Bond discounting conventions

Question

during the preparation for my thesis, I've come across some strange discrepancies between literature and the information I've been taught.

It comes down to the proper way of discounting cash-flows of a (semi-annual)coupon bond, which has been done during my lectures this way: $$P_{t}\left(\tau\right) = \sum_{i=1}^{n}\frac{C_i}{\left[1+i_{t}\left(t_i\right)\right]^{t_i}} + \frac{F}{\left[1+i_{t}\left(\tau\right)\right]^{\tau}}$$

However, most literature I've read uses slightly different approach: $$P_{t}\left(\tau\right) = \sum_{i=1}^{n}\frac{C_i}{\left[1+\frac{i_{t}\left(t_i\right)}{2}\right]^{2t_i}} + \frac{F}{\left[1+\frac{i_{t}\left(\tau\right)}{2}\right]^{2\tau}}$$

Let's calculate the first payment, after 0.5 years (zero rate is 5%): $$\frac{C}{\left(1+0.05\right)^{0.5}}\neq \frac{C}{\left(1+0.025\right)^{1}} $$

My question is, which method is correct? Is it the "Semmi-annual discounting of semmi-annual coupons" or "annual discounting"? Or are those two just different conventions?

Thanks for answer

Answer 1

You are basically just arguing semantics from two models, neither of which are necessarily precisely accurate. If you observe the assumptions regarding yield to maturity, you have;

1) Coupons can be reinvested at the same yield through the life of the bond,
2) The payment dates all have consistent amount of time between each one, i.e. nothing falls on a holiday or leap years are unconsidered etc.

I observe your $i()$ is a function of $t$ so it is probably bootstrapped but still you have small timing (business day) discrepancies in either case.

Of the two I prefer the per-annum-rate formula which uses $\frac{i(t)}{2}$, and depending upon the context these small discrepancies may be negligible - so much so that you can still derive many useful and theoretical results without worrying about a business day or whatever.

The most generic way I can see of expressing the price of the bond is to write;

$$P(\tau, C) = \sum_i^{N(\tau)} C_iv_i +F v_{N(\tau)} $$.

Now the number of coupons is a function of the maturity and the discount factors, $v_i()$, are derived by your model curve, however it is produced (and that model may factor timing of events like business dates for example). This is the typical method you apply in interest rate swaps too fyi.