Bond Valuation
Bond Valuation
Bond Valuation
Bond : It is an instrument of loan raised by the Govt. or a company against a specific interest rate and a promised date of repayment. Bonds are secured with specific collateral behind them as distinguished from debentures which are unsecured. Interest of the bond is generally lower than the debenture and is paid semi-annually.
Bond Terminology
Face value/Nominal value: This can be thought of as the principal amount on which interest is paid by the issuer. Issue price: The price at which the bond is issued to the lender. It may be at face value, may be at discount or may be at premium. Redemption value: Generally bonds are redeemed at face value on the maturity date, but some time it is also on premium. Coupon: Bond typically pay interest periodically at a prescribed rate of interest. The annual rate at which this interest is paid is known as coupon rate. Maturity: This is the future date of a bond on which the bond is repaid and extinguished. Some bonds do not repay the principal in one installment but spread it out over several years. In this case the date of last installment is taken as maturity date. Basis points: A basis point is simply one-hundredth of one percent change in interest rates and the difference between the two interest rate are usually stated in basis points.
Security
Unsecured
Value of Bond
The value of bond or any financial assets is equal to the sum of the present value of the cash flows expected from it. The value of bond is the total present value of the expected future cash flows, the cash flow from bond is consist of coupon payment till maturity plus the final payment of redemption value at the time of maturity.
Value of Bond Calculation of Bond Return Current yield: This is the interest received calculated as a percentage of bond s current price. Findings: (i) If the bond is selling at par then current yield would be equal to coupon rate. (ii) If the bond is selling at premium (discount) the current yield would be less (more) than coupon interest rate. Drawback: An important drawback is that it consider only coupon income as a source return to the investors, it ignores the capital gains or losses that would also accrue to them.
Yield to Maturity
In practice an investor considering the purchase of a bond not quoted promised rate of return. Instead the investor must use the bond price, maturity date and coupon payment to infer the return offered by the bond over its life. The YTM is defined as the interest rate that makes the present value of a bond payments equal to its price. This interest rate is often viewed as a measure of the average rate of return that will be earned on a bond if it is bought now and held until maturity. It is also viewed as effective rate of return expected by an investor of a bond if the bond is held to maturity.
Assumptions (YTM): 1. All coupon and interest payment are made on schedule. 2. The bond held till maturity. 3. The coupon payments are fully and immediately reinvested at precisely the same interest rate as the promised YTM. Yield to Call: Some bond carry a call feature that entitle the issuer to call/buyback the bond prior to the stated maturity. For such bonds it is a practice to calculate the YTC as well as YTM.
Decision Criteria: Higher the YTM better the bond, from the view point of the investors. Major drawbacks of YTM:
It is assumed that the cash flows are reinvested at the rate equal to YTM. This may not be true always. Valuation of Zero coupon bonds.
Trading Strategies
I want to maximize rates of return when interest rate change. If you expect a major decline in interest rate Your bond portfolio would be ..
Risk of Bonds
Default risk:: Arises when company default in paying interest or principal. Other things being equal, bonds which carries higher default risk traded at higher stated YTM. Interest rate risk: The change in interest rate in the general level of economy leads to increase in RRR & decrease in price. Inflation risk: Call risk: Issuer redeemed the bond before maturity. Liquidity risk: Barring some popular GoI Bonds the others are not actively traded in the secondary Market.
Duration
The duration of financial asset measures the sensitivity of the asset s price to interest rate movement. Summary statistic of the effective average maturity of a bond. Duration of bond is useful measure of the sensitivity of a bond s market price to interest rate (yield) movement. It is approximately equal to the percentage change in price for a given change in yield. Eg. A 10 years bond with a duration of 7 years means that it would fall approximately 7% in value if the interest rate increased by 1%.
t v PV (C )
t t !1
price
Developed by Frederick R. Macaulay, 1938 Where: t = time period in which the coupon or principal payment occurs Ct = interest or principal payment that occurs in period t i = yield to maturity on the bond
Fredrick Macaulay introduced the concept of Duration by taking weighted average maturity of the bond. Each weight factor shows the relative importance of each cash flow to the bonds value or market price. e.g. A company issues Rs.1000 bond with a coupon of 11% payable annually with a maturity of 6 years . Calculate the duration. Note If nothing is mentioned regarding YTM or required rate of return; Coupon rate will be taken as the proxy of YTM or RRR for discounting.
Properties of Duration
1. The duration of zero coupon bond is equals its time to maturity. 2. Holding maturity & YTM constant, a bond s duration is lower when a coupon rate is higher. 3. Holding coupon rate constant, a bond s duration generally increase with its time to maturity. 4. Holding other factor constant, the duration of a coupon bond is higher when the bond s yield to maturity is lower. 5. The longer the term to maturity of a coupon paying bond, the greater the difference between its duration and term to maturity. 6. The duration of perpetual bond is (1+YTM)/YTM.
Bond Duration in Years for Bonds Yielding 6 Percent Under Different Terms
COUPON RATES
Years to
Maturity 1 5 10 20 50 100 8 0.02 0.995 4.756 8.891 14.981 19.452 17.567 17.167 0.04 0.990 4.558 8.169 12.980 17.129 17.232 17.167 0.06 0.985 4.393 7.662 11.904 16.273 17.120 17.167 0.08 0.981 4.254 7.286 11.232 15.829 17.064 17.167
Source: L. Fisher and R. L. Weil, "Coping with the Risk of Interest Rate Fluctuations: Returns to Bondholders from Nave and Optimal Strategies," Journal of Business 44, n (October 1971): 418. Copyright 1971, University of Chicago Press.
Where: (P = change in price for the bond P = beginning price for the bond Dmod = the modified duration of the bond (i = yield change in basis points divided by 100
(P v100 ! Dmod v (i P
Modified Duration
Modified duration is a modified version of the Macaulay model. It shows how much the duration changes for each percentage change in yield. The formula is used to determine the effect that 1% change in interest rates will have on the price of the bond. A bond having a face value of Rs.500 maturity in 6 years pays a coupon of 12% paid annually. The market price of the bond is Rs.470. The YTM is 13.53%. Calculate Modified duration. Also calculate the % change in price of the bond if the YTM expected to increases by 1.5%.
Bond Convexity
Modified duration is a linear approximation of bond price change for small changes in market yields
(P v100 ! Dmod v (i P
However, price changes are not linear, but a curvilinear (convex) function
Modified Duration
Dmod dP di ! P
For small changes this will give a good estimate, but this is a linear estimate on the tangent line. MD is the slope of the curve at a given yield, mathematically is the first derivative of price with respect to yield divided by price.
Convexity
Convexity is a measure of how much a bond s price-yield curve deviates from the linear approximation of that curve. Convexity always is positive number, implying that the price-yield curve lies above the modified duration (tangent) line.
Determinants of Convexity
The convexity is the measure of the curvature and is the second derivative of price with resect to yield (d2P/di2) divided by price Convexity is the percentage change in dP/di for a given change in yield
d P 2 di Convexity ! P
Determinants of Convexity
Inverse relationship between coupon and convexity Direct relationship between maturity and convexity Inverse relationship between yield and convexity
Relative effect of these two factors depends on the characteristics of the bond (its convexity) and the size of the yield change Convexity is desirable???