Understanding Yield Spreads : Understanding Yield Spreads Presented by: CA Tarun Mahajan Prepared by: CA Tarun Mahajan
Yield Curve? : Yield Curve? Different maturity bonds may have different yields. If take Maturity on x-axis & Yield on Y-axis then the resulting graph is termed as Yield curve.
Yield curve can take any shape. Four common shapes of yield curve are as follows:
Normal or upward sloping
Inverted or downward sloping
Flat
Humped Prepared by: CA Tarun Mahajan
Normal (Upward Sloping) Yield Curve : Normal (Upward Sloping) Yield Curve Term to Maturity Yield Normal It Indicates that yield increases with maturity. Prepared by: CA Tarun Mahajan
Inverted (Downward Sloping) Yield Curve : Inverted (Downward Sloping) Yield Curve Term to Maturity Yield Inverted It Indicates that yield decreases with maturity Prepared by: CA Tarun Mahajan
Flat Yield Curve : Flat Yield Curve Term to Maturity Yield Flat It Indicates that yield is same for all the maturities. Prepared by: CA Tarun Mahajan
Humped Yield Curve : Humped Yield Curve Term to Maturity Yield Humped It Indicates that yield is totally independent of maturity. Prepared by: CA Tarun Mahajan
Slide 7 : Theories of Term Structure of Interest Rate
Pure Expectation Theory
Liquidity Preference Theory
Market Segmentation Theory Prepared by: CA Tarun Mahajan
Pure Expectation Theory : Pure Expectation Theory Yield of a longer maturity is an average of the short term rates that are expected in future.
If 1st year’s yield is 6% and 1 years yield, 1 year from now is 8% than current two years yield will be (1.06 x 1.08)1/2 – 1 = 7%.
If short term rates are expected to rise then long term yield will be more than short term yield and yield curve will be upward sloping and vice versa. Prepared by: CA Tarun Mahajan
Pure Expectation Theory: Shape of Yield Curve : Pure Expectation Theory: Shape of Yield Curve Prepared by: CA Tarun Mahajan
Liquidity Preference Theory : Liquidity Preference Theory Investors require a risk premium for holding longer term bond.
Higher the liquidity risk premium, higher will be the long term yield.
Here shape of yield curve will be a function of future expected interest rate & liquidity risk premium.
Even under this theory yield curve can take any of the four shapes. Prepared by: CA Tarun Mahajan
Liquidity Preference Theory : Liquidity Preference Theory Even if there is an expectation of declining short term rates in future, still yield curve may be upward sloping due to high liquidity risk premium. Pure Expectation Curve Liquidity Preference Curve Liquidity Premium Prepared by: CA Tarun Mahajan
Market Segmentation Theory : Market Segmentation Theory Different Parties in the market have preference for different maturities.
Yield at a particular maturity is dependent on demand & supply of funds at that level.
Preferred habitat theory says that though parties have preference for a particular maturity but they can be moved to other maturities by offering better yield.
Here also yield curve can take any shape. Prepared by: CA Tarun Mahajan
Market Segmentation Theory : Market Segmentation Theory Maturity Yield Short Term Long Term Medium Term Prepared by: CA Tarun Mahajan
Yield Spreads : Yield Spreads It means the difference in yield on two bonds.
Three different yield spread measures are as follows:
Absolute Yield Spread: It is simply the difference between yield on two bonds.
Relative Yield Spread: (High Yield/Low Yield)-1 or
(Absolute yield spread/ lower yield) – 1
Yield Ratio: High Yield / Low Yield
Example: yield on bond A & B are 5% & 6%. Hence absolute is 1%, relative is 20% and ratio is 1.20. Prepared by: CA Tarun Mahajan
Intermarket & Intramarket Spread : Intermarket & Intramarket Spread US bond market is segmented into following sectors on the basis of issuer:
Government Sector, Government Agencies, Municipal Sector, Corporate Sector, Mortgage Sector, Asset Backed Securities Sector, Foreign Sector etc.
Difference in yield between corporate bond & T-bill will be termed as intermarket spread
Difference in yield between two corporate bonds will be termed as intra market spread. Prepared by: CA Tarun Mahajan
Credit Spread : Credit Spread Means the difference between two issues that are similar in all respect except for credit rating. Say difference in yield between AAA and AA rated corporate bonds.
During expanding economy credit spread declines.
During economic contraction credit spread increases. This is due to greater probability of default. Prepared by: CA Tarun Mahajan
Yield Spread: embedded options & Liquidity : Yield Spread: embedded options & Liquidity An investor will demand more return for a callable bond. Hence callable bonds will have more yield spread as compared to option free bonds.
Put option gives investor an additional right. Hence he will demand a lower return. Therefore putable bonds will have lower yield spread.
Bonds having less liquidity will have higher yield spread. Prepared by: CA Tarun Mahajan
After tax yield & tax equivalent yield : After tax yield & tax equivalent yield If there are two bonds, one is taxable and other is tax free. Then to compare their yield:
either you will have to calculate after tax yield for taxable bond
After tax yield = Taxable yield x (1-tax rate)
Or you will have to calculate before tax yield (also called tax equivalent yield) for the tax free bond.
Tax equivalent yield = Tax free yield / (1-t) Prepared by: CA Tarun Mahajan
Interest rate policy tools available to Central Bank : Interest rate policy tools available to Central Bank Though the interest rates are determined by economic conditions but government also attempts to control it through following tools:
Discount rate: it is the rate at which banks can borrow reserves from FED. Lower discount rate results in lower interest rate.
Open Market Operations: means buying/ selling of T-Sec. by Fed in the open market. Buying T-Sec. results in lower interest rate.
Bank Reserves: % of deposit that bank must retain. Lower reserve results in lower interest rate.
Credit policy: asking banks to loosen their lending policy will result in lower interest rate. Prepared by: CA Tarun Mahajan
Valuation of Debt Securities : Valuation of Debt Securities Presented by: CA Tarun Mahajan Prepared by: CA Tarun Mahajan
Steps for valuation of bond : Steps for valuation of bond Estimate the cash flows over the life of the bond, i.e., coupon & principal.
Determine the appropriate discounting rate based on the risk associated with cash flows.
Calculate present value of cash flows and make a total.
For example value of a 5 year $1000 par value of 12% coupon bond at 10% discounting rate is $1076. Prepared by: CA Tarun Mahajan
Difficulty in estimating Cash Flows : Difficulty in estimating Cash Flows It is difficult to estimate likelihood of default and quantum of default for payment of coupons or principal.
It is difficult of estimate timing & amount of repayment of principal for callable, putable, pre-payable or accelerated sinking fund securities.
Coupon payment is difficult to estimate for floating rate securities.
Cash flows are difficult to estimate for convertible or exchangeable bonds. Prepared by: CA Tarun Mahajan
Appropriate Discounting Rate : Appropriate Discounting Rate For Treasury securities suitable discounting rate is risk free rate. Which is either a single rate for all the cash flows or a series of rates for different cash flows.
For other securities discounting rate is risk free rate plus a suitable risk premium. Higher the risk higher the premium. Prepared by: CA Tarun Mahajan
Discounting with spot rate yield curve : Discounting with spot rate yield curve Here an appropriate discounting rate is given for each individual cash flow, based on how far in the future will it be received.
These individual rates are called spot rates.
Example: 15% $100 par value bond maturing after 3 years is to be discounted with the following rates: year 1= 11%, year 2 = 12% and year 3= 13%. Prepared by: CA Tarun Mahajan
Arbitrage free bond valuation : Arbitrage free bond valuation Above value of $105.17 is called arbitrage free value of the bond.
If the actual value of bond is higher than 105.17 then we can make arbitrage profit by buying the individual strips and selling the bond and vice versa. Prepared by: CA Tarun Mahajan
Change in value with respect to change in interest rate : Change in value with respect to change in interest rate $1000 par value 10% coupon, 3 year bond with semiannual coupons. Prepared by: CA Tarun Mahajan
Change in value with respect to change in interest rate : Change in value with respect to change in interest rate We can observe that :
Bond price is inversely related with discounting rate.
Price yield curve is convex. Means with rise in yield bond price reduces slowly but with fall in yield bond price increases rapidly.
This convexity is good for bondholder. Prepared by: CA Tarun Mahajan
Price of a bond as it approaches maturity : Price of a bond as it approaches maturity Continuing with previous example we can calculate value and observe that it approaches par as the bond approaches maturity. Prepared by: CA Tarun Mahajan
Value of zero coupon bond : Value of zero coupon bond Value of a zero coupon bond is very simple to calculate. It contains only one cash flow at maturity hence we are required to discount that cash flow only.
Calculate value of a zero coupon $1000 maturity bond maturing in 5 years, if the yield is 10% p.a. compounded half yearly.
1000/ (1.05)10 = 614 Prepared by: CA Tarun Mahajan
Yield Measure, Spot Rates & Forward Rates : Yield Measure, Spot Rates & Forward Rates Presented by: Tarun Mahajan Prepared by: CA Tarun Mahajan
Yield Measure : Yield Measure Following are the various yield measures:
Current Yield
Yield to Maturity
Yield to Call
Yield to first par call
Yield to Refunding
Yield to Put
Cash Flow Yield
Realized Yield Prepared by: CA Tarun Mahajan
Current Yield : Current Yield Annual Cash Coupon
Current Bond Price
Calculate current yield for $1000 par value, 10% p.a. semi annual pay bond that is currently quoted at $1100
Current Yield = 100/1100 = 9.09% Prepared by: CA Tarun Mahajan
Yield to Maturity (YTM) : Yield to Maturity (YTM) Rate of return to the investor if he holds the bond till the date of its maturity
It is the discounting rate which equates the present value expected cash flows with the current market price of the bond. It is similar to IRR
It is calculated as a semi annually compounded rate and then multiplied by 2 to make is annualized.
This semi annually compounded rate is also termed as Bond Equivalent Yield (BEY). Prepared by: CA Tarun Mahajan
YTM: Example : YTM: Example Calculate YTM for 10% $1000 par value, 2 years bond, if it pay coupon semiannually and is currently traded at $950.
N=4, PV= -950, PMT=50, FV=1000,
CPT I/Y = 6.46, i.e.,
12.92% p.a. compounded semi annually Prepared by: CA Tarun Mahajan
YTM for zero coupon bonds : YTM for zero coupon bonds It can be calculated either as a annual pay YTM or semi annual pay YTM.
The convention is to quote it as BEY (semi annual pay YTM)
Calculate annual pay & semi annual pay YTM for a $1000 par value T-note maturing in 5 years and currently quoted at $613.91
FV = 1000, PV = -613.91, N = 5 (10)
CPT I/Y = 10.25% (10%) Prepared by: CA Tarun Mahajan
Yield to Call : Yield to Call It is the yield to investor if he holds the bond till the date of call.
Method of calculation is same as YTM. Here N= no. of periods till call and FV= call price.
If a bond contains a provision for call at par at some time in future, we can also calculate “Yield to first par call”.
A bond currently quoted at $1100 is callable at $1050 after 3 years, callable at par after 4 years and matures after 30 years. If the coupon is 10% p.a. payable semiannually, calculate all the yields. Prepared by: CA Tarun Mahajan
Slide 37 : Current yield = 100/100 = 9.09%
YTM = 9.02% p.a.
YTC = 8.34% p.a.
YTFPC = 7.09% p.a.
Yield to worst is the worst possible yield outcome. Here it is 7.09%
Yield to refunding is the yield till the date when refunding protection ends.
Yield to put is the yield till the date of put. It is normally higher than YTM. Prepared by: CA Tarun Mahajan
Cash Flow Yield (CFY) : Cash Flow Yield (CFY) It is calculated for mortgage backed and other amortizing securities.
This securities have monthly principal payment including prepayments
If we assume certain prepayment rates over the period of loan and incorporate these pre payments into out calculation, resulting yield is called Cash Flow Yield.
Monthly CFY can be converted into BEY as follows: BEY = [(1+monthly CFY)6-1] x 2 Prepared by: CA Tarun Mahajan
Realized Yield : Realized Yield Realized yield means the yield calculated considering reinvestment of coupons and intermediate repayment of principal.
Calculate RY for a 3 year 10% coupon (annual pay) bond quoted at 10% discount if the coupons can be reinvested at 6%.
YTM: N=3, PV= -900, PMT=100, FV=1000
CPT I/Y = 14.33% Prepared by: CA Tarun Mahajan
Realized Yield : Realized Yield RY: FV = 100(1.06)2 + 100(1.06)+110=
= 1318.36
PV = -900, CPT I/Y = 13.57%
We can observe that if the reinvestment rate is less the YTM, RY will also be less than YTM and vice versa.
Higher coupons & longer maturity bonds have more reinvestment risk.
A non callable zero coupon bond will have no reinvestment risk. Prepared by: CA Tarun Mahajan
Comparing bonds with different coupon frequencies : Comparing bonds with different coupon frequencies BEY = [(1+ annual YTM)1/2 -1 ] x 2 and
Annual YTM = (1+BEY/2)2 – 1
Co. A has 10% p.a. semi annual yield and co. B has 10.10% p.a. annual yield. Which is higher.
Annual: A= 10.25% and B 10.10%
Semiannual: A = 10% and B = 9.86%
In either way A is better. Prepared by: CA Tarun Mahajan
Slide 42 : Spot Rate Yield Curve Prepared by: CA Tarun Mahajan
Spot Rates : Spot Rates Spot rates are the discounting rate for single payments.
It can be viewed as the discounting rate on zero coupon bonds.
Spot rates for different maturities can be derived either YTMs or from forwards rates.
A curve between maturity & spot rates is called Spot rate curve. Prepared by: CA Tarun Mahajan
YTM to Treasury Spot rate Curve: Bootstrapping : YTM to Treasury Spot rate Curve: Bootstrapping 6 months Spot rate = 8%
It is same as 6 months YTM because there is only one payment for the 6 month Bond. Prepared by: CA Tarun Mahajan
Slide 45 : 1 year Spot Rate:
50/(1.04) + 1050/(1+S/2)2 = 1000
S = 10.05%
18 months Spot Rate:
60/(1.04) + 60/(1+1.05025)2+ 1060/(1+S/2)3 = 1000
S = 12.17% Prepared by: CA Tarun Mahajan
Spreads Nominal, Zero Volatility and Option Adjusted : Spreads Nominal, Zero Volatility and Option Adjusted Prepared by: CA Tarun Mahajan
Nominal Spread : Nominal Spread It is the simplest measure.
It is equal to YTM of the bond less YTM of similar maturity T-security.
Say Bond A has YTM of 12% & T-bond has YTM of 9%. Nominal Spread = 3%
YTM has a limitation that it ignores shape of spot yield curve and discount all the cash flows from same rate. Hence nominal spread suffers the same limtation. Prepared by: CA Tarun Mahajan
Zero Volatility Spread (ZVS) : Zero Volatility Spread (ZVS) It considers shape of the spot yield curve.
A 10% annual pay $1000 par value coupon bond has 3 year to maturity and current market value of $ 1028.59.
Treasury spot rates are 6%,7% & 8% respectively for 1st , 2nd and 3rd year.
If we add a mark up of 1% to theses rates and use that to discount bond cash flows:
100/1.07 + 100/1.082 + 1100/1.093 = 1028.59 Prepared by: CA Tarun Mahajan
Zero Volatility Spread : Zero Volatility Spread it makes present value of future cash flows equal to current market price of the bond.
This mark up of 1% is termed as Zero Volatility Spread.
ZVS can be calculated by doing trial & error. Prepared by: CA Tarun Mahajan
Option Adjusted Spread (OAS) : Option Adjusted Spread (OAS) Callable bonds have more yield hence more Z spread compared to option free bonds
Putable bonds have less yield hence less Z spread compared to option free bonds.
If we remove the effect of options from z spread to make it equivalent to an option free bond then the resulting figure is called Option Adjusted Spread.
OAS = ZS – cost of call option
OAS = ZS + cost of put option Prepared by: CA Tarun Mahajan
Forward Rates : Forward Rates Prepared by: CA Tarun Mahajan
Forward Rate : Forward Rate Forward rate means the rate of interest for lending/ borrowing in future.
1f2 means rate of interest for a one year loan to be taken 2 year from now.
Calculate value of a three year 10% coupon bond given 1f0 = 10%, 1f1=11%, 1f2=12%.
100/1.10 + 100/(1.10x1.11) + 1100/(1.10x1.11x1.12)
= $ 977.18 Prepared by: CA Tarun Mahajan
Deriving Spot Rate from Forward Rate : Deriving Spot Rate from Forward Rate Spot rate is equal to Geometric Means of short term spot rates.
In the previous example two year spot rate is:
(1.10x1.11)1/2 -1 = 10.5%
Three year spot rate is :
(1.10x1.11x1.12)1/3 -1 = 11%
We can also approximate spot rate by simply taking Arithmetic mean of forward rates. Prepared by: CA Tarun Mahajan
Deriving Forward Rate from Spot Rates : Deriving Forward Rate from Spot Rates We know that S2 = [(1+1f0) x (1+1f1)]1/2- 1
Now 1f0 = S1
Hence S2 = [(1+S1) x (1+1f1)]1/2- 1
By Solving [(1+ S2 )2/ (1+S1)] – 1 = 1f1
By extending the same logic we can say:
[(1+ S3 )3/ (1+S2)2] – 1 = 1f2
Calculate 1f3 if S3 = 10% & S4 = 11%.
Answer: 1f3 = 14.05% Prepared by: CA Tarun Mahajan
Deriving Forward Rate from Spot Rates : Deriving Forward Rate from Spot Rates A shortcut for calculating approximate value can be as follows:
S3 x 3 - S2 x2 = 1f2
In the previous example:
1f3 = 11x4 10x3 = 14% Prepared by: CA Tarun Mahajan
Measurement of Interest Rate Risk : Measurement of Interest Rate Risk Presented by: Tarun Mahajan Prepared by: CA Tarun Mahajan
Interest Rate and bond price : Interest Rate and bond price We know that there is an inverse relationship between interest rate (yield) and bond value. When interest rate falls bond price increases and vice versa.
There are two ways to calculate change in bond price when interest rate changes:
Full Valuation Approach
Duration & Convexity Approach. Prepared by: CA Tarun Mahajan
Macaulay’s Duration of Bond : Macaulay’s Duration of Bond According to Macaulay “Duration of a bond is that point of time at which if the entire amount (all the remaining coupons and principal amount) is paid lump sum, won’t result in loss of interest to either party.”
Duration = ? t x Wt
Here t = a year/ period; Wt = Weight of year t
= Present value of cash flow for that year Current market value of the bond Prepared by: CA Tarun Mahajan
Duration of 3 years, 7% coupon bond having current market value of 92.54 and Yield of 10% may be calculated as under: : Duration of 3 years, 7% coupon bond having current market value of 92.54 and Yield of 10% may be calculated as under: Here duration is 2.8. Prepared by: CA Tarun Mahajan
Effect of bonds characteristics on Duration : Effect of bonds characteristics on Duration Duration is higher for longer maturity bonds.
Duration is higher for lower coupon bonds. It is equal to maturity for zero coupon bonds.
Duration is lower for higher yield bonds. Because higher yield reduces present value hence weight of later years. Prepared by: CA Tarun Mahajan
Duration & price change : Duration & price change ?P/P = [-D/(1+Y)] x ?Y
?P/P = proportionate change in bond price
D= Duration
Y = Current yield on the bond
?Y = Change in yield
Minus sign indicates inverse relation between bond price and yield.
D/(1+Y) is also termed as modified duration (MD) hence D= -MD x ?Y Prepared by: CA Tarun Mahajan
Duration & price change : Duration & price change In the previous example if yield increase from 10% to 11% :
?P/P = - 2.8/1.10 x .01 =.025 =2.54%
New price = 92.54-2.54% = 90.19.
If we calculate bond price by full valuation, i.e., discounting bond cash flows @11%, we get P = Rs.90.23
This difference in value under two methods is due to convexity. Prepared by: CA Tarun Mahajan
Convexity : Convexity It is a measure of curvature of the price-yield curve. More curved the price-yield curve, greater the convexity.
A straight line has zero convexity.
Convexity is good for bond holders because when yield falls bond price increases sharply but when yield rises bond price decreased slowly. Prepared by: CA Tarun Mahajan
Convexity : Convexity Price Yield Actual Price Yield Curve Price based on
Duration only Duration underestimates
the price. Prepared by: CA Tarun Mahajan
Duration with convexity and price change : Duration with convexity and price change ?P/P = - [D/(1+Y)]x?Y + ½ Convexity x (?Y)2
Now if yield rises first part of RHS will be negative while second part will be positive hence net result will be less negative. i.e., less fall in price.
If yield falls first part will be positive and second will also remain positive hence more positive, i.e., more rise in price.
In the previous example if convexity is 9.85 then P = 90.23 Prepared by: CA Tarun Mahajan
Duration of Portfolio : Duration of Portfolio Duration of a bond portfolio is weighted average of duration of individual securities.
Dp= D1xW1+ D2xW2 +…….. + DnWn
Calculate duration of a bond portfolio consisting of bond A and bond B having market value of Rs.2 crores and 3 crores respectively, if their respective durations are 6 and 3.5
Dp = 6x0.4 + 3.5x0.6 = 4.5 Prepared by: CA Tarun Mahajan
Limitation of Duration : Limitation of Duration While we calculate price change using duration there is an in built assumption that yield will change by same rate for all the maturities.
In other words we assume that there will be a parallel change in the yield cure. (This assumption is unreliable especially for a portfolio)
If there is a non parallel shift in yield curve, and for bond with embedded options duration will be misguiding. Prepared by: CA Tarun Mahajan
Effective Duration (ED) : Effective Duration (ED) ED = (V- - V+) / (2V0 x ?Y)
= Average % price change
Change in Yield
Example: Price of a 10 year bond is Rs.930 it rises to 950 for a .25% fall in yield while it falls to 915 for a .25% rise in yield. Calculate effective duration.
Avg. % price change = (950-915)/ (2x930)=1.88
ED = 1.88/.25 = 7.53 Prepared by: CA Tarun Mahajan
Effective Duration : Effective Duration Effective duration is a better measure as compared to Macaulay’s Duration or Modified Duration.
Because it captures non parallel shift in price-yield curve as well as effect of embedded options.
However it will perform best for yield changes close to yield changes used in calculation of effective duration. Prepared by: CA Tarun Mahajan
Price Value of a Basis point (PVBP) : Price Value of a Basis point (PVBP) It means dollar change in bond price for one basis point (.01%) change in yield.
It is Bond Value x Duration x .0001
For example a bonds value is $100,000 and its duration is 6.
its PVBP = 100,000 x 6 x.0001 = $60 Prepared by: CA Tarun Mahajan