# In for the duration

27 June 2013

For decades it has been a safe bet that anyone who mentioned “bond duration” at a social occasion, even among investors, was a bit of a geek. But if long-term yields keep rising, that will change.

Duration is a sophisticated measure of how soon bond investors get their money back. A bond’s maturity is a simple count of the time left until the principal is repaid. But that is too simple, as it takes no account of the cash flow the bond investor receives in the meantime. Duration captures both maturity and interim interest payments.

The technical definition of duration is the weighted average maturity of all the bond’s discounted cash flows, both interest and principal. It equates to the amount of time an investor needs to wait until the purchase price equals the interest actually received, plus interest earned on those interest payments, plus the present value of the future repayment of the principal.

That’s a bit complicated, and there is a formula to go with it. But the rules of thumb are simple: the longer the maturity, the more duration is pushed up, and the higher the interest payments, the more duration comes down.

Right now a new benchmark U.S. 10-year Treasury yields 2.5 percent and has a duration of 8.9 years. If the yield were 3 percentage points higher, at 5.5 percent, the duration of new bonds would be eight years. The higher coupon means the redemption payment in year 10 contributes less to the present value of the cash flows.

Duration is more than a way to get an accurate handle on the economic life of a bond. It is also a neat tool to calculate the effect of changes in prevailing yields on particular bond prices. For every percentage point change in the yield, the bond price moves in the other direction by 1 percent per year of duration. Take, for example, a bond with a duration of eight years. If the interest rate falls by 2 percentage points, the bond price increases by 16 percent. So when rates increase, longer durations bring bondholders worse losses, and bring greater gains to bond issuers.

The mathematical explanation of the happy coincidence of years and percentage price changes involves differentials – definite geek-stuff – and needs a few footnotes about converting so-called Macaulay duration into so-called Modified Duration. Whatever. Investors need only be grateful. It helps them get a grip on what higher yields would mean.

Consider the quite different debt structures of the sovereign debt of the G7 countries. Barclays Capital calculates the overall duration of British debt is 9.5 years. Japan is in second place at 7.9 years, while Germany, France, Italy and Canada are between 6 and 7. The United States is at the bottom at 5.1 years.

The UK Debt Management Office, a government agency, decided to push out its maturities shortly after bond yields started falling in the disinflationary years. It has not stopped. This week it sold a 55-year bond, the longest maturity issued since 1937. If yields keep rising at anything like the pace since May – up from 1.7 to 2.4 percent on 10-year gilts – the policy will pay off handsomely for British taxpayers.

Duration quantifies the possible gain for the UK, since the taxpayer is up by exactly as much as the bondholders are down. A 3 percentage point increase in UK yields would mean an approximate 30 percent drop in the total market value of gilts, on the formula.

A chart in latest annual report of the Bank of International Settlements provides a helpful way of imagining the impact on investors: the loss for bondholders would equal about 25 percent of GDP. The loss from a similar yield increase in France, with almost the same ratio of debt to GDP but a shorter duration, would be 13 percent. In the United States, which has more debt but much shorter duration, it would be only 8 percent.

The other side of the story

It looks like the British were very clever in their borrowing strategy while the Americans were quite misguided. But it’s not quite so simple. Most of the gains to the UK Treasury are also losses to UK taxpayers, since 70 percent of gilts are held domestically.

The British investors which bought the disproportionate supply of long-term debt – including many pension funds and insurers under pressure from British regulators – are not going to be pleased. Worse, banks which fund their long-term holdings with short-term deposits could face big losses, although they all claim to have hedged their way to safety.

The UK government could end up rescuing some of the victims of its long-duration policy. That would probably cause more economic and political damage than higher interest payments ever would have. The Americans’ short duration might actually prove to be solid long-term thinking.