LO 12.4: Evaluate the im pact o f changes in maturity, yield, and volatility on the

LO 12.4: Evaluate the im pact o f changes in maturity, yield, and volatility on the convexity o f a security.
The convexity effect can be measured by applying a special case of Jensens inequality as follows:
Example: Applying Jensens inequality
Assume that next year there is a 50% probability that 1-year spot rates will be 10% and a 50% probability that 1-year spot rates will be 6%. Demonstrate Jensens inequality for a 2-year zero-coupon bond with a face value of $ 1 assuming the previous interest rate expectations shown in Figure 1.
Answer:
The left-hand side of Jensens inequality is the expected price in one year using the 1-year spot rates of 10% and 6%.
0.5 x $1
(1.10) T 0.5 x $1
(1.06)
= $0.92624
The expected price in one year using an expected rate of 8% computes the right-hand side of the inequality as:
____________________________$1____________________________ 0.5×1.10 + 0.5×1.06
$1 1.08
0.92593
Thus, the left-hand side is greater than the right-hand side, $0.92624 > $0.92593.
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If the current 1-year rate is 8%, then the price of a 2-year zero-coupon bond is found by simply dividing each side of the equation by 1.08. In other words, discount the expected 1-year zero-coupon bond price for one more year at 8% to find the 2-year price. The price of the 2-year zero-coupon bond on the left-hand side of Jensens inequality equals $0.85763 (calculated as $0.92624 / 1.08). The right-hand side is calculated as the price of a 2-year zero-coupon bond discounted for two years at the expected rate of 8%, which equals $0.85734 (calculated as $1 / 1.082).
The left-hand side is again greater than the right-hand side, $0.85763 > $0.85734.
This demonstrates that the price of the 2-year zero-coupon bond is greater than the price obtained by discounting the $1 face amount by 8% over the first period and by 8% over the second period. Therefore, we know that since the 2-year zero-coupon price is higher than the price achieved through discounting, its implied rate must be lower than 8%.
Extending the above example out for one more year illustrates that convexity increases with maturity. Suppose an investor expects the spot rates to be 14%, 10%, 6%, or 2% in three years. Assuming each expected return has an equal probability of occurring results in the decision tree shown in Figure 3.
Figure 3: Risk-Neutral Decision Tree Illustrating Expected 1-Year Rates for Three Years
The decision tree in Figure 4 uses the expected spot rates from the decision tree in Figure 3 to calculate the price of a 3-year zero-coupon bond.
The price of a 1-year zero-coupon bond in two years with a face value of $1 for the upper node is $0.89286 (calculated as $1 / 1.12). The price of a 1-year zero coupon bond in two years for the middle node is $0.92593 (calculated as $1 / 1.08). The price of a 1-year zero coupon bond in two years for the bottom node is $0.96154 (calculated as $1 / 1.04).
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The price of a 2-year zero-coupon bond in one year using the upper node expected spot rates is calculated as:
[0.5 x ($0.89286 / 1.10)] + [0.5 x ($0.92593 / 1.10)] = $0.82672
The price of a 2-year zero-coupon bond in one year using the bottom node expected spot rates is calculated as:
[0.5 x ($0.92593 / 1.06)] + [0.5 x ($0.96154 / 1.06)] = $0.89032
Lastly, the price of a 3-year zero-coupon bond today is calculated as:
[0.5 x ($0.82672 / 1.08)] + [0.5 x ($0.89032 / 1.08)] = $0.79493
Figure 4: Risk-Neutral Decision Tree for a 3-Year Zero-Coupon Bond
$0.79493
$1
$1
$1
$1
To measure the convexity effect, the implied 3-year spot rate is calculated by solving for r(3)in the following equation:
0.79493
1
(1 + r(3))
r (3) = 3/o j5493 1 = 0-0795 or 7.95%
Notice that convexity lowers bond yields and that this reduction in yields is equal to the value of convexity. For the 3-year zero-coupon bond, the value of convexity is 8% 7.95% = 0.05% or 5 basis points. Recall that the value of convexity for the 2-year zero-coupon bond was only 1.84 basis points. Therefore, all else held equal, the value of convexity
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increases with maturity. In other words, as the maturity of a bond increases, the price-yield relationship becomes more convex.
This convexity occurs due to volatility. Thus, we can also say that the value of convexity increases with volatility. The following decision trees in Figures 5 and 6 illustrate the impact of increasing the volatility of interest rates. In this example, the 1-year spot rate in one year in Figure 5 ranges from 2% to 14% instead of 4% to 12% as was shown in Figure 1.
Figure 5: Risk-Neutral Decision Tree Illustrating Volatility Effect on Convexity
(25% joint probability)
(50% joint probability)
(25% joint probability)
Using the same methodology as before, the price of a 2-year zero-coupon bond with the listed expected interest rates in Figure 5 is $0,858.
Figure 6: Price of a 2-Year Zero-Coupon Bond with Increased Volatility
This price results in a 2-year implied spot rate of 7.958%. Thus, the value of convexity is 8% – 7.958% = 0.042% or 4.2 basis points. This is higher than the previous 2-year example where the value of convexity was 1.84 basis points when expected spot rates ranged from 4% to 12%, instead of 2% to 14%. Therefore, the value of convexity increases with both volatility and time to maturity.
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R i s k P r e m i u m

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