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 - Fake Bonds

Why are yield translators relevant to the current thread on understanding convertible bonds?  The reason is that if you want to have a model which gives you an estimate of the price of a convert, you need to have a yield curve in place so that you can find out the value today of a bunch of future payments over the coming years of the life of the convertible you're looking at.  Many of the points of a yield curve are invented or interpolated by a so-called bootstrapping algorithm.  But they're bootstrapped around a few real market facts, real market rates, currently trading that very moment in the market.  From this smattering of real-world points, a whole curve gets magic'ed into existence.  And as I previously mentioned, those real world points, those real world markets - cash markets, government bill and bond markets, swap markets, Eurodollar futures markets (all of which I'll come back to), each has their own history, their own typical loan durations, typical rate quoting conventions.  And where you have a panoply of disparate rate conventions that you'd like to pull together into a single coherent picture of yields, then that's exactly where your yield translators come in.

To flesh this out a bit, I'd like to create a couple of artificial contracts, with many real world details trashed for the purposes of clarity.  Then what I'd like to do is show you how a yield curve works on getting a present value for those made-up contracts of mine.  I'll just initially pluck a yield curve or two out of thin air.  After you see how it is used, then we'll turn our attention to creating a real, honest to goodness, no scrimping yield curve, with a view to having it help us price a convert.  We can use the family of fake bonds to see what a difference the various shapes of yield curve make on valuation, perhaps see when it pays to have accurate yield curves, and when it doesn't really pay to have accuracy.

First up, I'd like in my family of fake bonds a contract which just has a single redemption payment in a year's time.  Then one with a single payment in two, and so on for a ten year horizon.  So we have our first ten family members.  But converts often pay a coupon, so I'd like my eleventh to have twice yearly payments of 4% annualised, running for five years, and ending with a full redemption payment.  Number twelve is the same coupon-bond like payment history, but with 8% annualised.  And finally, I'd like a 6% bi-annual coupon, running for a ten year period, with a redemption at the end.  In all cases, I'd like the face value of these bonds to be £1,000,000.  By the way, this is unrealistic, since usually the face value is £100 or £1,000.  But if we wanted £1,000,000 worth of exposure then we'd just buy 1,000 or them, or 100 of them, respectively.  Why not just make it simple, and let us assume we're buying one of them, and the face value is £1,000,000.

The first ten I'll call fake zero coupon bonds (I'll explain the terminology later, for now it is just a name).  11 I'll call my fully sweetened convert.  12 I'll call my lightly sweetened convert.  13 I'll call my straight bond.

 Now, each of the 13 contracts embody 13 loan you've made (or acquired) to some institution or body who you regard as unimpeachably trust-worthy.  Who do you have in mind?  A family member?  A big bank?  A company with lots of cash?  A company with a long history?  A local state? A government?  A government from a particular time in history?  Perhaps a shell company whose only purpose in life is to fund your coupon payments and your final redemption out of a pot of cash it already have stored safely?  Think about it, and whatever works for you, that's how credit-worthy our fake family of issuers are.  These future cash payments, in other words, are just about as certain is it is possible to be with respect to future cash flows.  This is a pragmatic point I'm making here about certainty.  We're not talking philosophical certainty but a much more contingent and localised certainty.

The final piece of damage I'll inflict upon reality is to assume the world really does operate 24 hours per day, 365 days per year - namely everybody works weekends, and there are no public holidays.