Index

Concept of Fixed Income

Interest Rate Movements

Bond Price and Interest Rates

Bond Maturity and Interest Rates

Concept Of Duration

Yield Curves
Concept of Fixed Income
Fixed income returns are steady returns on your investments. Let’s take a simple example. You have placed Rs 100 as a fixed deposit with the State Bank of India (SBI). The period of the deposit is three years and you will receive an interest of 8.5% on your deposit every year for the next three years. Your returns from the fixed deposit will not change for the next three years and you will receive Rs 8.50 every year as interest on the fixed deposit. At the end of three years you get back your initial investment of Rs 100.
Four fixed income concepts emerge from the simple fixed deposit example. The fixed deposit can also be looked at as a bond issued by the SBI. The first concept is the principal or face value. You deposited Rs 100 with the SBI which is the principle. In fixed income security terms, Rs 100 is the face value of the bond issued by SBI. The second concept is the interest rate or coupon. SBI pays you an interest of 8.5% per year on the principal of Rs 100. In fixed income terms, 8.5% is the coupon payable every year on the bond of face value of Rs 100. The third concept is the period of the fixed deposit or maturity. The fixed deposit you placed with the SBI is for a fixed period of three years. In fixed income terms, the bond has a maturity of three years. Hence whether it is a fixed deposit or a bond, for an investment of Rs 100 you receive Rs 8.5 every year for three years, at the end of which you get your Rs 100 back.
The fourth concept is the concept of credit risk. SBI is rated AAA by rating agencies which is the highest safety rating. The highest safety rating implies that you will get back your principle and receive your interest from SBI. SBI is also the largest bank in India and is owned by the Government of India which increases its safety factor. Your principle of Rs 100 and interest payment of Rs 8.5 every year is certain due to SBI being a strong entity. The same will not be certain if you have opened a fixed deposit with a bank that is not given the highest rating by credit rating agencies.
We learnt about the concepts of fixed income in the first tutorial by taking an example of a simple fixed deposit. You placed a principle of Rs 100 in a three year maturity fixed deposit with the SBI (State Bank of India). You received 8.5% per year for three years as interest on your fixed deposit. You got back your money from the SBI at the end of three years. The investment in fixed deposit could be equated to a bond with a face value of Rs 100 paying a coupon of 8.5% per annum and maturing at the end of three years. The difference between the fixed deposit and the bond is the value of the bond when interest rates change.
Interest rate change and its effect on the fixed deposit
A principle of Rs 100 invested in SBI for three years at 8.5% interest rates will generate Rs 8.5 every year for three years. At the end of the third year you get back your principle of Rs 100. The principle remains static throughout the three years, and neither depreciates nor appreciates in value even if interest rates change during the deposit period. For example, after placing the deposit at 8.5% for three years, you go next year to SBI for a three year deposit. SBI offers you 9.5% interest rate for a three year deposit. The higher interest rate offered implies that interest rates have gone up in the market. The fact that interest rates have risen does not affect the principle value of your fixed deposit investment though you have lost the opportunity to invest at higher rates.
Interest Rate Movements
Interest rate change and its effect on the bond
Investment in a bond with similar terms as the fixed deposit exposes you to interest rate risk. You are holding one SBI bond of face value of Rs 100 with an annual coupon rate of 8.5% and having a maturity of three years and interest rates rise. The price of the bond will increase depending on the rate of increase in interest rates. For example if you want to sell the bond next year to place the money in a three year fixed deposit paying 9.5% interest, you will get back only Rs 98/ which is the price of the bond at the time of selling the bond. The price of the bond has decreased due to the rise in interest rates. The face value of the bond remains same at Rs 100 but the value of the bond is lower due to rise in interest rates.
The price of the bond moves up when interest rates fall. If the interest rate on the fixed deposit falls to 7.5% after one year, the price of the bond will then become Rs 102. You have gained Rs 2 on the face value of the bond due to the fall in interest rates. The value of the fixed deposit does not change.
The reason why the price of the bond rises or falls with a rise or fall in interest rates is due to the time value of money. The bond is valued using a discounted cash flow (DCF) method, of which we will talk about in the next tutorial.
The concepts you have learned here is that if interest rates rise or fall your fixed deposit value remains the same while your bond value changes as per the rate of increase or decrease in interest rates.
Bond Price and Interest Rates
Let us take an example of a three year SBI (State Bank of India) bond. The bond is rated AAA which is the highest safety rating given by rating agencies. The terms of the bond are
Issuer: SBI
Rating: AAA
Issue date: 1^{st} August 2011
Maturity period: 3 years
Maturity date: 1^{st} August 2014
Face value of the bond: Rs 100
Coupon rate: 9.5%
Coupon payable: Annually on 1^{st} August every year
Amount payable on maturity: Rs 100
If you invest in this bond and hold it till maturity you will receive 9.5% on your investment every year for three years and at the end of three years you get back your original investment. The face value of the bond is Rs 100 and if you invest Rs 100,000/ you get 1000 bonds (Rs 100,000/Rs 100).
The bond is listed on the NSE (National Stock Exchange) and is tradable. You can sell your holding in the SBI bond partly or wholly. We will elaborate on buying and selling of bonds in later tutorials.
You do not want to hold the SBI bond for three years and after one year of investment you want to sell the bond. Let us assume you sell the bond on the 1^{st} of August 2012 after receiving the first coupon payment of 9.5%. In your case you will receive Rs 9500 as you have invested Rs 100,000 in the SBI bond.
What will you receive after one year? Let us take three scenarios 1) Interest rates are the same at 9.5%. 2) Interest rates have moved up by 100bps (1%) to 10.5% and 3) Interest rates have moved down by 100bps (1%) to 8.5%. 1bps (basis point)= 0.01%
1.Interest rates are the same at 9.5%
Bond price = Rs 100. You get back your investment of Rs 100,000 (1000 bonds * Rs 100)
Total returns including interest received = Rs 109,500 or 9.5%.
2. Interest rates are higher by 100bps.
Bond price = Rs 98.27 You get back Rs 98,270 (1000 bonds * Rs 98.27)
Total returns including interest received = Rs 107,770 or 7.7%
3. Interest rates are lower by 100bps
Bond prices= Rs 101.77 You get back Rs 101,770 (1000bonds * Rs 101.77)
Total returns including interest received= Rs 111,270 (11.27%)
We can see that if interest rates fall your returns are higher than the coupon returns while if interest rates rise your returns are lower than the coupon returns. You should be buying three year SBI bonds if you believe interest rates will fall and you should be investing in SBI fixed deposits instead of SBI bonds if you believe interest rates will rise.
Bonds prices are inversely related to interest rates. The simple equation below gives you the relationship.
SBI three year maturity bond with coupon of 9.5%.
Bond price = 9.5/(1+9.5%)^1 + 9.5/(1+9.5%)^2 + 109.5 /(1+9.5%)^3
where 9.5 is the coupon on face value of 100, 9.5% is the discount factor (or prevailing interest rates for a three year maturity bond) and 109.5 is the coupon + face value received on maturity.
If you plug this equation in an excel sheet and change the discount factor from 9.5% to 10.5% or 8.5% you will see that the bond prices fall when interest rates rise and bond prices rise when interest rates fall.
Bond Maturity and Interest Rates
Let us take the example of the SBI (State Bank of India) bond, which we took in tutorial 3. The bond is rated AAA which is the highest safety rating given by rating agencies. The terms of the bond are
Issuer: SBI
Rating: AAA
Issue date: 1^{st} August 2011
Face value of the bond: Rs 100
Coupon rate: 9.5%
Coupon payable: Annually on 1^{st} August every year
Amount payable on maturity: Rs 100
We will find out the relationship between bond maturity and interest rates by keeping the discount factor (prevailing interest rate in the market) constant and changing the maturity of the SBI bond. We will keep the discount factor constant at 8.5% and take three different tenors for the bond, three years, five years and ten years.
1. SBI bond maturity: 3 years. Discount factor 8.5%
Bond price = Rs 102.53. The three year SBI bond with a coupon rate of 9.5% gains in value by Rs 2.53 for a 100bps (1bps=0.01%) fall in interest rates.
2. SBI bond maturity: 5 years. Discount factor 8.5%
Bond price = Rs 103.90. The five year SBI bond with a coupon rate of 9.5% gains in value by Rs 3.90 for a 100bps fall in interest rates
3. SBI bond maturity: 10 years. Discount factor 8.5%
Bond price= Rs 106.51. The ten year SBI bond with a coupon rate of 9.5% gains in value by Rs 106.51 for a 100bps fall in interest rates.
Tabulating the results of the change in bond prices with a change in maturity keeping the discount factor constant we see that bond prices increase with an increase in maturity when interest rates fall.
The inverse is true when interest rates rise bond with longer maturity see a sharper fall in prices than bond with shorter maturity. Table 2 gives the prices of three, five and ten year maturity SBI bonds at a discount factor of 10.5%.
The higher the maturity of a bond, the more the rise or fall in price of the bond due to change in interest rates.
Two main concepts in fixed income are a) Bond prices change with change in interest rates and b) The extent of change in bond prices to interest rates depend on maturity of the bond. Using this concept, fixed income investors invest in short maturity bonds when interest rates rise and invest in long maturity bonds when interest rates fall.
Concept Of Duration
Duration gives the approximate percentage change in a bond’s price to change in interest rates. A duration of 6.5 years for a ten year bond suggest that for every 100bps change in the bond’s yield, the price of the bond will rise or fall by 6.5%.
What is duration? Duration can be defined as the time taken by a bond to pay back its purchase price. Coupon bearing bonds have durations that are lower than the maturity of the bond while zero coupon bonds have durations that are equal to the maturity of the bond
The reason it takes less time to get back the purchase price in a coupon bearing bond is due to the fact that cash flows keep coming in to the bond holder at regular intervals. A zero coupon bond holder does not receive any cash flow before the maturity of the bond and he has to wait till the bond maturity to get his payments.
Macaulay duration is the most widely used form of calculating duration of a bond. Macaulay Duration is calculated as follows:
Duration = (PVC1*t1+ PVC2*T2+ —PVCnM*tn)/P
PV= Present Value of future cash flows
C1,C2—CN= Coupons
t1,t2—tn= time periods
P=Price of bond
Example of duration
Bond A has maturity of five years and pays annual coupon of 9.5%. The face value of the bond is Rs 100. The duration of the bond assuming bond yield is 9% is as follows.
PV of cash flows of the bond = 9.5/ ((1.09)^(1))+ 9.5/((1.09)^(2))± 9.5/((1.09)^(3))+ 9.5/((1.09)^(4))+ 109.5/((1.09)^(5)) = 8.71+7.99+7.33+6.73+71.16 = Rs 101.95
Duration = 8.71*1+7.99*2+7.33*3+6.73*4+71.16*5= 8.71+15.99+22.00+26.92+355.83= 429.47
Duration in years= 429.47/101.95= 4.2 years.
The buyer of Bond A can expect to be repaid in 4.2 years time and has an interest rate sensitivity of 4.2% for every 100bps change in bond yields.
Duration thumb rules and caution
The higher the coupon on the bond the lower the duration with all other factors remaining constant
The higher the yield to maturity on the bond the lower the duration
Zero coupon bonds have duration that equals maturity of the bond
Duration is only an approximation of bonds sensitivity to interest rates, there are other factors that change the sensitivity including factors such as liquidity, spreads and yield curve movements.
Modified duration
Modified Duration (MD) gives the change in price of a bond to a 100bps change in yield. MD measures the sensitivity of bond prices to change in yields.
MD is calculated as follows
MD= Duration / ((1+(Yield/no of coupons per year))
Taking the example of Bond A. MD of Bond A= 4.2/(1+(9%/1))= 3.86
Bond A’s price will change by Rs 3.86 for a 100bps change in yields.
The difference between duration and MD is that duration is measured in years and gives the payback period for initial cash flows while MD is an absolute measure that gives the bond price sensitivity to change in bond yields.
Convexity
In the bond market to capture price volatility duration is good measure when the changes in yield are small. However if the yield changes are high then we use the measure of convexity along with duration. Convexity is a measure of the curvature of the price/yield relationship.
Conceptually it is defined as
C = (1/P)*(d^{2}P/dy^{2})
P=Price of bond
Y=Yield of bond
(d^{2}P/dy^{2}) = Second partial derivative w.r.t the yield.
Formula for convexity measure is given as
C= (P_{+}+P_{–} – 2*P_{0})/ (P_{0*}(Dy)^{2}) ………………………………………….(1)
P+ = Price of bond when change in yield positive
P = Price of bond when change in yield negative
P0 = Current price
Dy = change in yield
How Convexity Works?
Three bonds and different price movements for change in yields
1. Coupon= 7.16%, Yield= 8.16%, Maturity= 20 May 2023 Settlement Date=27 May 2013
Price =93.25 Duration= 7.17 MDuration= 6.89
2. Coupon= 10%, Yield= 8.16%, Maturity= 20 May 2023 Settlement Date=27 May 2013
Price =112.39 Duration= 6.73 MDuration= 6.41
3. Coupon= 6%, Yield= 6%, Maturity= 20 May 2023 Settlement Date=27 May 2013
Price =100 Duration= 7.64 MDuration= 7.41
(Price in Rs,Duration in years)
If yield is increased by 100bps then
For 1^{st} bond Price =87.094 (6.60%) Duration= 7.06(0.13) MDuration= 6.75(0.11)
For 2^{nd} bond Price =105.4149(6.21%) Duration= 6.62(0.11) MDuration= 6.33(0.14)
For 3^{rd} bond Price =92.90(7.001%) Duration= 7.53 (0.10) MDuration= 7.28(0.13)
(Price in Rs,Duration in years)
If yield is decreased by 100bps then
For 1^{st} bond Price =99.99(7.21%) Duration= 7.28(0.10) MDuration=7.03(0.13)
For 2^{nd} bond Price =120.01(6.77%) Duration= 6.84(0.11) MDuration= 6.61(0.14)
For 3^{rd} bond Price =107.86 () Duration= 7.74(0.09) MDuration=7.55(0.13)
(Price in Rs,Duration in years)
Convexity of bonds are as
1^{st} bond = 61.63
2^{nd} bond = 56.06
3^{rd} bond = 68.62
Lower coupon bonds give the highest price gain and price losses when yield changes are high.
Note that convexity is the second partial derivative of the bond valuation equation w.r.t. the yield. Hence, convexity is the rate of change of duration with respect to the change in yield.
The convexity of the price/YTM graph reveals two important insights:
The price rise due to a fall in YTM is greater than the price decline due to a rise in YTM, given an identical change in the YTM
For a given change in YTM, bond prices will change more when interest rates are low than when they are high.
Yield Curves
Yield curves plot the yield and maturity of fixed income securities that have similar characteristics and similar credit risk profiles. Government bond yield curves plot government bonds that are risk free in nature (with many caveats including bonds issued in local currency) while corporate bond yield curves plot bonds that carry similar ratings of AAA, AA, and BBB etc.
The normal shape of a yield curve is upward sloping i.e. bonds with shorter maturities carry lower yields than bonds with longer maturities. The US treasury yield curve is upward sloping as seen in chart 1.
The US treasury yield curve is upward sloping due to the fact that yields trend higher as one goes along the maturity scale. One year US treasury yields are at 0.15%, five year treasury yields are at 0.55%, ten year treasury yields are at 1.41% and thirty year treasury yields are at 2.49%.
Why is upward sloping the normal shape for a yield curve?
The reason why yield curves are upward sloping is that expectations of future short term rates are higher. The expectations built in an upward sloping yield curve is that the economy will grow down the line and with growth there will be inflation. Hence investors demand a higher yield to hold longer maturity securities given the growth expectations for the future.
The second reason why yield curves slope upwards is that there is uncertainty on how factors such as inflation will behave in the future and this uncertainty is priced in higher yields on longer maturity bonds.
Other shapes of yield curves
Yield curves can also be flat or inverted. Bonds across maturities priced at almost similar yields lead to flat yield curves. Short maturity bond yields priced higher than long maturity bond yields result in inverted yield curves. Chart 2 and chart 3 are examples of flat and inverted yield curves respectively.
The reason why a yield curve is flat is that there is uncertainty in an economy and investors do not know which way the economy will go. An inverted yield curve suggests the opposite of normal yield curve, the economy is expected to slacken rather than grow leading to falling inflation expectations down the line.
In theory different shapes of yield curves have explanations of economic growth and risk preference. However in practice shape of yield curves depend on many factors including central bank actions, liquidity and foreign investor behaviour. We will look into the more practical aspects of yield curves in the next tutorial.