrowing is at historically cheap levels for the Government. A 0% real yield means that Government can borrow over the long-term with only inflation to worry about (in real terms).

    Ultimately, the widely reported metric of economic growth reported is GDP growth in real terms. So to the extent that real yields are negative, issuing debt to fund investment now would be expected to increase real GDP growth (all else being equal) – and that’s before you take into account any multiplier effects.

    For 20 - 30 year maturities the real yield (i.e. after inflation is removed) on index-linked Gilts is currently slightly negative. However, the annual interest payments the Government would have to make is determined by the coupon rate at issuance, which is typically around 0.625% of notional for a 30 year maturity (increased in line with RPI inflation). These coupons are paid on the notional, rather than the market value of the index-linked Gilt, so the annual interest payment are probably around the 0.5% you quoted (currently, market price > notional).

    That said, who knows what real yields will be in say 30 years time when the Government came to re-financing to pay of the notional (increased by RPI inflation). Clearly Gilt yields are at historic lows due to a combination of events working in the same direction: i.e. 'safe haven’ flows, short-term interest rates (base rates) expected to be low for longer, and the impact of Quantitative Easing.

    On balance it does seem like a good opportunity at present for the Government to ignore the argument to borrow a little more.

    1. Sorry, my last para should say:

      "On balance it does seem to me that there is a very good opportunity at present for the Government to borrow 'a little more' to invest, to help the UK out of this malaise”.

      The UK Government can presently borrow at approx historically low rates, where the (real) interest cost for new borrowing is approximately zero % in today’s money.

      Of course, at the extreme, if the UK Government MASSIVELY increased its borrowing, then the marginal cost of borrowing would likely go up as the bond markets would lose some confidence that the UK Gov could service its debts.

      Jonathan - I think you should be both depressed (for DS) and encouraged at the same time that the next generation of "interested amateurs" understand the basic issues of borrowing.

      Over & out.

  8. Sorry Jonathan, but I think your proposal amounts to a reckless gamble. The trick is here: "What about when the debt comes due? could refinance the debt. Assuming real interest rates are the same then as now....".

    But you can't assume that real interest rates will be the same. Assuming that the economy has recovered by then (and if not your policy proposal will have failed anyway), it seems likely that the large stock of government debt that we will have accumulated during the downturn will be far more expensive to finance when the private sector offers more numerous and less risk lending opportunities, and we will need even more extreme austerity to avoid a debt trap.

    Yes, debt is attractively cheap right now, but I am afraid that we are already full up.

  9. The lender automatically and electronically deducts the loan value plus the interest and other financial charges.

  10. I agree. Borrowing funds from others does not have a huge impact on fiscal sustainability. payroll solutions