Fixed Income

Bonds

 
Of all the fixed Income Instruments FD, Certificate of deposits, Commercial Papers, PTC bonds have always been the center of focus be it Government or corporate bonds.

What is a bond?

By definition, bonds are debt obligations issued by government/Corporate entities. When you buy a bond, you are loaning money to the issuer in exchange for a set number of interest payments over a predetermined period. At the end of that period, the bond reaches its maturity date, and the full amount of your original investment is returned to you.
 
Tracking bonds can be about as thrilling as watching a chess match, whereas watching stocks can have some investors as excited as NFL fans during the Super Bowl. However, don't let the hype (or lack thereof) mislead you. Both stocks and bonds have their pros and cons. This article will explain the advantages of bonds and offer some reasons as to why you may want to include them in your portfolio.   

Those just entering the investment scene are usually able to grasp the concepts underlying stocks and bonds. Essentially, the difference can be summed up in one phrase: debt versus equity. That is, bonds represent debt and stocks represent equity ownership.
 
This difference brings us to the first main advantage of bonds: In general, investing in debt is safer than investing in equity. The reason being the priority that debt holders have over shareholders. If a company goes bankrupt, debt holders are ahead of shareholders in the line to be paid. In a worst-case scenario, such as bankruptcy, the creditors (debt holders) usually get at least some of their money back, while shareholders often lose their entire investment. 

Second advantage is getting income in the form of coupons which is periodical. Other advantages include Tax benefits in Tax exempt bonds. Inflation adjusted return in inflation linked bonds.
 
The various notions in a bond are Coupon (same as periodic interest payments can be annual semi-annual, inflation linked and so on), YTM (yield to maturity is the discounting factor or the opportunity cost), Accrued interest (it is the interest buyer owes to the seller and considered an outflow in bond calculation), Dirty Price (clean price + accrued interest used for Total consideration calculation).

Duration and Convexity

Definition: The duration of a financial asset that consists of fixed cash flows, for example a bond, is the weighed average of the times until those fixed cash flows is received. When an asset is considered a function of yield, duration also measures the price sensitivity to yield, the rate of change of price with respect to yield or the percentage change in price for a parallel shift in yields.
 
Duration is an estimated measure of the price sensitivity of a bond to a change in interest rates. It can be stated as a percentage or in dollar amounts. It can be helpful to "shock" or analyze what will happen to a bond when market rates increase or decrease.
Types of Duration Calculation:

Macaulay Duration: The weighted average term to maturity of the cash flows from a bond. The weight of each cash flow is determined by dividing the present value of the cash flow by the price and is a measure of bond price volatility with respect to interest rates.

Macaulay duration can be calculated by:

Modified Duration: This is a formula that expresses the measurable change in the value of a security in response to a change in interest rates. It is calculated as follows:

In which:
n = number of coupon periods per year
YTM = the bond's yield to maturity
 
Example:
Let's assume that the calculation yields a duration of 6.14, this means that if interest rates change, the value of the bond will change by 6.14%. If there is a 50 basis point change, the value will change by 3.07% and for a 25 basis point change would equal a 1.53% change.


Effective Duration: A duration calculation for bonds with embedded options. Effective duration takes into account that expected cash flows will fluctuate as interest rates change.
 
Example:
Stone & Co. bonds are selling at 95, yielding 5.25%
Let's assume that yields increase by 25bps, causing the price to decline to 93.
Therefore, the price changes by 2.1%. Now let's assume that yields decrease by 25bps, causing the price to increase to 98. As a final step, just average the two percentage price changes for a 1 basis points move in rates.

Answer:
Duration = Price if yield decline - Price if yield increase / 2 * (initial price) *change in yield in decimals
 
As such: 98-93/ 2*95*.0025 = 10.52
 
Approximate Percentage Price Change of a Bond Given a Change in Duration
Let's continue with the above duration of 10.52. This would equal a percentage price change of 10.52 % for a change of 100 basis points in either direction. If the basis points change were 50, then the percentage price change would be 5.26% (10.52/2). If it were a 25bps change, the value would be 2.63% (10.52 / 4).

Approximate New Price of a Bond Given the Duration and New Yield Level
Let's return once again to working with a duration of 10.52. This time, we'll add a total market value of the Stone & Co bonds of $10,000,000.

Assume that the rates change by 100 bps. This would cause the value of the bonds to change by $ 1,052,000 ($10,000.000 *.1052). This is also known as dollar duration. The price will then range from $11,052,000 to $8,948,000.

If rates increase by 50 basis points, however, the dollar change would be $526,000 giving the bonds a price range of $ 10,526,000 to $ 9,474,000.

Convexity is a measure of the sensitivity of the duration of a bond to changes in interest rates, the second derivative of the price of the bond with respect to interest rates (duration is the first derivative). In general, the higher the convexity, the more sensitive the bond price is to the change in interest rates.

Why bond convexities may differ

The price sensitivity to parallel changes in the term structure of interest rates is highest with a zero-coupon bond and lowest with an amortizing bond (where the payments are front-loaded). Although the amortizing bond and the zero-coupon bond have different sensitivities at the same maturity, if their final maturities differ so that they have identical bond durations they will have identical sensitivities. That is, their prices will be affected equally by small, first-order, (and parallel) yield curve shifts. They will, however, start to change by different amounts with each further incremental parallel rate shift due to their differing payment dates and amounts.

For two bonds with same par value, same coupon and same maturity, convexity may differ depending on at what point on the price yield curve they are located.

Suppose both of them have at present the same price yield (p-y) combination; also you have to take into consideration the profile, rating etc., of the issuers: let us suppose they are issued by different entities. Though both bonds have same p-y combination bond A may be located on a more elastic segment of the p-y curve compared to bond B.

This means if yield increases further, price of bond A may fall drastically while price of bond B won’t change that is, bond B holders are expecting a price rise any moment and are therefore reluctant to sell it off, whereas bond A holders are expecting further price-fall and ready to dispose of it.

This means bond B has better rating than bond A. So the higher the rating or credibility of the issuer the less the convexity and the less the gain from risk-return game or strategies; less convexity means less price-volatility or risk; less risk means less return.