### Bond Price and Yield relationship

We do not make any investments without a return. Bonds are also a type of investment which gives you certain return.

The total return that an investor gets / realises on a bond has a fancy name called as Yield. When i say Total return, it includes both Coupon payments and also the capital appreciation. Now one may ask a question here as to what is the capital appreciation in a Bond. It just pays off the Face value at the maturity. But the answer is that say you purchase a 1Y bond of face value 100 at 94 and it pays you a coupon of 10% p.a and you hold the bond for a year till maturity. so at the end of 6 months you get your first coupon of 5$ and at the end of 1st year, you get the 2nd coupon of 5$ and also the FV of 100$. However, you have paid only 94$ to purchase this bond. so this increased return of 6$ that you receive (100-94) is also a component of the Yield. So now the first sentence will make more sense in terms of total return.

Yield to Maturity is the total return that one gets on a bond if he hold the bond until the end of its life time.

Instead of dwelling much more into how to calculate YTM and what are the underlying assumptions of YTM etc, we will shift our focus to one important concept that everyone needs to know when it comes to bond yields which is the Price Yield relationship. So what is so special about this price Yield relationship?

**Fact**: **The bond price and bond Yields are Inversely related. Yes, you read it right! when the price of the Bond goes UP, the bond Yield goes DOWN and vice versa.**

Now, why is there such a relationship between price and Yields. why are they opposite polls ? i will try to give you 3 types of intuitive explanation for this question. Firstly, the common sense explanation, Secondly, the arbitrage explanation and thirdly, the mathematical explanation.

Lets start with common sense explanation.

I started with the statement that Bond yield is the total return that you make on your bond investment. So when you pay less for the investment of same worth,the more you gain and more your return is. This proves that when you pay less for the bond (i.e when the bond price goes down) you get a higher return (the Bond yield goes up). For example assume that a bond with Face value 100 is trading at 95. now one should pay $95 to purchase a bond worth 100 and say he gets a yield of (hypothetical) 5%. Now if the Bond price goes further down to $92, the investor needs to pay only 92 to get a bond worth of 100 than 95 earlier. so his gains / returns from this bond increases which means that the Yield increases as the price falls.

Now that we have had a basic reasoning of why there is an inverse relationship between bond yield and price, lets deepen our understanding by understanding the arbitrage explanation.

We have all heard the term, There is No Free Lunch. That is what arbitrage free market is all about. We all get paid of the risk that we take. or if I may put the statement in more relevant way , for 2 different investments with same risk and other profiles, the return should be same. So What happens when the returns are not same and how does this link to the inverse relationship between Bond price and Yield?

When there exists a difference in the returns between 2 instruments of same risk and other profiles, there will exist an arbitrage opportunity. This arbitrage opportunity does not exists in practical life or if it exists, is very short lived. Let’s explain this with an example.

Say Company X issues a bond with coupon of 6% and 1 year later, the interest rate in market rises and a similar company Y with similar rating risk and other profiles issues a similar bond but with a coupon of 7%. Now as a rationale investor, which bond do we buy? the obvious answer is Company Y bond because that yields more. Now so what happens to company X’s bonds. the Bonds price adjusts itself ( i.e. the bond’s price fall) so that it can also attract buyers. So now as a rationale investor, you are indifferent between both the bonds of Company X and Company Y because even though company X pays lesser coupons compared to company Y, the company X’s bonds are cheaper which increases the yield to make it equal to the yield from Company Y’s bonds. This example explains why there is an inverse relationship between bond yield and price.

Lastly, we look at the mathematical explanation for this inverse price and yield relationship.

The bond’s theoretical current price is arrived at by discounting the coupon and FV at Yield to Maturity. So now the YTM sits in the denominator in the bond pricing equation. now imagine what happens to the end result when the denominator increases ? the end Value falls. That is exactly something that happens with Yield and price. in the Bond pricing equation, if the denominator (Yield) increases, the end result (Bond price) falls and Vice versa.