## Sunday, 5 July 2015

### 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%