标题: Reading 66: Introduction to the Measurement of Interest Rate [打印本页]
作者: honeycfa 时间: 2010-4-25 20:02 标题: [2010]Session 16-Reading 66: Introduction to the Measurement of Interest Rate
LOS d Compute and interpret the effective duration of a bond, given information about how the bond's price will increase and decrease for given changes in interest rates.
A bond's yield to maturity decreases from 8% to 7% and its price increases by 6%, from $675.00 to $715.50. The bond's effective duration is closest to:
Effective duration is the percentage change in price for a 1% change in yield, which is given as 6.
作者: honeycfa 时间: 2010-4-25 20:03
An investor finds that for every 1% increase in interest rates, a bond’s price decreases by 4.21% compared to a 4.45% increase in value for every 1% decline in interest rates. If the bond is currently trading at par value, the bond’s duration is closest to:
Duration is a measure of a bond’s sensitivity to changes in interest rates.
Duration = (V- – V+) / [2V0(change in required yield)] where:
V- = estimated price if yield decreases by a given amount
V+ = estimated price if yield increases by a given amount
V0 = initial observed bond price
Thus, duration = (104.45 – 95.79)/(2 × 100 × 0.01) = 4.33. Remember that the change in interest rates must be in decimal form.
作者: honeycfa 时间: 2010-4-25 20:03
An international bond investor has gathered the following information on a 10-year, annual-pay U.S. corporate bond:
- Currently trading at par value
- Annual coupon of 10%
- Estimated price if rates increase 50 basis points is 96.99%
- Estimated price is rates decrease 50 basis points is 103.14%
The bond’s duration is closest to:
Duration is a measure of a bond’s sensitivity to changes in interest rates.
Duration = (V- – V+) / [2V0(change in required yield)] where:
V- = estimated price if yield decreases by a given amount
V+ = estimated price if yield increases by a given amount
V0 = initial observed bond price
Thus, duration = (103.14 ? 96.99) / (2 × 100 × 0.005) = 6.15. Remember that the change in interest rates must be in decimal form.
作者: honeycfa 时间: 2010-4-25 20:04
If bond prices fall 5% in response to a 0.5% increase in interest rates, what is the bond's effective duration?
Approximate percentage price change of a bond = - (duration) (delta y) =
-5 = - (duration) (0.5) = 10.
作者: honeycfa 时间: 2010-4-25 20:04
When interest rates increase, the duration of a 30-year bond selling at a discount:
The higher the yield on a bond the lower the price volatility (duration) will be. When interest rates increase the price of the bond will decrease and the yield will increase because the current yield = (annual cash coupon payment) / (bond price). As the bond price decreases the yield increases and the price volatility (duration) will decrease.
作者: honeycfa 时间: 2010-4-25 20:05
A bond with a yield to maturity of 8.0% is priced at 96.00. If its yield increases to 8.3% its price will decrease to 94.06. If its yield decreases to 7.7% its price will increase to 98.47. The effective duration of the bond is closest to:
The change in the yield is 30 basis points.
Duration = (98.47 ? 94.06) / (2 × 96.00 × 0.003) = 7.6563.
作者: honeycfa 时间: 2010-4-25 20:05
A non-callable bond with 18 years remaining maturity has an annual coupon of 7% and a $1,000 par value. The current yield to maturity on the bond is 8%. Which of the following is closest to the effective duration of the bond?
First, compute the current price of the bond as:
FV = $1,000; PMT = $70; N = 18; I/Y = 8%; CPT → PV = –$906.28
Next, change the yield by plus-or-minus the same amount. The amount of the change can be any value you like. Here we will use ±50 basis points.
Compute the price of the bond if rates rise by 50 basis points to 8.5% as:
FV = $1,000; PMT = $70; N = 18; I/Y = 8.5%; CPT → PV = –$864.17
Then compute the price of the bond if rates fall by 50 basis points to 7.5% as:
FV = $1,000; PMT = $70; N = 18; I/Y = 7.5%; CPT → PV = –$951.47
The formula for effective duration is:
(V- – V+) / (2V0Δy)
Therefore, effective duration is:
($951.47 – $864.17) / (2 × $906.28 × 0.005) = 9.63.
作者: honeycfa 时间: 2010-4-25 20:05
Calculate the effective duration for a 7-year bond with the following characteristics:
- Current price of $660
- A price of $639 when interest rates rise 50 basis points
- A price of $684 when interest rates fall 50 basis points
The formula for calculating the effective duration of a bond is:
where:
-
V- = bond value if the yield decreases by ?y
-
V+ = bond value if the yield increases by ?y
-
V0 = initial bond price
-
?y = yield change used to get V- and V+, expressed in decimal form
The duration of this bond is calculated as:
作者: honeycfa 时间: 2010-4-25 20:06
Consider an annual coupon bond with the following characteristics:
- Face value of $100
- Time to maturity of 12 years
- Coupon rate of 6.50%
- Issued at par
- Call price of 101.75 (assume the bond price will not exceed this price)
For a 75 basis point change in interest rates, the bond's duration is:
Since the bond has an embedded option, we will use the formula for effective duration. (This formula is the same as the formula for modified duration, except that the “upper price bound” is replaced by the call price.) Thus, we only need to calculate the price if the yield increases 75 basis points, or 0.75%.
Price if yield increases 0.75%: FV = 100; I/Y = 6.50 + 0.75 = 7.25; N = 12; PMT = 6.5; CPT → PV = 94.12
The formula for effective duration is
|
Where: |
| V- |
= call price/price ceiling |
|
V+ |
= estimated price if yield increases by a given amount, Dy |
|
V0 |
= initial observed bond price |
|
Dy |
= change in required yield, in decimal form |
Here, effective duration = (101.75 – 94.12) / (2 × 100 × 0.0075) = 7.63 / 1.5 = 5.09 years.
作者: honeycfa 时间: 2010-4-25 20:06
Assume that the current price of a bond is 102.50. If interest rates increase by 0.5% the value of the bond decreases to 100 and if interest rates decrease by 0.5% the price of the bond increases to 105.5. What is the effective duration of the bond?
The duration is computed as follows:
| Duration = |
105.50 ? 100 |
= 5.37 |
| 2 × 102.50 × 0.005 |
作者: honeycfa 时间: 2010-4-25 20:06
A 30-year semi-annual coupon bond issued today with market rates at 6.75% pays a 6.75% coupon. If the market yield declines by 30 basis points, the price increases to $1,039.59. If the market yield rises by 30 basis points, the price decreases to $962.77. Which of the following choices is closest to the approximate percentage change in price for a 100 basis point change in the market interest rate?
Approximate % change in price =
(price if yield down – price if yield up) / (2 × initial price × yield change expressed as a decimal).
Here, the initial price is par, or $1,000 because we are told the bond was issued today at par. So, the calculation is: (1039.59 – 962.77) / (2 × 1000 × 0.003) = 76.82 / 6.00 = 12.80.
作者: honeycfa 时间: 2010-4-25 20:07
Joshua Reynaldo is a fixed income portfolio manager for Golden Apple Capital Management. The portfolio is valued at $900 million, of which $840 million is currently invested. Fiona Campbell, the firm’s strategist, is becoming concerned about the possibility of an increase in interest rates. Reynaldo agrees, and this makes him nervous because the effective duration of his current portfolio investments is 10.315. However, his portfolio is presented to clients as a long-term fund, so there are limits to how short he can make the duration of the portfolio and still stay within the investment policy guidelines.
Reynaldo needs to invest the $60 million cash currently in the portfolio, and wants to do it in a way that will minimize the portfolio’s downside risk in a rising rate environment. He considers two different bonds. Both trade at their $1,000 par value and make coupon payments semiannually.
The first bond is a 12-year issue of Yorkville Technologies. Reynaldo likes the bond because of its attractive 5.9% coupon. He is concerned, however, because Yorkville is only rated Baa and Campbell is expecting a deterioration in credit quality as part of her economic outlook.
The second bond is also a 12-year maturity, but issued by Mountain States Electric & Gas, an Aaa utility. The 5.2% yield is not as attractive as the lower quality issue, but the Mountain States bond would represent a safe haven if credit spreads begin to widen, as both he and Campbell expect. Reynaldo’s only concern about the Mountain States bond is that it is callable any time at 102.
Discussing these possibilities with Campbell, Reynaldo tells her, “I ran my calculations assuming rates rise or fall by 50 basis points, and found that the effective convexity of the Mountain States bond is ten times the effective convexity of the Yorkville bond.” Campbell adds, “But the signs are opposite – the Mountain States bond has negative convexity and the Yorkville bond has positive convexity.”
Reynaldo continues, “I haven’t done a full valuation yet, but using my figures for duration and assuming convexity is 46, it looks like a 100 basis point rise in rates would cause the price of the Yorkville bond to fall by 6.73%.” Campbell, looking over his shoulder at his calculations, adds, “The dollar value of an 01 for the Yorkville bond is only 0.063, though.”
Reynaldo decides to invest in the Yorkville bond.
Which statement about how duration tends to predict price changes for large swings in yield is most accurate? Duration:
|
A) |
overestimates the increase in price for increases in yield. | |
|
B) |
overestimates the increase in price for decreases in yield. | |
|
C) |
underestimates the increase in price for decreases in yield. | |
For large swings in yield, duration tends to underestimate the increase in price when yield decreases and overestimate the decrease in price when yield increases. This is because duration is a linear estimate and does not account for the curvature in the price/yield relationship.
Using a 50 basis point change in interest rates, what is the difference in effective duration between the Mountain States bond and the Yorkville bond? The effective duration of the:
|
A) |
Mountain States bond is 0.34 lower than the effective duration of the Yorkville bond. | |
|
B) |
Yorkville bond is 2.21 higher than the effective duration of the Mountain States bond. | |
|
C) |
Mountain States bond is 0.34 higher than the effective duration of the Yorkville bond. | |
In order to calculate effective duration, we first need to know the bond price if interest rates rise or fall by 50 basis points.
For the Yorkville bond:
N = 24; PMT = (0.059 coupon × $1,000 par value / 2 payments per year =) 29.50; FV = 1,000
If rates rise by 50 basis points, I = ((5.9% + 0.50 =) 6.4% / 2 payments per year =) 3.2%; PV = -958.56.
Since the bond has a par value of $1,000, the estimated price will be (958.56 / 1,000 × 100 =) 95.86.
If rates fall by 50 basis points, I = ((5.9% ? 0.50 =) 5.4% / 2 payments per year = ) 2.7%; PV = -1043.74.
Since the bond has a par value of $1,000, the estimated price will be (1043.74 / 1,000 × 100 =) 104.37.
Now that we have the prices, we can use the formula for effective duration (ED):
ED = (104.37 – 95.86) / (2 × 100 × 0.005)
ED = 8.51 / 1
ED = 8.51
For the Mountain States bond:
N = 24, PMT = (0.052 coupon × $1,000 par value / 2 payments per year =) 26.00, FV = 1,000
If rates rise by 50 basis points, I = ((5.2% + 0.50 =) 5.7% / 2 payments per year =) 2.85%; PV = -956.97.
Since the bond has a par value of $1,000, the estimated price will be (956.97 / 1,000 × 100 =) 95.70.
If rates fall by 50 basis points, I = ((5.2% ? 0.50 =) 4.7% / 2 payments per year =) 2.35%; PV = -1045.46.
Since the bond has a par value of $1,000, the estimated price will be (1045.46 / 1,000 × 100 =) 104.55
However, since the bond is callable at 102, the price will be 102, not 104.55.
ED = (102 – 95.70) / (2 × 100 × 0.005)
ED = 6.30 / 1
ED = 6.30
The ED of the Yorkville bond is (8.51 – 6.30 =) 2.21 higher than the ED of the Mountain States bond.
If Reynaldo must invest his $60 million cash in either the Yorkville bond or the Mountain States bond, or some combination of the two, what is the lowest value he can achieve for the effective duration of the total portfolio?
If he purchases the Yorkville bond:
Portfolio duration = (w1 × ED1) + (w2 × ED2)
Portfolio duration = ((840 / 900) × 10.315) + ((60 / 900) × 8.51)
Portfolio duration = (0.933 × 10.315) + (0.067 × 8.51)
Portfolio duration = 9.624 + 0.570
Portfolio duration = 10.194
If he purchases the Mountain States bond:
Portfolio duration = (w1 × ED1) + (w2 × ED2)
Portfolio duration = ((840 / 900) × 10.315) + ((60 / 900) × 6.30)
Portfolio duration = (0.933 × 10.315) + (0.067 × 6.30)
Portfolio duration = 9.624 + 0.422
Portfolio duration = 10.046
Note, however, that we did not need to calculate the duration of the portfolio if he purchases the Yorkville bond. Since we know that the Mountain States bond has lower effective duration than the Yorkville bond, we know that the lowest effective duration for the total portfolio would be achieved by investing all $60 million in the Mountain States bond.
Regarding the statements made by Reynaldo and Campbell about the expected price change in the Yorkville bond:
For the Yorkville bond:
Percentage price change = (-8.51 × 0.010 × 100) + (46.0 × 0.0102 × 100)
Percentage price change = -8.51 + 0.46
Percentage price change = -8.05
Reynaldo’s statement is incorrect.
To calculate the dollar value of an 01 we need to know the price of the bond if interest rates rise (or fall) by 1 basis point:
N = 24; PMT = (0.059 coupon × $1,000 par value / 2 payments per year =) 29.50; FV = 1,000
If rates rise by 1 basis point, I = ((5.90 + 0.01 =) 5.91% / 2 payments per year =) 2.955%; PV = -999.149, for a price of 99.915.
PVBP = 100 – 99.915 = 0.085
Campbell’s statement is also incorrect.
Reynaldo and Apple are training a new analyst, Norah Spears. They ask Spears what she knows about duration and convexity. Spears replies with four statements:
|
Statement 1: |
Modified duration is a better measure than effective duration for bonds with embedded options. |
|
Statement 2: |
The convexity adjustment corrects for the error embedded in the duration. |
|
Statement 3: |
Modified duration ignores the negative convexity of a callable bond. |
|
Statement 4: |
Convexity of option-free bonds is always added to duration to modify the errors in calculating price volatility. |
Which of the following regarding Spears’ statements is most accurate?
|
A) |
Spears is correct with respect to Statement 2, but incorrect with respect to Statement 4. | |
|
B) |
Spears is correct with respect to all four statements. | |
|
C) |
Spears is correct with respect to Statement 3, but incorrect with respect to Statement 1. | |
Effective duration is a better measure than modified duration for bonds with embedded options because modified duration does not explicitly recognize the change in cash flows that will occur in a bond with embedded options as yield changes. Therefore, Statement 1 is incorrect. The other four statements made by Spears are correct.
作者: honeycfa 时间: 2010-4-25 20:07
A non-callable bond with 4 years remaining maturity has an annual coupon of 12% and a $1,000 par value. The current price of the bond is $1,063.40. Given a change in yield of 50 basis points, which of the following is closest to the effective duration of the bond?
First, find the current yield to maturity of the bond as:
FV = $1,000; PMT = $120; N = 4; PV = –$1,063.40; CPT → I/Y = 10%
Then compute the price of the bond if rates rise by 50 basis points to 10.5% as:
FV = $1,000; PMT = $120; N = 4; I/Y = 10.5%; CPT → PV = –$1,047.04
Then compute the price of the bond if rates fall by 50 basis points to 9.5% as:
FV = $1,000; PMT = $120; N = 4; I/Y = 9.5%; CPT → PV = –$1,080.11
The formula for effective duration is:
(V-–V+) / (2V0Δy)
Therefore, effective duration is:
($1,080.11 – $1,047.04) / (2 × $1,063.40 × 0.005) = 3.11
作者: honeycfa 时间: 2010-4-25 20:08
A non-callable bond has an effective duration of 7.26. Which of the following is the closest to the approximate price change of the bond with a 25 basis point increase in rates using duration?
The formula for the percentage price change is: –(duration)(Δy). Therefore, the estimated percentage price change using duration is: –(7.26)(0.25%) = –1.82%.
作者: honeycfa 时间: 2010-4-25 20:08
The price of a bond is equal to $101.76 if the term structure of interest rates is flat at 5%. The following bond prices are given for up and down shifts of the term structure of interest rates. Using the following information what is the approximate percentage price change of the bond using effective duration and assuming interest rates decrease by 0.5%?
Bond price: $98.46 if term structure of interest rates is flat at 6%
Bond price: $105.56 if term structure of interest rates is flat at 4%
The effective duration is computed as follows:
Using the effective duration, the approximate percentage price change of the bond is computed as follows:
Percent price change = -3.49 × (-0.005) × 100 = 1.74%
作者: honeycfa 时间: 2010-4-25 20:08
A bond with a semi-annually coupon rate of 3% sells for $850. It has a modified duration of 10 and is priced at a yield to maturity (YTM) of 8.5%. If the YTM increases to 9.5%, the predicted change in price, using the duration concept decreases by:
Approximate percentage price change of a bond = (-)(duration)(Δy)
Δy = 9.5% ? 8.5% = 1%
(-10)(1%) = -10%
($850)(-0.1) = -$85
作者: honeycfa 时间: 2010-4-25 20:09
A bond has the following characteristics:
- Modified duration of 18 years
- Maturity of 30 years
- Effective duration of 16.9 years
- Current yield to maturity is 6.5%
If the market interest rate decreases by 0.75%, what will be the percentage change in the bond's price?
Approximate percentage price change of a bond = (-)(effective duration)(Δy)
= (-16.9)(-0.75%) = +12.675%
作者: honeycfa 时间: 2010-4-25 20:09
Par value bond XYZ has a modified duration of 5. Which of the following statements regarding the bond is TRUE? If the market yield:
|
A) |
increases by 1% the bond's price will increase by $50. | |
|
B) |
increases by 1% the bond's price will decrease by $50. | |
|
C) |
increases by 1% the bond's price will decrease by $60. | |
Approximate percentage price change of a bond = (-)(Duration)(Δy)
(-5)(1%) = -5%
($1000)(-0.05) = –$50
作者: honeycfa 时间: 2010-4-25 20:09
Given a bond with a modified duration of 1.93, if required yields increase by 50 basis points, the expected percentage price change would be:
Approximate percentage price change of a bond = (-)(duration)(Δ y)
(-1.93)(0.5%) = -0.965%
作者: honeycfa 时间: 2010-4-25 20:10
What happens to bond durations when coupon rates increase and maturities increase?
As coupon rates increase, duration: As maturities increase, duration:
As coupon rates increase the duration on the bond will decrease because investors are recieving more cash flow sooner. As maturity increases, duration will increase because the payments are spread out over a longer peiod of time.
作者: honeycfa 时间: 2010-4-25 20:10
A non-callable bond with 10 years remaining maturity has an annual coupon of 5.5% and a $1,000 par value. The current yield to maturity on the bond is 4.7%. Which of the following is closest to the estimated price change of the bond using duration if rates rise by 75 basis points?
First, compute the current price of the bond as: FV = 1,000; PMT = 55; N = 10; I/Y = 4.7; CPT → PV = –1,062.68. Then compute the price of the bond if rates rise by 75 basis points to 5.45% as: FV = 1,000; PMT = 55; N = 10; I/Y = 5.45; CPT → PV = –1,003.78. Then compute the price of the bond if rates fall by 75 basis points to 3.95% as: FV = 1,000; PMT = 55; N = 10; I/Y = 3.95; CPT → PV = –1,126.03.
The formula for effective duration is: (V-–V+) / (2V0Δy). Therefore, effective duration is: ($1,126.03 – $1,003.78) / (2 × $1,062.68 × 0.0075) = 7.67.
The formula for the percentage price change is then: –(duration)(Δy). Therefore, the estimated percentage price change using duration is: –(7.67)(0.75%) = –5.75%. The estimated price change is then: (–0.0575)($1,062.68) = –$61.10
作者: honeycfa 时间: 2010-4-25 20:10
An $850 bond has a modified duration of 8. If interest rates fall 50 basis points, the bond's price will:
ΔP = (-)(P)(MD)(Δi)
ΔP = (-)(8)(850)(-0.005) = +$34
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