Assessing Fixed Income Portfolio Risk Using Duration and Convexity

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1 Assessing Fixed Income Portfolio Risk Using Duration and Convexity G.Kalaiarasan, S.Srinivasan Department of Mathematics and Actuarial Science Abstract B. S. Abdur Rahman University,Chennai,India The price of fixed income security is a function of the promised payments and interest rate expected by investor. The promised payments are fixed and price of bond change with changes in interest rate in market. Thus, the investor in fixed income security must need to quantify their investment s or portfolio s sensitivity to interest rate i.e. interest rate risk. This paper will empirically shows duration plus convexity will effectively capture the interest rate risk involved in fixed income securities. Introduction: In investment management, the most important decision is the allocation of funds among asset classes. The two major asset class are equities and fixed income securities. Our focus in this paper is on one of the major asset classes: fixed income securities and its portfolio 1. Fixed income securities play a critical role in the portfolio of individual and institutional investors. In its simplest form. Afixed income security is a financial obligation of an entity 2 that promise to pay a specified sum of money at specified future dates. The entity that promises to make is 125 G.Kalaiarasan, S.Srinivasan referred as issuer of the security and counter party is referred as holder of the security. Fixed income security includes bonds, mortgage- backed securities, asset-backed securities and bank loans. In this paper we will use the terms fixed income securities and bonds interchangeably. In its simplest form, a bond is a financial instrument in which issuer agrees to pay interests 3 and principle at or by specified dates. The term to maturity of a bond indicates the time period over which the bondholder can expect to receive interest payment and the number of years before the principle will be

2 paid. The yield offered on a bond depends on the term to maturity and its relationship can be examined by constructing yield curve.the par value of a bond is the amount that the issuer agrees to repay the bondholder at or by the maturity date. The bond may trade below or above its par value. When a bond trades below its par value, it is said to be trading at a discount. When a bond trades above its par value, it is said to be trading at a premium. The reason why a bond trading at above or below its par value will be discussed later in this paper. Valuation: Valuation is the process of determining the fair value of a financial asset. The fundamental principle of financial valuation is that its value is equal to the present value of its expected cash flows. The future cash flows in a bond is its coupons and par value. Thus, the valuation of fixed income involves the following three steps: Step 1: Estimate the expected cash flows. Step 2: Determine the appropriate market Required rate of interest (return) that should be used to discount cash flows. 126 G.Kalaiarasan, S.Srinivasan Step 3: Calculate the present value of expected cash flow using market required rate of interest determined. 1 Portfolio is collection of assets; in this case asset is fixed income securities 2 Entity can be Central government, state government and any corporate. Usually, the individual cannot issue fixed income securities. 3 Usually interest is termed as coupon and it is coupon rate par value. 4 Market rate of interest is also termed as yield to maturity The price of a bond with maturity of T may be expressed as the present value of coupons (C) and par value (Par) discounted at the market required rate of interest (Y) as follows: (1) Thus, the price of a bond is a function of the promised payments and the market required rate of interest. Since the promised payments are fixed, bond prices change in response to changes in the market determined required rate of interest. Interest Rate Risk: The risk that an investment s value will change due to change in market required rate of interest. Interest rate risk affects the value of bonds more directly than stocks, and it is a major risk to all bondholders.

3 Axis Title Price of a bond International Journal of Engineering Technology, Management and Applied Sciences If we change interest rate(y) in (1), the price of a bond will change. The price of the bond will change in the opposite direction to change in interest rates. That is, when interest rate rise, a bond price will fall; when interest rate fall, a bond price will raise. Exhibit 1 shows the price of bond change with change in interest rate One of the factors affecting the interest rate risk is the maturity of a bond. With a longer term, the bond is more likely to subject to change in price due to interest rate change; the reason will be discussed later. The other factor affecting the interest rate risk is the coupon rate 5. A higher coupon rate allows the price of the bond to be recovered in a shorter time frame and thus expose an investor to less interest rate risk Interest Rate There are four measures of bond price sensitivity that are commonly used. They are simple maturity, macaulay duration (effective 127 G.Kalaiarasan, S.Srinivasan maturity), modified duration, and convexity. Each of these provides a more exact description of how a bond price changes relative to changes in the required rate of return. 5 The coupon rate is the interest rate that issuer of a bond agrees to pay. Maturity: Simple maturity is just the time left to maturity on a bond. It is straightforward and requires no calculation. The longer the time to maturity the more sensitive a particular bond is to changes in the required rate of return. There is a greater probability that interest rates will rise within a longer time period than within a shorter period Exhibit 2 shows the interest rate sensitivity of bonds based on its term to maturity Interest Rate Figure 2 compares the interest rate sensitivity of 50 year term and 20 year term

4 bonds. As discussed, 50 year term bond exibit high sensitivity to interest change than 20 year term bond. Another important thing to observe from figure 1 & 2 is, the interest rate sensitivity curve of a bond is covex in shape. In figure 2, 50 year term bond is more convex than 20 year bond.thus in general, the interest rate sensitivity of a bond increases with increase in convexity of curve and vise versa. The maturity of a bond can be used as a measure of interest rate sensitivity of a bond only when its cash flows are continously compounded. Many of the cash flows occur before the actual maturity of the bond and the relative timing of these cash flows will affect the pricing of the bond. Thus, the maturity can be an effective tool only in case of zero coupon bonds 6 and rest other types of bonds need advanced tools like duration and convexity to measure its interest rate sensitivity. Macaulay Duration :( Effective maturity) Frederick Macaulay 7 in 1938 suggested that investors use the effective maturity of a bond as a measure of interest Rate sensitivity. He called this duration and defined it as a value-weighted average of the timing of the cash flows. 128 G.Kalaiarasan, S.Srinivasan Macaulay Duration takes the present value of each payment and divides it by the total bond price, P. By doing this, we can ascertain a percentage (w t ), of the total bond value that is received in each period, t. where CF t denotes cash flow at t and P is the price of the bond 6 Zero coupon bond does not pay any periodic coupons. The holder realizes interest by buying the bond substantially below its par value. 7 Fredick Macaulay, Some Theoretical Problems Suggested by the Movement of Interest Rates, Bond Yields,and Stock Prices in the United States since 1856 (New York: National Bureau of Economic Research, 1938). The duration 8 or effective maturity for the bond can be estimated by multiplying the weight (W t), times the time, t and then summing all of the weighted values. This measure takes into account the relative timing of the cash flows. Calculation of the Macaulay Duration measure is fairly straightforward but can be somewhat tedious 9. Like in case of maturity, a bond with high duration is more sensitive to change in interest(riskier) than a bond with less duration. Exhibit 3 shows calculation of duration for a bond.

Price of bond International Journal of Engineering Technology, Management and Applied Sciences Modified Duration: Modified duration gives more direct measure of the relationship between changes in

5 Price of bond International Journal of Engineering Technology, Management and Applied Sciences Modified Duration: Modified duration gives more direct measure of the relationship between changes in interest rates and changes in bond prices. The basic pricing formulation for bonds is Where CF t is cash flows at time, t and P is the price of the bond. The change in the bond price when the interest rate change can be found by differentiation of P with respect to Y. The above expression can be written as, % Δ in bond price = - Modified Duration times the change in interest rate. Exhibit 4 shows estimated bond prices using modified duration and comparing it with actual prices of a bond 8 Duration, Macaulary duration and Effective duration can be used interchangeably. 9 Excel offers a worksheet function DURATION(.), which calculates the Macaulay Duration. To get percentage change in the price of a bond with change in interest rate, divide both side of equation by P Interest rate Estimating a bond price using By rearranging above, we get Above figure 3 shows, modified duration assumes that the price changes are linear with respect to changes in the interest rate. The curved line is the actual price curve. The straight line is the price relationship using 129 G.Kalaiarasan, S.Srinivasan

6 modified duration. Everywhere the actual price curve is above the modified duration relationship. The difference(error) is always positive, i.e., actual calculated price is greater than the new price using the modified duration relationship. In addition, the percentage changes in price are not symmetric. The percentage decrease in price for a given increase in yield is always less than the percent increase for the same decrease in yield. This property is referred to as convexity. Note that the two prices are quite close for small changes in the interest rate but the difference (error) grows as the change in interest rate becomes bigger. Convexity: From Figure 3 it is clear that the modified duration relationship does not fully capture the true relationship between bond prices and interest rate, if the change in interest rate is high. The modified duration relationship failed to capture the convex nature of interest rate sensitivity of a bond price. The modified duration can also be interoperated as first order derivative of bond price with respect to interest rate as respect to interest rate(convexity) can be implied. This can be done by using second order Taylor s series expansion. Letting P = f(x), we have Rearranging above equation and dividing both sides by P(Y) to get percentage change in bond price with change in interest rate. Since modified duration is not appropriate, the second order derivative of bond price with 130 G.Kalaiarasan, S.Srinivasan Where,

7 Price of bond International Journal of Engineering Technology, Management and Applied Sciences Thus the above expression can be written as, % Δ in Price = - Modified Duration (ΔY) + (1/2) Convexity (ΔY) 2. Exhibit 5 shows estimated bond prices using duration and convexity and comparing it with actual prices of a bond The figure 5 shows, the convexity combined with duration considerably reduced the difference between estimated bond price and its actual price.thus, duration plus convexity is an effective tool to measure the interest rate sensitivity of a bond, even if its term to maturity is large. Portfolio: Interest Rate Estimating the bond price using Actual price Estimated price A portfolios duration can be obtained by calculating the weighted average duration of the bonds in the portfolio. Mathematically, a portfolios duration can be calculated as follows: W 1 D 1 + W 2 D 2 + W 3 D W k D k Where, W i = market value of bond i / market value of the portfolio. D i = duration of bond i. K = number of bonds in the portfolio. Duration and convexity have traditionally been used as tools for immunization of portfolio or asset-liability management. To avoid exposure to interest rate risk, an organization (such as an insurance company or defined benefit pension plan) with significant fixed income exposures might structure its assets so that their duration matches the duration of its liabilities so the two offset. This technique is called duration matching. Even more effective (but less frequently practical) is duration-convexity matching, in which assets are structured so that durations and convexities match. Conclusion: Interest rates are constantly changing and 131 G.Kalaiarasan, S.Srinivasan

8 add a level of uncertainty to fixed-income investing. Duration and convexity allow investors to quantify this uncertainty and are useful tools in the management of fixedincome portfolios. References: 1, Saket Vasisth., 2010, Duration and Convexity, Actuary India Magazine 2, Miles Livingston, Lei Zhou., A Highly Accurate Measure of Bond Price Sensitivity to Interest Rates, Journal of fixed income. 3, Weil, Roman L., 1973, Macaulay s Duration: An Appreciation, Journal of Business. 4,Duration and Convexity at riskglossary.com. 5, Use Duration And Convexity To Measure Bond Risk at investopedia.com. 6, Use Duration And Convexity To Measure Bond Risk n_convexity.shtml. 7, Litterman, R. and J. Scheinkman.,1991, Common Factors Affecting Bond Returns, Journal of Fixed Income. 9, Grantier, Bruce, 1988, Convexity and Bond Performance: The Benter the Better, Financial Analysts Journal. Exhibit 1 This table gives the price sensitivity of a 30- year zero coupon bond Interest rate(%) Price of bonf(rs.) Exhibit 2 This table gives the price sensitivity of a 50- year and a 20-year zero coupon bonds. 8,Risks of Investing in Bondsinvestinginbonds.com 132 G.Kalaiarasan, S.Srinivasan

9 Interest rate(%) Price of 20 yr bond(rs.) Price of 50 yr bond(rs.) Exhibit 3 Table shows calculation of duration for 7 year par bond with 30% coupon rate. Exhibit 4 This table gives the actual price and estimate price using duration of a 30-year par bond with 5% coupon rate. Interest rate(%) Actual price(rs.) Estimated price(rs.) t(b) Cash Flow Present value(a) Exhibit 5 W t (A)*(B) Duratio n 3.64 This table gives the actual price and estimate price using duration and convexity of a 30- year par bond with 5% coupon rate Interest rate(%) Actual price(rs.) Estimated price(rs.) G.Kalaiarasan, S.Srinivasan

INVESTMENTS. Instructor: Dr. Kumail Rizvi, PhD, CFA, FRM

INVESTMENTS. Instructor: Dr. Kumail Rizvi, PhD, CFA, FRM INVESTMENTS Instructor: Dr. KEY CONCEPTS & SKILLS Understand bond values and why they fluctuate How Bond Prices Vary With Interest Rates Four measures of bond price sensitivity to interest rate Maturity