## Saturday, 16 February 2013

### Dressing and undressing

Sticking to describing packages of cash flows discounted at an appropriate rate, there's a clothed-naked metaphor which springs to my mind.  The clothed element is the financial cultural description of the cash flows - this first one is an up-front fee, this next set are interest plus repayment of capital, that one is a final fee, etc.   The naked element strips away the semantics, leaving just the cash flows, the dates and the discount curve.  In a similar vein, the clothed element of rate interpretation contains cultural artefacts around day-count conventions, business day convention, the practice of holiday dates and weekends, accrued interest calculations, clean and dirty pricing.  And the naked being continuously compounded, year-fraction equivalent.  The operations team, finance team, legal team, trading group, management, they all live in the clothed realm.  The quants and the technologists who implement the quant algorithms live in the naked realm.

The phenomena of this kind of nakedness are cleaner, fewer, more complex to understand.  And the process of dressing and undressing, while prone to many errors and omissions is a technique it pays to do well.

### Copping a feel of bond floors

It is nice to have some present value experience, or rules of thumb.  Given that one of the long term goals of this blog, and of me personally, is to become very experienced in pricing convertible bonds, then it is  useful to know what the bounds on a bond's present value tend to be.

Here's a rough and ready view.  Convertibles last 5 years.  Interest rates are usually in the 4%  ($r=0.04$) ballpark.   Like bonds, they are usually quoted with their price (and some other analytics) in par format, which for the present purposes is like pretending that they always have a face value $F$ of 100 units of the convertible's currency.

If you pretended that a security was nothing other than a single cash flow at the end of the five years ($t=5$), then the present value would be $100 e^{rt}$ or 82, approximately.  This number is usually referred to as the investment value or the bond floor of the convertible, since it is, in general, the present value of all the fixed income side of the instrument, that is to say, ignoring volatility and optionality components.

Many convertibles have get out clauses, both for the investor and for the issuer.  The net result is that, under favourable market conditions, the convertible might only last 3 years.  Again at 4%, this would give a bond floor of about 89.

When you've only a year left then the bond floor drifts up to 96, on its way to par, which is in this  theoretical and overly simple case, the final repayment price.  The closer in time you get to expiry, the closer the simple bond floor goes toward 100.

If the prevailing discount rate is much lower, say at 1%, then you'd get 5 year present values of 95.  In summary, the five year bond present value for interest rates 1,2,3,4,5,6 and 7 percent are, respectively, 95,90,86,82,78,74,70.  That same range of rates applied to a single cash flow only three years hence, where there isn't so much of a compounding effect, produces these bond floors: 97,94,91,89,86,84,81, which all deviate less from par than the five year instrument.

Below is a somewhat prettier table showing this.  Discount rate on left and years running along the top. There's synmmetry in here since really all iso-values of $-rt$ give the same discount factor.  This is, of course, just a visualisation of how the natural exponential $e^x$ plays out when $x$ is made up of two factors, $-r$ and $t$.

While I'm at it, taking representative discount rates of 1% 4% and 7%, how many years before the present value of 100 drops by half? 70 years, 18 years and 10 years respectively.  Likewise for a drop to one hundredth of its value (that is before the present value of 100 becomes 1)? 420, 105 and 60.

Adding intervening cash-flows by way of interest payments is more of the same, just an extra wrinkle.

If a nation state wanted to get rid of half the public debt in a decade, then one way to achieve it is to have a nominal discount rate of 7%.  If we call that 7% 2% real and 5% inflation, then a central bank just needs 5% inflation for a decade to wipe out half of the debt holders.  If a nation's debt holders are all domestic debt holders, then you've effected a transfer tax from the average lender to the average debtor, a kind of Jubilee.  A foreigner looking at your country might demand more of his currency now for your currency as a result of this worry.  If the debt holders are largely foreign, you are imposing the cost on them, which will have its own macro-economic consequences.