# Duration and yield

I have some basic questions about mainly duration and yield.

1) Almost no-one defines what yield they are talking about when talking about duration and discount rate, I've seen some talk about interest rate some yield etc I'm assuming (from wiki on duration also) that usually people are talking about yield to maturity. Is this the same for general yield curves also do they mean yield to maturity?

2) I'm aware that Macaulay duration is the weighted present value time to receipt of cash flows but I'm unsure as to what that actually means from a practical stand point. Say a maturity is 5 years and the duration is 4.5 what does that actually tell me, I'm slightly confused as i believe it means that it would effectively take 4.5 years to earn the present value of all cash flows but i will only earn all the money back at maturity as the final payment is the biggest. Maybe an example of portfolio immunization might help here , why is it better to match durations over maturity for example?

3) When we say that bond prices rise when interest rates fall, are we talking about the base rates, like the bank of England rate rising?

4) For modified duration I'm unsure as to how this measures the sensitivity in price to interest rates. I've looked at many pages but they just put the equation and say it does, Wikipedia is quite good but I'm still unsure as to how exactly dividing through by 1 plus the rate gives you sensitivity to the interest rate. I really want to know the mechanics behind it however

5) I've read examples explaining that when interest rates rise for example 1% for example, the price must fall to increase the yield by 1% in order to make up for it. If this is the case then surely we can just work out the price from that, how does duration play into it or is this just a too simple example.

I realise these are very basic questions but I'd like to know how these all fit together as opposed to just learning the equations and i find some definitions on the internet very confusing. Sorry if some are difficult questions to understand.

• If I were you I would focus on Modified Duration, and its interpretation in terms of the sensitivity of bond price to a (small) change in yield. That is the key concept. The Macaulay duration (developed earlier) is no longer all that interesting nowadays, and its interpretation in terms of weighed average of cash flow times is of no practical use IMHO. If I was writing a modern textbook I would downgrade Macaulay duration to a footnote or even omit it entirely. Apr 12, 2019 at 22:02
• Thanks Alex, do you mean yield to maturity when you say change in yield,? Apr 12, 2019 at 22:08
• Yes, change in YTM. The bond price is a function of the YTM and by taking the derivative of this function with respect to y we can derive the Modified Duration. In the process of taking the derivative it just so happens that a factor 1/(1+y) comes out, but that is of no great significance. It is just there. Apr 12, 2019 at 22:14
• Thanks this actually really helped in the end, i think i was looking for some intuitive link between Macaulay duration and the 1+r to make it modified duration but just deriving it actually makes sense Apr 13, 2019 at 15:36