标题: Reading 66: Introduction to the Measurement of Interest Rate [打印本页]
作者: honeycfa 时间: 2010-4-26 11:12 标题: [2010]Session 16-Reading 66: Introduction to the Measurement of Interest Rate
LOS g: Describe the convexity measure of a bond.
A 7% coupon bond with semiannual coupons has a convexity in years of 80. The bond is currently priced at a yield to maturity (YTM) of 8.5%. If the YTM decreases to 8%, the predicted effect due to convexity on the percentage change in price would be:
Convexity adjustment: +(Convexity)(change in i)2
Convexity adjustment = +(80)(-0.005)(-0.005) = +0.0020 or 0.20% or +20 basis points.
作者: honeycfa 时间: 2010-4-26 11:13
Why is convexity a good thing for a bond holder? Because when compared to a low convexity bonds a high convexity bond:
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is usually underpriced. | |
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is more sensitive to interest rate changes, increasing the potential payoff. | |
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has better price changes regardless of the direction of the yield change. | |
Relative to a bonds with low convexity, the price of a bond with high convexity will increase more when rates decline and decrease less when rates rise.
作者: honeycfa 时间: 2010-4-26 11:13
How does the convexity of a bond influence the yield on the bond? All else the same, for a bond with high convexity investors will require:
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a higher or lower yield depending on the bond's duration. | |
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Convexity is to the advantage of the bond holder because a high-convexity bond's price will decrease less when rates increase
作者: honeycfa 时间: 2010-4-26 11:13
For a given change in yields, the difference between the actual change in a bond’s price and that predicted using the duration measure will be greater for:
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a bond with less convexity. | |
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a bond with greater convexity. | |
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Duration is a linear measure of the relationship between a bond’s price and yield. The true relationship is not linear as measured by the convexity. When convexity is higher, duration will be less accurate in predicting a bond’s price for a given change in interest rates. Short-term bonds generally have low convexity.
作者: honeycfa 时间: 2010-4-26 11:13
With respect to an option-free bond, when interest-rate changes are large, the duration measure will overestimate the:
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A) |
increase in a bond's price from a given increase in interest rates. | |
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final bond price from a given increase in interest rates. | |
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fall in a bond's price from a given increase in interest rates. | |
When interest rates increase by 50-100 basis points or more, the duration measure overestimates the decrease in the bond’s price.
作者: honeycfa 时间: 2010-4-26 11:14
Convexity is more important when rates are:
Since interest rates and the price of bonds are inversely related, unstable interest rates will lead to larger price fluctuations in bonds. The larger the change in the price of a bond the more error will be introduced in determining the new price of the bond if only duration is used because duration assumes the price yield relationship is linear when in fact it is a curved convex line. If duration alone is used to price the bond, the curvature of the line magnifies the error
作者: honeycfa 时间: 2010-4-26 11:14
If a bond has a convexity of 120 and a modified duration of 10, what is the convexity adjustment associated with a 25 basis point interest rate decline?
Convexity adjustment: +(C) (Δi)2
Con adj = +(120)(-0.0025)(-0.0025) = +0.000750 or 0.075%
作者: honeycfa 时间: 2010-4-26 11:15
For a given bond, the duration is 8 and the convexity is 50. For a 60 basis point decrease in yield, what is the approximate percentage price change of the bond?
The estimated price change is -(duration)(?y) + (convexity) × (?y)2 = -8 × (-0.006) + 50 × (-0.0062) = +0.0498 or 4.98%.
作者: honeycfa 时间: 2010-4-26 11:15
A bond has a duration of 10.62 and a convexity of 91.46. For a 200 basis point increase in yield, what is the approximate percentage price change of the bond?
The estimated price change is:
-(duration)(?y) + (convexity) × (?y)2 = -10.62 × 0.02 + 91.46 × (0.022) = -0.2124 + 0.0366 = -0.1758 or –17.58%.
作者: honeycfa 时间: 2010-4-26 11:15
If a Treasury bond has a duration of 10.27 and a convexity of 71.51. Which of the following is closest to the estimated percentage price change in the bond for a 125 basis point increase in interest rates?
The estimated percentage price change = the duration effect plus the convexity effect. The formula is: [–duration × (Δy)] + [convexity × (Δy)2]. Therefore, the estimated percentage price change is: [–(10.27)(1.25%)] + [(71.51)(0.0125)2] = –12.8375 + 1.120% = –11.7175%.> >
作者: honeycfa 时间: 2010-4-26 11:15
Consider a bond with a duration of 5.61 and a convexity of 21.92. Which of the following is closest to the estimated percentage price change in the bond for a 75 basis point decrease in interest rates?
The estimated percentage price change is equal to the duration effect plus the convexity effect. The formula is: [–duration × (Δy)] + [convexity × (Δy)2]. Therefore, the estimated percentage price change is: [–(5.61)(–0.0075)] + [(21.92)(-0.0075)2] = 0.042075 + 0.001233 = 0.043308 = 4.33%.
作者: honeycfa 时间: 2010-4-26 11:16
A bond has a convexity of 25.72. What is the approximate percentage price change of the bond due to convexity if rates rise by 150 basis points?
The convexity effect, or the percentage price change due to convexity, formula is: convexity × (Δy)2. The percentage price change due to convexity is then: (25.72)(0.015)2 = 0.0058.
作者: honeycfa 时间: 2010-4-26 11:16
A bond has a modified duration of 6 and a convexity of 62.5. What happens to the bond's price if interest rates rise 25 basis points? It goes:
ΔP/P = (-)(MD)(Δi) +(C) (Δi)2
= (-)(6)(0.0025) + (62.5) (0.0025)2 = -0.015 + 0.00039 = - 0.01461
作者: honeycfa 时间: 2010-4-26 11:16
A bond’s duration is 4.5 and its convexity is 43.6. If interest rates rise 100 basis points, the bond’s percentage price change is closest to:
Recall that the percentage change in prices = Duration effect + Convexity effect = [-duration × (change in yields)] + [convexity × (change in yields)2] = (-4.5)(0.01) + (43.6)(0.01)2 = -4.06%. Remember that you must use the decimal representation of the change in interest rates when computing the duration and convexity adjustments.
作者: honeycfa 时间: 2010-4-26 11:16
An investor gathered the following information about an option-free U.S. corporate bond:
- Par Value of $10 million
- Convexity of 45
- Duration of 7
If interest rates increase 2% (200 basis points), the bond’s percentage price change is closest to:
Recall that the percentage change in prices = Duration effect + Convexity effect = [-duration × (change in yields)] + [convexity × (change in yields)2] = [(-7)(0.02) + (45)(0.02)2] = -0.12 = -12.2%. Remember that you must use the decimal representation of the change in interest rates when computing the duration and convexity adjustments.
作者: honeycfa 时间: 2010-4-26 11:17
Assume that a straight bond has a duration of 1.89 and a convexity of 15.99. If interest rates decline by 1% what is the total estimated percentage price change of the bond?
The total percentage price change estimate is computed as follows:
Total estimated price change = -1.89 × (-0.01) × 100 + 15.99 × (-0.01)2 × 100 = 2.05%
作者: honeycfa 时间: 2010-4-26 11:17
Which of the following statements about the market yield environment is most accurate?
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As yields increase, bond prices rise, the price curve flattens, and further increases in yield have a smaller effect on bond prices. | |
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For a given change in interest rates, bond price sensitivity is lowest when market yields are already high. | |
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Positive convexity applies to the percentage price change, not the absolute dollar price change. | |
The price volatility of noncallable (option-free) bonds is inversely related to the level of market yields. In other words, when the yield level is high, bond price volatility is low and vice versa.
The statement beginning with, As yields increase. . . should continue . . .bond prices fall. Positive convexity (bond prices increase faster than they decrease for a given change in yield) applies to both absolute dollar changes and percentage changes.
作者: honeycfa 时间: 2010-4-26 11:17
A bond has a modified duration of 6 and a convexity of 62.5. What happens to the bond's price if interest rates rise 25 basis points? It goes:
?P = [(-MD × ?y) + (convexity) × (?y)2] × 100 >>
?P = [(-6 × 0.0025) + (62.5) × (0.0025)2] × 100 = -1.461%
作者: honeycfa 时间: 2010-4-26 11:18
A bond has a modified duration of 7 and convexity of 50. If interest rates decrease by 1%, the price of the bond will most likely:
Percentage Price Change = –(duration) (?i) + convexity (?i)2
therefore
ercentage Price Change = –(7) (–0.01) + (50) (–0.01)2=7.5%.
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