# Hypothetical yield for a government perpetuity

I understand that with interpolation or bootstrapping one can determine spot rates given other spot rates, however how would you go about establishing what a hypothetical perpetual bond issued by, say, the US government given the current rates should yield?

To do this, you would have to start making some assumptions about the distribution of the yield curve. It's not unreasonable to suppose that the market has little visibility on what the Fed Funds rate will beyond 5, let alone 10, let alone 20 years time. So the curve beyond the belly will increasingly reflect term premium, with diminishing returns (in yields). Given these diminishing returns/yields, there will be an asymptotic peak that should reflect the perp yield.

This was certainly the case eg in the UK which had perpetual Consols (to finance the Napoleonic Wars) and War Loan (to finance WWI) issued for decades/centuries until they were recalled about a decade ago, when yields fell below coupon. So you have decades/centuries of this basic model not being wrong, when such perps existed.

The question then is what model you would wish to impose on the yield curve.

Personally, I would assume that anything under 5 years could be influenced by market expectations about central bank policy rates. So in its simplest form, you might start to estimate an infinite yield using only the 5Y, the 10Y and the 20/30Y. But you could obviously choose to include the 7Y if you wish, or every bond issued >5Y if that takes your fancy.

So every bond yield >5Y can be decomposed into forwards. My 10Y is a 5Y and a 5Y5Y. My 20Y is a 5Y15Y and a 5Y, or a 10Y10Y and a 10Y. My 30Y is a 5Y25Y plus 5Y, 10Y20Y plus 10Y, or a 20Y10Y plus 20Y. And so on. Double the number of bonds in your sample, you quadruple the differences in the structure of their yields.

Decomposing yields thus allows you to model the forward timing versus the then yield of any bond in your sample. Take TR=(1+Y)^T and decompose it into (!+shorter-y)^y * (1+forward)^f. Log these to make these additive, and thus regressible against T and log(T). Now you can infer what a 100Y or a 200Y or the 100Y100Y forward bond "would" yield (assuming term premium is time-indifferent); which should be little different between three; and thus if 100Y approximates 200Y closely thus, both approximate a fair yield on a perp.

best, DEM

• Thank you for taking the time to explain this Jan 26, 2021 at 18:47

UK used to have perps outstanding ("consols"; see for e.g. https://en.wikipedia.org/wiki/Consol_(bond)). If you want a real-world example of where they traded relative to the rest of the government curve, I suggest taking a look at those.

• Thank you for the link Jan 26, 2021 at 18:47