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%
