Three pounds in six months

I walk into my bank and give them £97.  Six months later I walk in again and they tell me I have £100 in there.  What just happened?  Well, I made £3 in 6 months.  But in terms of yield.  Let's tell a number of yield stories.  But before I do, remember they're all tied back to these same basic facts, 97, 6 months, 100.  97, 0.5, 100. 

Story 1.  I got simple interest.  Let's calculate the holding period yield.  Remember the discussion about holding period yield.  The holding period is 0.5 years.  The holding period yield is (100-97)/97 or 3.092783%  The fact that I observed just two cash flows has made this simple interest story seem plausible.  Put the story in reverse. The bank said to me six months ago: give us £97 and we will make you a 3.092783% return on your money at the end of 6 months on a simple interest basis.

Someone might want to know what that return might look like if it was over a year instead of half a year.  Simple interest rates can be considered to scale linearly with time.  In reality things are more complicated.  In reality, you must consider implicit compounding in this re-basing operation.  But simply assuming linear scaling is an acceptable approximation for some circumstances.  So with twice as much time we would assume we would make 6.18556701%  The nominal period of that 6.18556701% rate is now on an annualised basis. Put that story in reverse. The bank said to me: give us £97 and will give you a return of 6.18556701% on an annualised basis, for a term of 6 months.

Story 2. My interest was being monthly compounded.  Well let's leverage off what we found out in story 1 to work out what the monthly compounding rate would be which can make 97 grow to 100 in 6 months.  We say $(1+0.03092783) =  (1+\frac{r}{6})^{6 \times 0.5}$ .  In other words the return is 6.1228718698%

Story 3.  They were rather kindly performing a continuous compounding for me.  The continuously compounded rate, quoted on an annualised basis, is 6.091841496%

All three of these are expressing a financially meaningful return for the 97, 0.5, 100 observed facts of the original thought experiment.

Story 4.  This investment was in US T Bills, and the bank discount rate implied by the move of +£3 over six months (let's say 182 days)  is 3/100 x 360/182, which is 5.934065934%.  

Story 5.  The investment was in money market instruments.  The money market equivalent yield is 3/97 x 360/182 or 6.1175937%

Anatomy of a convert - time and time again

In a previous posting I showed all the ways simple discrete compounded and continuously compounded rates could be inter-converted.  However, now I'd like to state that all these types of rate can really be seen as variants of the discrete compounding $(1+\frac{r}{n})^{nt}$.  It is good that this can be seen as the most fundamental representation, in my mind, since it works so well as a contributory definition of capital as a property which produces other properties.

You already know, thanks to Jacob Bernoulli, that the limit of discrete compounding is $e^{rt}$.  All you need to see is that, when you let $n=\frac{1}{t}$ then the discrete compounding formula $(1+\frac{r}{n})^{nt}$ becomes $(1+rt)^{\frac{t}{t}}$ which is identical to the simple interest formula $(1+rt)$.  Great; so we have a continuum of compounding frequencies, running from 1 to $\infty$.

What are you doing when you set $n=\frac{1}{t}$?  Well, remember $t$ is the term of the loan or bond and $n$ is usually the number of times per year (i.e. per unit t=1) you compound.  So in the general case you will have discretely compounded $nt$ times.  But if you only want to compound once at the end, then setting $n=\frac{1}{t}$ is the way to do it.  Also notice the similarity with geometric and arithmetic means here.  With arithmetic means, between the start observation $t_0$ and the end observation $t_N$ you have a series of in between observations $t_1, t_2,...$ and you can work out a series of returns $t_{i+1}/t_i-1$ and then calculate the arithmetic average of these returns.  Likewise you can calculate the single value $t_N/t_0-1$.

Anyway, back to the fixed income analysis of converts.  As you'd imagine with any loan, both parties probably have in mind some notional loan term.  You can imagine a retail client approaching a bank manager.  One of the first questions is bound to be for how long do you need this loan?  After that some haggling will result in a rate.  Understand that the rate in question is actually the remaining pair $r$,$n$.  You always need to know the $n$ to understand the value of $r$.  If you really wanted to lay this out sequentially, then you could say, in any negotiation about a loan you first set the term, $t$, then set the compounding framework $n$ for understanding finally the rate $r$.  A more prosaic interpretation is to say that $n$ selects the formula you use to plug your $r$ and $t$ into.

Notice the choice of $n$ here doesn't have any implication for actual cash flow transactions - the borrower could actually keep a hold of the interest until a final payment.  Or on the same analysis, he could pay out on the compounding dates into your account and you'd be free to do with the interest anything you wanted, including spending it unwisely.  It doesn't alter the theoretical analysis.   Payment dates are just a best considered on a different schedule to the compounding schedule implied by $n$.  

The world of fixed income is overwhelmingly interested in $n={\frac{1}{t},1,2,4,12,365,\infty}$, namely simple interest, annualised compounding, bi-annual compounding, quarterly, monthly, daily and continuous compounding.  At a pinch you could reduce it further to $n={\frac{1}{t},1,2,\infty}$

But this is not all.  It would be if all fixed income markets quoted securities in one of the rate formalisms covered by $n={\frac{1}{t},1,2,4,12,365,\infty}$.  They don't.  Often they quote some other market variable, and you need to do some unpacking of that market quote.  That unpacking is, naturally, a function of the various fixed income markets themselves - and there is additionally some regional variation in conventions/usage patterns.

In the next posting, I'll talk about the day count conventions, the digital-to-analogue converters of time and the final wrinkle we need to iron out before we can move on to look at real market quotes and begin to get into the details of building a yield curve.

Anatomy of a convert - yield translators

In my last post, I mentioned that I'd examine the ways in which the various levels of rate compounding could be inter-translated.  But first, why is there a need to do this at all?  The answer is to facilitate comparison between various bonds (and swaps, for that matter).

Usually, if two financial instruments can be valued to a present value cash amount, then this explicitly facilitates the comparison.  Good enough for many instruments.  With fixed income instruments like convertibles (actually, any bonds), there's a second dimension people like to compare across disparate instrument types - the yield.  At its most general, you'd like to know the internal rate of return so you can more directly compare one bond to another - including also to a so-called risk free government security.  If you had the internal rate of return of a convertible, then you can directly read off how much greater this bond's internal rate of return is compared with, for example, British gilts by simply subtracting the gilt rate from the bond internal rate of return.  This then gives you an idea of how much additional return you can expect from that bond.

Always remember in what follows, these are just in essence quoting conventions you'll be translating.  And you will need to do this because there are many bond markets out there each with its own historically determined quoting convention which traders in that market obey when quoting rates.  So inter-market comparisons require yield translation.  For any rate quoted with an expectation of the simple interest formula being applied to the principal, there will always be a corresponding other rate which will give you exactly the same final amount, but with $n$ discrete compoundings per year, or with continuous compounding.  Think about this a moment.  I've heard people say that continuous compounding is somehow an approximation when used in academic finance, since real instruments are simple or discretely compounded.  You should now realise, if you didn't already, that for each and every discrete fixed interest instrument on the planet, there's a continuously compounded rate which will present value the cash flows to precisely the same value as the discrete formula.

Remember too, your basic atomic operation on a single bond payment period is that of simple interest.  You're likely to see simple interest in instruments which last a short enough time to have only a single payment period.  This time round, I'll introduce the subscript $s$ to indicate all the variables of this formula relate to simple interest but I'll still assume the principal is £1 and that the rate is annualised over a term running from now and lasting for $t_s$ years.  You will get back $(1+r_st_s)$.  Continuous compounding returns you $e^{r_ct_c}$.  Finally, I'll give two versions of discrete compounding, with different compounding frequencies $n_1$ and $n_2$ and times $t_1$ and $t_2$.  This will allow me to show you how to translate from one discrete convention into a separate one.  Since I'm using 1 and 2 as the subscript, I'll continue for discrete compounding, using $r_1$ and $r_2$ for the discrete-to-discrete case.  If I'm just referring to a single discrete compounding, I'll drop the subscript.  In which case £1 discretely compounfed $n$ times per year for $t$ years will return you $(1 + \frac{r}{n})^{nt}$ in the end.

1. Simple to continuous.   $(1+r_st_s) = e^{r_ct_c}$ which means $ r_ct_c = \ln( 1+r_st_s)$ and as a result  $ r_c = \frac{\ln( 1+r_st_s)}{t_c}$
2. Continuous to simple.  Here you start from the same equation as 1 above but quickly move to $r_s =  \frac{e^{r_ct_c}-1}{t_s}$
3. Simple to simple.  Why not, eh, for completeness?  This is easy,   $(1+r_1t_1) = (1+r_2t_2)$ and the 1s drop off leading you to notice that you're just time scaling one rate into another $r_1=r_2 \frac{t_2}{t_1}$
4. Continuous to continuous.  Start with $  e^{r_1t_1} =   e^{r_2t_2}$, take logs and you're back to the situation of simple to simple, where you're just scaling one rate into another in proportion to the time ratio  $r_1=r_2 \frac{t_2}{t_1}$
5. Simple to discrete.   First as usual the equation, $(1+r_st_s)  = (1+\frac{r}{n})^{nt}$  Then take logs, $\ln(1+r_st_s) = nt \ln (1+\frac{r}{n})$ and divide through by $nt$ so that you get $\ln (1+\frac{r}{n}) = \frac{ \ln(1+r_st_s) }{nt}$.  Raise to the power of $e$ to get $ (1+\frac{r}{n}) = e^{\frac{ \ln(1+r_st_s) }{nt}}$.  After that, take the 1 across, then multiply by $n$ so that $r= n(e^{\frac{ \ln(1+r_st_s) }{nt}}-1)$
6. Discrete to simple, starts at the same place as 5, but is much easier since $ r_s= \frac{(1+\frac{r}{n})^{nt}-1}{t_s}$ right away.  
7. Continuous to discrete.  Equate   $   e^{r_ct_c}  = (1+\frac{r}{n})^{nt}$.  Taking logs you see that $r_ct_c =  nt \ln (1+\frac{r}{n})$  and so $ \ln (1+\frac{r}{n}) = \frac{r_ct_c}{nt}$.  Finally $r=n(e^{ \frac{r_ct_c}{nt}}-1)$
8. Discrete to continuous.  This time the equation in 7 becomes $r_ct_c = \ln{ (1+\frac{r}{n})^{nt} }$ and so straight away $r_c = \frac{ \ln{ (1+\frac{r}{n})^{nt}} }{t_c}$ and if you want to do the conversion as quickly as possible you'll eliminate the power so that  $r_c = \frac{ nt \ln{ (1+\frac{r}{n})} }{t_c}$
9. Discrete to discrete. Start with  $(1+\frac{r_1}{n_1})^{n_1t_1} =  (1+\frac{r_2}{n_2})^{n_2t_2}$.  Take logs.  $  n_1t_1 \ln(1+\frac{r_1}{n_1}) =   n_2t_2 \ln(1+\frac{r_2}{n_2})$.  So  $   \ln(1+\frac{r_1}{n_1}) =  \frac{ n_2t_2}{ n_1t_1 } \ln(1+\frac{r_2}{n_2})$ and when you raise to $e$ again you get  $(1+\frac{r_1}{n_1}) =  e^{\frac{ n_2t_2}{ n_1t_1 } \ln(1+\frac{r_2}{n_2})}$.  In that case   $r_1 =  n_1(e^{\frac{ n_2t_2}{ n_1t_1 } \ln(1+\frac{r_2}{n_2})}-1)$.  As you can see, this is bound to be the most computationally demanding converter.

Anatomy of a convert - on the interest of interest

In my last post, I glossed over one extra possibility - that in your sequence of cash payments strung out over a number of back to back time periods (for example your £5 per month over 12 consecutive months), after having received the first payment by the end of the first month, then during the second and subsequent months not only do you earn a return on the £1,000 initially invested, but you also earn interest on the £5 which by rights is now yours.  The presence of this additional method of accruing returns is called compound interest.  The compound moment is the moment when your interest payment comes due and is immediately available to earn interest for you.  The more frequently that compounding occurs, the more valuable its effect.  It can happen not at all (referred to as simple interest), with a certain finite frequency, or in the limit, with infinite frequency.

In all cases I've come across, when you drill down to the most atomic interest payment period, then that interest calculation period is always simple, never compound.  Only when you have a string of two or more interest periods is the possibility of compounding even possible.  So think of all kinds of compounding as the application of simple interest, but with a changed amount of principal at  the start of the later simple interest period.  You can see this clearly from the maths.

Simple interest expressed as an annualised $r$ applied for $t$ years on a nominal £1 amount results in $(1+rt)$ at the end of the period.  So if a bank gives you a promise to return 6% to you for a month, if you give them your £1,000 then you should expect $1000 \times (1+0.06 \times \frac{1}{12})$ back, which is £5.  Compound interest is just this simple interest repeated with a new principal of £1,005.

If you compound $n$ times per year over the period $t$ then your return on £1 for an annualised $r$ will be $(1+\frac{r}{n})^{nt}$  Why not just consider $n$ to be the number of compounding periods, and drop the $t$ - you could, but don't forget the $r$ is usually expressed on an annualised basis, and if you made $n$ be the entire number of compounding periods and $r$ be the full term rate, not an annualised rate, then you'd get the easier to understand equation $(1+\frac{r}{n})^{n}$ and this is clearer because you see it is just the product of $n$ separate applications of a simple interest formula, where the time period simple interest is just $\frac{r}{n}$.   Imagine a juicy deal where you get 100% return annualised, and compounded for $n$ time perdiods as before.  The cash back on £1 would then be $(1+\frac{1}{n})^n$.  Now $\lim_{n\rightarrow \infty}(1+\frac{1}{n})^n = e$, the Euler constant (approximately 2.718).  In other words, something very useful occurs - you get to use $e$ instead of discrete compounding.  Why is this useful - well, as you'll see later, this kind of compounding is often assumed in the academic literature since the operations of integration and differentiation are well understood on $e$ and are noticibly easier to work with than integrations of awkward polynomials.  If you compound more and more frequently you eventually reach a limit.  Compounding in the limit is called continuous compounding.  So if someone gave you that juicy deal of 100% annualised but didn't tell you how often the compounding was, then he's underspecified the contract - since you could be getting anything from £2 to £2.71 back at the end of the year.  Quite a difference.  The moral is, unless you know the degree of compounding on a multi-period interest payment, then the contract is underspecified.

Continuous compounding will appear again when we talk about yield curves.  Simple interest is more likely to be seen with very short duration kinds of bond - mostly short term government bonds and so-called cash instruments.  Finally, corporate bonds - convertible and otherwise - are often paid out twice a year.  But they go as cash to the bond holder, who's free to do anything they want with the cash - for example re-invest it in this bond, invest it in a so-called risk free government bond, put it in a savings account or stuff it under the mattress, to name but a few.  So in valuing a bond of any kind, this needs to be taken into consideration.

Next up, I'll show different kinds of direct translation from one rate regime to another, all of which will be practically useful when it comes finally to valuing a convertible.