## Saturday, 6 October 2012

### Certificates of Deposit : market price = fair value

If the simple CD is tradeable, when what is its price?  Well, that's two questions:  what is its fair price at any moment during its life, and second, what is its market price.  Well, remember, it is a two step process - first you get the future value of that single cash flow you'll get on expiry (par plus whatever interest you earned) for some given principal amount $P$.  There's nothing really in that which could vary over time.  The value at expiry is the same regardless of how close to expiry you are.

What might change is the yield you'd use to discount this singular known future value.  Why? Well, imagine you bought a 1 year CD for principal $P$ and a nominal yield $y$, which was the going rate when you entered the market.  The market being that set of institutions from that country who are also offering 1 year CDs on principals the size of $P$.  Now where does this number come from?  Well, as you'd imagine, it is sensitive to short term interest rates.  So Imagine you just bought this new CD and got a rate of $y$ when that very moment the domestic central bank raised the short term policy rate by a whole percentage point.  Well, the CD market would adjust and offer the marginal next customer a higher rate of return $y^\prime$.  So when you come to present value the same fixed future value $P(1+y)$ you get $P \times \frac{1+y}{1+y^\prime}$ since the fraction is less than 1, which results in some amount less than $P$.  That is, the fair value of your security, this instrument which was going to give you $P(1+y)$ in a year, is now worth less than $P$.  Another way of saying this is that the value of the CD is sensitive to fluctuations in interest rates in the economy.  It has interest rate risk.  That new value, call it $P^\prime$, is the new fair value of the CD.

The only moving part here is $y^\prime$, the single prevailing rate you discount your future payment.  This, in a sense, is also the market price.  Now this is unlike more complex securities in a number of ways.  Often other securities have more moving parts, but you'll always just have a single market price.  But for now, enjoy the simplicity of the relationship.  Regardless of how the market actually quotes this rate, whether they tell you it as the current value $P^\prime$, whether it is quoted as $y^\prime$ itself, whether it is $100-y^\prime$ or any other transformation, the bottom line is, that market quote can be transformed into $y^\prime$.  Now imagine I had two CDs, each with different nominal yields $y_1$ and $y_2$, on identical principals $P$ and expiry 1 (year).   Clearly they'll be worth different amounts in any given prevailing market environment $y^\prime$ and time to expiry $t$.

Just for now, let's pretend the market quotes the market yield as the current cash value of an invested principal $P=1$.  That is, pretend the market price of a CD is expressed as $P^\prime = \frac{1+y}{1+y^\prime}$.  This market price is then synonymous with the fair value of the instrument, which is also $P^\prime$.  That identity relationship doesn't often happen with other financial instruments.  With other instruments, there's a gap between the market price and the fair value.

In the next post I'd like to introduce you to the second of the great risks in finance, already present in this simplified product.

### Certificates of Deposit : first pass at a quantitative analysis

CDs are pretty simple money market instruments, or so the story goes.  Well, what better place to start looking at issues in the quantitative analysis of fixed income products.  In my mind, the end goal is the analysis of the queen of all securities - the convertible bond - that fiendishly complex hybrid of fixed income and equity derivatives.  But one step at a time.

In this blog, I pretend that there's even less complexity in CD instruments than there is.  But first, I'd like to make a clarification on the difference between CDs and your local bank account, something I touched on in the last post.  CDs are bearer instruments - whoever has the legal ownership of them could in theory sell them on (or buy them) in a secondary market.  Now, while this may not happen with every CD, it certainly cannot happen to your own bank current or savings account.

On a first pass, taking the easiest fake CD I can think of, you would interpret the nominal annualised yield of the CD, call it $y%$, as  the yield on some principal $P$ on a simple interest basis, for a CD which lasted one year.  You expect to receive  $P \times (1+y) \times 1$.  That is, the future value of the single cash flow as seen from a valuation date of day 1 of the life of the CD.  As a last step you'd look to find the present value of that future cash flow.  Well, the discount rate to use for the future cash flow $P \times (1+y) \times 1$ arriving in your bank account would be the effective yield of other CDs just like this out there in the world.  I.e. $y$ (on the relatively modest assumption that when the original deal was struck and $y$ was the first CD's annualised yield, it was so since that's what the prevailing rate in the market for CDs just like this was trading at, and the period of time you'd discount it for would be the distance between the valuation date (which is still day 1 of the instrument) until that future cash payment (1 year).  This present valuing back to the first day of the CD leads you back to a PV of $P$.  This makes intuitive sense, since in a way you just bought it and paid $P$ for it, so it isn't too surprising $P$ is the fair value.

Under the normal course of events, as time passes, the present value of the CD moves from $P$ towards the value of $P \times (1+y)$.  That is to say, on the last day, regardless of prevailing interest rates, you will indeed receive $P \times (1+y)$ and that cash flow would then be an immediate cashflow which didn't require further present valuing.  Another way of thinking of this, of making it more natural, is to realise that, the closer you get to the payoff day, the more it'll cost you to buy an instrument which will be worth $P \times (1+y)$.  It starts off costing you $P$ and day by day you find purchasing that same cashflow moving monotonically from $P$ to $P \times (1+y)$.

In future postings, this all gets more complex along a couple of dimensions: day count conventions vary from the implicit ACT/ACT in the above analysis; intervening coupons may be paid out, with compounding effects, the product itself may be contain more structure in the terms and conditions, the PV discounting factor can change on a moment by moment basis through the life of the CD.  But for now, the above represents the simplest possible CD analysis on the simplest possible CD.  It would be good to see how this ground-zero CD ticks over time.