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**Unit II: Valuation of Securities Lesson No. 2.2– Bond Valuation**

MBA & MBA – Banking and Finance (Term-IV) Course : Security Analysis and Portfolio Management Unit II: Valuation of Securities Lesson No. 2.2– Bond Valuation

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BOND PRICING The value of a bond is equal to the present value of the cash flows expected from it. Requisites to determine value of a bond: > estimate of expected cash flows > estimate of the required return Assumptions: > The coupon rate is fixed for the term of the bond > The coupon payments are made every year > Each coupon payment is receivable exactly a year later than the preceding payment. > The bond is redeemed at par on maturity.

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**VALUATION of BONDS P= ∑ C + M t=1 (1+r)t (1+r)n**

Where P= Value (in Rs.) n= number of years C= annual coupon payment (in Rs.) r= periodic required return M= maturity value t= time period when the payment is received

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PROBLEMS Find the value of 12%, Rs.1,000 par value bond with a maturity period of 10 years, and a required rate of return of 13%. A Rs. 100 par value bond bearing a coupon rate of 12 per cent will mature after 5 years. What is the value of the bond, if the discount rate is 15%? A Rs. 100 par value bond bearing a coupon rate of 12 per cent will mature after 7 years. What is the value of the bond, if the discount rate is 14%? 12 percent

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**BOND VALUE with Multiple Compounding in a year**

dn P= ∑ C/d M t=1 (1+r/d)t (1+r/d)dn Where P= Value (in Rs.) n= number of years C= annual coupon payment (in Rs.) r= periodic required return M= maturity value t= time period when the payment is received d= number of compounding periods in a year

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PROBLEMS Find the value of 12%, Rs.1,000 par value bond on which interest is payable semi-annually with a maturity period of 10 years, and a required rate of return of 14%. A Rs. 100 par value bond bears a coupon rate of 14% and matures after 5 years. Interest is payable semi-annually. Compute the value of the bond if the required rate of return is 16%. A Rs. 100 par value bond bears a coupon rate of 12% and matures after 6 years. Interest is payable semi-annually. Compute the value of the bond if the required rate of return is 16%, compounded semi-annually.

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**Current Yield= Annual interest**

BOND YIELDS CURRENT YIELD > Relates the annual coupon interest to the market price. >It is expressed as: Current Yield= Annual interest Price >Find the current yield of a 12%, Rs.1,000 par value bond with a maturity period of 10 years, and selling for Rs. 950. > This method does not consider capital gain (or loss) and also ignores time value of money.

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YIELD TO MATURITY The Yield To Maturity (YTM) of a bond is the interest rate that makes the present value of the cash flows receivable from owning the bond equal to the price of the bond. It is computed as: n P= ∑ C M t=1 (1+r)t (1+r)n Where P= Price of the bond n= number of years left to maturity C= annual interest (in Rs.) M= maturity value (in Rs.)

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YIELD TO MATURITY A 9% Rs. 1,000 par value bond, maturing after 8 years. The bond is currently selling at Rs.800. What is the YTM of the Bond? Approximating YTM: (Short-cut Method) YTM ≈ C+ ( M-P)/n 0.4M +0.6P Where YTM= yield to maturity P= Present Price of the bond n= number of years left to maturity C= annual interest (in Rs.) M= maturity value (in Rs.)

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PROBLEMS The market price of a Rs. 1,000 par value bond carrying a coupon rate of 14% and maturing after 5 years is Rs What is the Yield to Maturity (YTM) on this bond? What is the approximate YTM? The market value of Rs. 1,000 par value bond, carrying a coupon rate of 12% and maturing after 7 years, is Rs What is the YTM on this bond? The market value of Rs. 100 par value bond, carrying a coupon rate of 14% and maturing after 10 years, is Rs. 80. What is the YTM on this bond?

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YIELD TO CALL A Callable Bond entitles the issuer buy back the bond prior to the stated maturity date (in accordance to a schedule that specifies a call price for each call date). Yield to Call is calculated as: n* P= ∑ C M* t=1 (1+r)t (1+r)n* Where M*= Call Price (in Rs.) n*= number of years until the assumed call date

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**REALISED YEILD TO MATURITY**

Realized Yield to Maturity is computed when the assumption of rate of re-investment of the interest received equal to the Yield to Maturity does not hold true. The future cash flows may be invested at different rates. The present market price of a 15% Rs 1,000 par value bond maturing after 5 years is Rs Find the realized YTM in case the re-investment rate applicable to the future cash flows of this bond is 16%. The realized YTM is the value of r*in the following equation: Present market price (1+r*)5= Future value 850 (1+r*)5 = 2032

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PROBLEMS The market price of a Rs. 1,000 par value bond carrying a coupon rate of 14% and maturing after 5 years is Rs What will be the realized yield to maturity if the re-investment arte is 12%. A Rs par value bond carries a coupon rate of 10% (payable annually) and has a remaining maturity of 4 years. The bond is presently selling for Rs The re-investment rate applicable to future cash inflows of the bonds is 9% per annum. What will be the realized yield to maturity?

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**RISK IN DEBTS Bonds are associated with the following risks:-**

1. Interest Rate Risk:- Also referred to as Market Risk, is measured by the percentage change in the value of a bond in response to a given interest rate change. Duration is a precise measure of interest rate sensitivity.

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**According to the Fisher effect, (1+r)= (1+a)(1+α) r = a + α +a α **

2. Inflation Risk:- Interest rates express the rate of exchange between current and future rupees i.e. they are defined in nominal terms. However, the interest rate must indicate the rate of exchange between current and future goods and services i.e. the real interest rate. To compute nominal interest rate, the real interest rate should be adjusted for the expected inflation. According to the Fisher effect, (1+r)= (1+a)(1+α) r = a + α +a α r – nominal interest rate a – the real interest rate α – expected inflation rate When the inflation is higher than expected, the borrower gains at the expense of the lender and vice-versa. Inflation risk is greater for long-term bonds.

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**3. Real Interest Rate Risk:- Shifts in supply and/or demand for**

3. Real Interest Rate Risk:- Shifts in supply and/or demand for funds affect the real rate of interest. 4. Default Risk:- Also known as Credit risk, it refers to the risk accruing from the fact that a borrower may not pay interest and/or principal on time. This risk is gauged by the rating assigned to the debt instrument by an independent credit rating agency.

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5. Call Risk:- A bond may have a call provision that gives the issuer the option to call the bond before its scheduled maturity. The issuer would generally exercise the call option when interest rate declines. It exposes the investors to Call risk, as they will have to accept a lower yield when they reinvest the amount received on premature redemption. 6. Liquidity Risk:- Given the poor liquidity in the debt market, investors face difficulty in trading debt instruments, particularly when the holding is large.

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BOND DURATION Bond Duration is the average amount of time required by a security to receive the interest and principal. It is the weighted average measure of bond’s life, where the various time periods in which the bond generates cash flows are weighted according to the relative sizes of the present value of those flows. The duration of a bond, in effect, represents the length of time that elapses before the “average” rupee of present value from the bond is received. It compares the sensitivity of the instruments to changes in interest rates. Bond Duration is used to make a comparison across bonds at different coupon rates. The bond duration helps in determining the need for additional cash flows or surplus cash positions. As duration increases, the risk of recovering the full value of the bond also increases.

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Types of Duration Macaulay duration(D) - It is useful in immunisation of interest rate risk. Modified Duration (D*)– It is an extension of the Macaulay duration and is a useful measure of the sensitivity of a bond’s price to interest rate movements.

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**COMPUTATION OF BOND DURATION**

Formula for Duration (D) is, C1 x C2 x …….. + Ct x t (1+r) (1+r)2 (1+r)t P0 Where, D= Bond duration C=Cash flow r= Current yield to maturity t= Number of years P0= Sum of present values of cash flows/current price of bond.

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**Weighted Present value**

Compute the Bond duration for the following: Face Value- Rs.1,000 Years to Maturity= 4 years Interest rate= 7% p.a. Yield to Maturity= 6% Time (Weights) Cash flow Present Value Ct *(P.V. factor) Weighted Present value 1 70 66.01 2 62.30 124.6 3 58.79 176.37 4 1070 847.55 3390.2 Total P0 = Duration = = years

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**Duration is a key concept in the bond analysis for the following reasons:-**

It measures the interest rate sensitivity of a bond It is a useful tool for immunising against interest rate risk

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**PROPERTIES OF DURATION**

The following rules relate to duration: The duration of a zero coupon bond is the same as its maturity For a given maturity, a bond’s duration is higher when its coupon rate is lower For a given coupon rate, a bond’s duration generally increases with maturity Other things being equal, the duration of a coupon bond varies inversely with its yield to maturity

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**5. The duration of a level perpetuity is:**

(1 + yield)/yield For example, at a 9% yield, the duration of a perpetuity that pays Rs.100 per year forever will be equal to : (1.09/0.09) = 12.11 From this rule it is clear that maturity and duration can be substantially different. While the maturity of the perpetuity is infinite, the duration of the bond at 9% yield is only years.

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**6. The duration of a level annuity approximately is:**

1 + yield Number of payments yield (1 + yield)No. of payments –1 For example, a 15 year annual annuity with a yield of 10% will have a duration of : = years – 1

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**1 + y (1+y) + T (c - y) y c[(1 + y) T – 1] + y**

7. The duration of a coupon bond approximately is: 1 + y (1+y) + T (c - y) y c[(1 + y) T – 1] + y Where y is the bond’s yield per payment period, T is the number of payment periods, and c is the coupon rate per payment period.

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**DURATION AND VOLATILITY**

The proportional change in the price of a bond in response to the change in its yield is as follows: ∆P = D X ∆(1 + y) P y where ∆P/P = proportional price change D = duration of the bond y = yield However, the above relationship can be better explain with the help of modified duration (D*) Modified duration. D* = D {Modified Duration = Macaulay Duration/ (1+y)} 1+Y ∆(1+Y) = ∆Y

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Hence, ∆P = D X ∆(1 + y) P y ∆P = D X ∆(1 + y) P y ∆P = D* ∆y P Thus the proportionate change in bond price is equal to the product of modified duration and the change in the yield of the bond.

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**DURATION AND IMMUNISATION**

If the interest rate goes up, it has two consequences:- The capital value of the bond falls Return on reinvestment of interest income improves If the interest rate declines, it has two consequences:- The capital value of bond rises The return on reinvestment of interest income decreases Thus interest rate change has two effects in opposite directions. An investor who wants to immunise (or protect) his bond portfolio against interest rate risk must ensure that the duration of his bond portfolio is set equal to holding period for the bond portfolio.

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**PROBLEMS The following data is available for a bond:**

Face value Rs.1000 Coupon (interest rate) 16% payable annually Years to maturity 6 years Redemption value Rs.1000 Current market price Rs What is the yield to maturity, duration and volatility of this bond?

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**PROBLEMS Consider two bonds, P and Q Bond P Bond Q**

Face value 1, ,000 Coupon (interest rate) 16% payable 12% payable annually annually Years to maturity Redemption value 1, ,000 Current market price Rs Rs 716 What are the yields to maturity, durations and volatilities of these bonds?

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PROBLEMS A zero coupon bond of Rs 10,000 has a term of maturity of 8 years and a market yield of 10% at the time of issue. (a) What is the issue price? (b) What is the duration of the bond? (c) What is the modified duration of the bond? (d) What will be the percentage change in the price of bond, if the yield declines by 0.5 percentage points ( 50 basis points)?

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**4. A 10 year annual annuity has a yield of 9 percent**

4. A 10 year annual annuity has a yield of 9 percent. What is its duration? 5. A 10 percent coupon bond has a maturity of 12 years. It pays interest semi-annually. Its yield to maturity is 4 percent per half-year period. What is its duration?

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Convexity Convexity is the rate at which price variation changes for a change in yield. Owing to the price yield relationship, for a given rise or fall in yield, the gain in price for a drop in yield will be greater than the fall in price due to an equal rise in yield. This “upside capture, downside protection” is what convexity accounts for . Mathematically, modified duration is the first derivative of price w.r.t. yield, and convexity is the second derivative of price w.r.t. yield. Hence, Convexity can also be stated as the first derivative of modified duration

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**Convexity accounts for the curvature of the line**

Convexity accounts for the curvature of the line. The convexity formula measures the rate of change of modified duration as yield rates change, fully accounting for the dynamic relationship between prices and interest rates. By using convexity in the yield change calculation, a much closer approximation can be achieved. Convexity = Modified duration / (1 + y )

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**Features of Convexity Particulars Bond A Bond B**

Bond’s price and yield are inversely related. The rise in bond price would cause a fall in yield and vice-versa-e.g. Particulars Bond A Bond B Par Value Rs 1,000 Rs 1,000 Coupon Rate 10% 10% Maturity Period 2 years 2 years Market price Rs Rs Yield 18% 8% Even though the bonds are of same maturity and coupon rate, the difference in the market price leads to difference in the yield. The bond with low price has high yield because with lesser amount of money more return is earned.

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**The relationship between bond’s price and yield is not linear**

The relationship between bond’s price and yield is not linear. The quantum increase in the bond’s price for a given decline in yield is higher than the decline in bond’s price for a similar amount of increase in bond’s yield. e.g. Take a case of 10% bond with five years to maturity having a face value of Rs 1,000. If the yield declines to 8%, bond price will be Rs = Rs 100(PVIFA 8%, 5 years) + Rs 1000(PVIF 8%, 5 years) = Rs 100 X Rs 1000 X = Rs

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**If the yield increases to 12%,then bond price will be Rs 927.88. **

= Rs 100 (PVIFA 12%, 5 years) + Rs 1000 (PVIF 12%, 5 years) = Rs 100 X Rs 1000 X = Rs Thus fall in yield has resulted in rise of Rs ( ) but the rise in yield caused a fall of Rs (1000 – ) in the price. The above relation is referred to as Convexity. It is applicable to all types of bonds. Degree of Convexity differs from bond to bond depending upon the size of the bond, the years to maturity and the current market price.

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**BOND PORTFOLIO MANAGEMENT STRATEGIES**

Types of strategies: Passive or buy and hold strategy Semi-active strategy/immunization Active strategy Passive strategy: A passive strategy generally involves a buy and hold philosophy wherein the investor’s objective are to achieve broad diversification, predictable returns and low management cost.

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**Semi-active management strategy / immunization:**

Interest rate risk comprises two risks - price risk and reinvestment risk. A bond investor will like to eliminate these two risks derived from changing interest rates. Eliminating these risks from a bond portfolio is referred to as immunization. A portfolio of bonds is immunized from the interest rate risk if the duration of the portfolio is equal to the desired holding period.

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Active strategy: Such strategies involve decisions or judgments regarding market rates and the shape of the yield curve. These strategies have assumed crucial significance at present in view of accelerating inflation, volatility in interest rates and unsatisfactory stock market returns. Investors have begun to focus on correctly positioning their portfolios maturity structure, coupons, and quality to benefit from changes in the general level of interest rates.

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**TYPES OF ACTIVE STRATEGY:**

Forecasting interest rate changes :– Bond prices and interest rates are inversely related. If investor expects interest rates to fall, he should buy bonds and vice-versa. Exploiting mispricings among securities :– Bond portfolio managers regularly monitor the bond market to identify relative mispricings. They try to exploit such opportunities by engaging in bond swaps, purchase and sale of bond, to improve the rate of return.

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**The most popular bond swaps are as follows:**

Pure yield pick up Swap:- A swap that involves a switch from a lower yield bond to a higher yield bond of almost identical quality and maturity. Substitution Swap:- A swap meant to take advantage of a yield spread between two bond issues, which is more than what is warranted by the difference in quality and maturity of the issues. Tax Swap:- A swap that involves selling of an existing bond, at a capital loss, using the capital gains in other securities, and purchasing another bond with near identical features.

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