# Bond arbitrage in practice

If we have the following term structure for riskless bonds:

$$\begin{array} {|c|c|} \hline \text{Maturity} & \text{\1 Zero-Bond price}\\ \hline \text{0 years} & \ 1.00 \\ \hline \text{1 years} & \ 0.97 \\ \hline \text{2 years} & \ 0.93 \\ \hline \text{3 years} & \ 0.89 \\ \hline \text{4 years} & \ 0.90 \\ \hline \end{array}$$ an arbitrage profit is possible by "shorting the 4-year zero-bond and longing the 3-year zero-bond".

Yet the question remains how this maneuver should work out in reality?

If we argue like this:

The trader writes a 4-year zero-bond of $$\ 1{,}000{,}000$$ nominal value. By selling this bond he gains $$\ 900{,}000$$. With this money he can buy the 3-year zero-bond for $$\ 890{,}000$$, leaving him $$\ 10{,}000$$. After three years, the 3-year ZB matures, paying him $$\ 1{,}000{,}000$$. This money is exactly what's needed to pay the obligations resulting from the 4-year zero-bond. So the net profit is $$\ 10{,}000$$.

...the question arises why the trader can sell riskless bonds?

If we argue like this:

Second idea: The trader borrows a 4-year zero-bond of $$\ 1{,}000{,}000$$ nominal value. By selling this bond he gains $$\ 900{,}000$$. With this money he can buy the 3-year zero-bond for $$\ 890{,}000$$, leaving him $$\ 10{,}000$$. After three years, the 3-year ZB matures, paying him $$\ 1{,}000{,}000$$.

... how does the story end? We have to get the 4-year zero-bond back, right? So we assume it's availabe for $$\ 1{,}000{,}000$$? Or can we hold on to the $$\ 1{,}000{,}000$$ for the year and then say that paying the borrower back the nominal value is good enough?

• Well, in reality, that arbitrage opportunity would never exist, so the whole matter is rather a theoretical example. – Daneel Olivaw Dec 16 '19 at 17:17
• For your second argument, I guess you could just use the \$1m you get after 3y to pay your counterparty \$1m 1y later (after all, for the counterparty, the bond he lent to you is simply a claim to get \\$1m after 4 years, as long as you pay him that sum he is expecting he should be happy). However, that means you've basically borrowed the bond for a duration of 4y, and that has a cost; that would be extremely expensive and would probably kill off any profit from the arbitrage opportunity. – Daneel Olivaw Dec 16 '19 at 17:21
• "Why the trader can sell riskless bonds?" well as you point out in reality nobody can really issue riskless bond apart from a State, which can always issue additional currency to pay you back. So again this looks rather like a theoretical example to help explain what an arbitrage is. – Daneel Olivaw Dec 16 '19 at 17:28
• This is not arbitrage. You are simply suggesting that the negative 3y1y rate is not possible, and you call it arbitrage because you believe it should be greater than 0. If you execute this trade you are running market risk and if the forecast forward rate ended up being even more negative than currently predicted you would ultimately lose money. An 'arbitrage' is a generated profit in a completely risk neutral sense. – Attack68 Dec 16 '19 at 21:28
• @Attack68 you can find this as an example for arbitrage in textbooks, Wikipedia. And can't we assume that we can simply hold the money as cash in the case of negative interest? How does arbitrage theory deal with negative interest rates anyway? How are the risks that come from holding cash handled? – Amaterasu Dec 17 '19 at 8:51