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 Macaulay Duration (Effective Maturity) Modified Duration Convexity Understand the term structure of interest rates and the determinants of bond yields 2
VALUING A BOND Bond Value = PV of coupons + PV of par Bond Value = PV of annuity + PV of lump sum PV C (1 1 r) Bond Value 1 C (1 1- C 2 r) r 2 1 (1 r) t... 1,000 (1 F (1 r) t r) C N N 3
VALUING A BOND Example If today is October 1, 2007, what is the value of the following bond? An IBM Bond pays $115 every September 30 for 5 years. In September 2012 it pays an additional $1000 and retires the bond. The bond is rated AAA (AAA YTM is 7.5%) Cash Flows Sept 08 09 10 11 12 115 115 115 115 1115 4
VALUING A BOND Example continued If today is October 1, 2007, what is the value of the following bond? An IBM Bond pays $115 every September 30 for 5 years. In September 2012 it pays an additional $1000 and retires the bond. The bond is rated AAA (AAA YTM is 7.5%) PV 115 1.075 115 115 115 1,115 2 3 4 1.075 1.075 1.075 1.075 5 $1,161.84 5
WHY BONDS PRICES FLUCTUATE? The price of a bond is a function of the promised payments and the market required rate of return. Since the promised payments are fixed, bond prices change in response to the changes in the market determined required rate of return. Bond price = f (required rate of return) 6
Bond Price, % HOW BOND PRICES FLUCTUATE? 115.00 110.00 105.00 100.00 95.00 90.00 85.00 80.00 0 1 2 3 4 5 6 7 8 9 10 Interest Rates, % Required rate of return YTM 7
HOW BOND PRICES FLUCTUATE? Bond prices or present values decrease as rates increase. It means, if we increase our yield above the coupon, the present value (price) must decrease below par. On the other hand, if we decrease our yield below the coupon, the present value (price) must increase above par. 8
WHY THE RELATIONSHIP IS INVERSE? Think that the yield-to-maturity is the interest rate required on newly issued debt of the same risk and that debt would be issued so that the coupon = yield. Then, suppose that the coupon rate on your bond is 8% and the yield is 9%. Which bond you would be willing to pay more for? You would pay more for new bond since it is priced to sell at $1,000, the 8% bond must sell for less than $1,000. 9
WHY THE RELATIONSHIP IS INVERSE? Another way to look at it is that return = dividend yield + capital gains yield. The dividend yield in this case is just the coupon rate. The capital gains yield has to make up the difference to reach the yield to maturity. Therefore, if the coupon rate is 8% and the YTM is 9%, the capital gains yield must equal approximately 1%. The only way to have a capital gains yield of 1% is if the bond is selling for less than par value. (If price = par, there is no capital gain.) 10
HOW TO MEASURE BOND PRICE SENSITIVITY TO YIELD CHANGES? Four measures of bond price sensitivity to interest rate 1) Maturity 2) Macaulay Duration (Effective Maturity) 3) Modified Duration 4) Convexity 11
MATURITY: SENSITIVITY MEASURE (1) Simple maturity is just the time left to maturity on a bond. We generally think of 5-year bonds or 10-year bonds. 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. 12
MATURITY: SENSITIVITY MEASURE (1) Consider two zero coupon bonds, each with a face value of $1,000. Bond A matures in 10 years and has a required rate of return of 10%. The price of Bond A is $376.89, where Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.91 or 13
MATURITY: SENSITIVITY MEASURE (1) If the required rate of return for each bond was to increase by 100 basis points to 11%, the prices would then be $342.73 for Bond A and $585.43 for Bond B. This translates into a -9.1% change in price for Bond A and -4.6% for Bond B. Thus, for zero coupon bonds simple maturity can be used to compare price sensitivity 14
MACAULAY DURATION: SENSITIVITY MEASURE (2) The relationship between price and maturity is not as clear when you consider non-zero coupon bonds. For a coupon-paying bond, 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. In order to deal with this, Frederick Macaulay 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. 15
MACAULAY DURATION: SENSITIVITY MEASURE (2) Consider a six- year bond with face value of $1,000, and a 6.1% coupon rate (semi-annual payments). If the current yield to maturity is 10%, the value of the bond is found by discounting each of the semi-annual payments 16
MACAULAY DURATION: SENSITIVITY MEASURE (2) Macaulay Duration takes the present value of each payment and divides it by the total bond price, P. By doing this, one has a percentage, w t, of the total bond value that is received in each period, t. The duration or effective maturity for the bond could then be estimated by multiplying the weight, w t, times the time, t and then summing all of the weighted values, or 17
MACAULAY DURATION: SENSITIVITY MEASURE (2) 18
MACAULAY DURATION: SENSITIVITY MEASURE (2) The semi- annual duration for this bond is 10.014 sixmonth periods. We usually use annual duration and we annualize the semi-annual duration simply by dividing by 2 (the number of six month periods in a year). In this case, the annualized duration would be 5.007 years. Note that the Macaulay Duration for a 5- year zero coupon bond is the same as the simple maturity, 5.0 years. Hence, we can expect that the original 6-year, 6.1% coupon bond when interest rates change to behave in a manner similar to a 5-year zero coupon bond, since their effective maturity (Macaulay Duration) is essentially the same. 19
MODIFIED DURATION: SENSITIVITY MEASURE (3) If we want a more direct measure of the relationship between changes in interest rates and changes in bond prices, we can use Modified Duration. Modified Duration, D, is defined as the following where P is the bond price, P is the change in bond price and y is the change in the required rate of return (yield to maturity). 20
MODIFIED DURATION: SENSITIVITY MEASURE (3) We know price of a bond is: Taking the derivative of P with respect to y, Inserting this into the formula for Modified Duration yields 21
MODIFIED DURATION: SENSITIVITY MEASURE (3) Rearranging the previous formula slightly Comparing this to the definition of Macaulay Duration and using that definition we can write Modified Duration as 22
MODIFIED DURATION: SENSITIVITY MEASURE (3) While it is easy calculate Modified Duration once you have Macaulay Duration the interpretations of the two are quite different. Macaulay Duration is an average or effective maturity. Modified Duration really measures how small changes in the yield to maturity affect the price of the bond. 23
MODIFIED DURATION: SENSITIVITY MEASURE (3) 24
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MODIFIED DURATION: SENSITIVITY MEASURE (3) Modified Duration assumes that the price changes are linear with respect to changes in the yield to maturity. From last table, the true relationship between the bond's price and the yield to maturity is not linear. The Column with the differences is always positive and increases as we move away from a yield to maturity of 10%. The actual relationship between the bond price and the yield to maturity is: 26
MODIFIED DURATION: SENSITIVITY MEASURE (3) 27
Non-linearity of relationship between bond prices and yield: The curved line is the actual price curve. The straight line is the price relationship using Modified Duration. Everywhere the actual price curve is above the Modified Duration relationship. This is exactly what we saw in Table. The difference was always positive, i.e., actual calculated price was 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 yield to maturity but the difference grows as the change in yield to maturity becomes bigger. 28
CONVEXITY: SENSITIVITY MEASURE (4) Modified Duration relationship does not fully capture the true relationship between bond prices and yield to maturity. In order to more fully capture this, practitioners use Convexity. The definition of Convexity is The actual definition of Convexity that we can use is 29
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CONVEXITY: SENSITIVITY MEASURE (4) We can annualize the semi-annual convexity of 110.88 by dividing it by 2 2 or 4. Here it would be 27.72. Convexity is useful to practitioners in a number of ways. First it can be used in conjunction with duration to get a more accurate estimate of the percentage price change resulting from a change in the yield. The formula is: Adjustment factor 31
CONVEXITY: SENSITIVITY MEASURE (4) Adding the convexity adjustment corrects for the fact that Modified Duration understates the true bond price. For example, in our example, at a yield of 12% the percentage price change using only Modified Duration was -9.54%, while the actual was -9.01%. If we use the Convexity value we just calculated, the predicted percentage price change would be This is -8.99%, which is much closer to the actual percentage price change of -9.01%. 32
CONVEXITY: SENSITIVITY MEASURE (4) Convexity provides insight into how a bond will react to yield changes. Earlier we saw that the price reaction to changes in yield is not symmetric. For a given change in yield, bond prices drop less for a given increase in yield and increase more for the same decreases in yield. The downside is less and the upside is more. This is clearly a desirable property. The higher the Convexity of a bond the more this is true. Thus, bonds with high convexity are more desirable. 33
TERM STRUCTURE OF INTEREST RATES Term structure is the relationship between time to maturity and yields, all else equal It is important to recognize that we pull out the effect of default risk, different coupons, etc. Yield curve graphical representation of the term structure Normal upward-sloping, long-term yields are higher than short-term yields Inverted downward-sloping, long-term yields are lower than short-term yields 34
FIGURE UPWARD-SLOPING YIELD CURVE 35
FIGURE DOWNWARD-SLOPING YIELD CURVE 36
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FACTORS AFFECTING BOND YIELDS Default risk premium remember bond ratings Liquidity premium bonds that have more frequent trading will generally have lower required returns Anything else that affects the risk of the cash flows to the bondholders will affect the required returns 38
ASSIGNMENT What is Effective Duration? Give an example to show its calculation What is the difference and similarity between Modified Duration and Effective Duration? Where it is appropriate to use Modified duration and where Effective duration is more suitable? 39
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