I finally know what distinguishes man from the other beasts: financial worries.

- Jules Renard

As we’ve discussed a number of different times, the application of the right financial concept the wrong way or to the wrong situation is not a recipe for long-term happiness in your finance career. However some mistakes are worse than others and there’s one happening now in public finance we think deserves pointing out. I’m referring to the debate over what the *right* discount rate is for evaluating present value (PV) savings in a refunding analysis.

Motivating the discussion, the Commonwealth of Massachusetts in a recent request for proposals (Section F, Q5) asked whether they should evaluate refunding savings based upon dscount factors derived from non-call spot rates from their own borrowing curve. Public finance convention since its dawn is to use discount factors derived from either the Arbitrage Yield or sometimes the True Interest Cost (TIC) or All-in TIC. We wrote a first cut discussion of this question in a blog post earlier this year and for the uninitiated there are some pretty simple conclusions worth looking at there. (HINT FOR BANKERS: your pv savings will look better on “scoop and chuck” refundings if you discount using spot rates and not the arb yield!)

In our Structure software we've allowed discounting at a fixed yield or at zero rates derived from the input curves for over 2 years. But is one of these answers really *right *in some pure, financial theoretic sense? Over the last 40 years, the now famous concepts brought to financial theorists by Black-Scholes-Merton have been generalized to show that if we can reliably replicate the cash flows from one financial product with a self-financing hedging strategy using one or more basic underlying instruments, then the value of the original product must be the cost of that strategy. This theory is predicated on an assumption that if the product in question was offered at a price in the market lower than the cost of the hedging strategy, active market players would buy the product and hedge it using the strategy and lock in a riskless profit. More succinctly, the theory assumes that markets are free from arbitrage profits. This is a seemingly benign assumption when lots of profit-maximizing market actors are around but one which has broken down with increasing frequency particularly in times of market/liquidity stress.

But what does this arbitrage-free stuff have to do with public finance refundings? Well it is *exactly* this no-arbitrage stuff, applied to public finance issuer’s debt portfolios, that underlies the argument that discounting using spot zero rates is somehow *right* and discounting using a single rate like the arb yield is wrong. But does this no-arbitrage condition apply in the public finance environment? Is the public finance treasurer running a matched book of assets and liabilities? Are an issuer’s sold (short) bond positions somehow hedged? They of course are not. In the parlance of modern finance any environment which falls outside the realm of no-arbitrage is called “speculative.” Is the refunding decision speculative? As unappreciated as I may be in saying this, of course it is. Guarantees are few and far between these days (death and taxes?), and there’s no guarantee for the issuer that having waited another month or year that the refunding may not be significantly better. It is completely speculative which is partly why it’s so difficult to make.

So given we’re clearly talking about a speculative, completely arbitrage-rich environment which any Street quant worth his PhD would call the real world, as opposed to the risk neutral (arbitrage free) one, what’s the theoretically correct, and financially *right* set of discount factors to use in public finance refunding analyses? The answer in the end isn’t that satisfying though I think it jibes with people’s own intuition. The right answer is not a single set of discount factors, but rather every possible set of discount factors for every possible real-world future interest rate scenario. True PV savings is actually stochastic and an entire continuum of numbers created based upon the whole continuum of real, possible future rates. Even with great public finance rate simulation software available, we doubt calculating distributions of PV savings numbers to capture its stochastic nature is going to catch on like wildfire in public finance, but it **is** the theoretically correct answer. All of the other alternatives are just simplifying models and assumptions, some with more appeal than others. And as always with financial models, buyer beware.

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