7. Desigualdades variacionales
7.1. El problema de desigualdad variacional
D
E
Interest Rate Risk
Duration and Convexity
August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA
Approximate Modified Duration
An alternative approach is to approximate modified duration directly:
where:
P– Price of the bond after a decrease in yield P+ Price of the bond after an increase in yield.
Let’s assume a yield change of 50 bps. for your 4-year, 5% annual coupon paying bond. The approx. modified duration is: 0
P
)
Yield
(
2
P
P
Duration
Modified
.
Approx
547
.
3
100
)
005
.
0
(
2
247
.
98
794
.
101
Duration
Modified
.
Approx
Interest Rate Risk
Duration and Convexity
5
Example:
An option-free bond has a remaining maturity of 5 years and a coupon of 4.5% which is paid semi-annually. The yield to maturity of the bond is 4.8%. Calculate the approximate modified duration and (based on the duration) the new bond price if interest raise by 50 basis points.
Interest Rate Risk
Duration and Convexity
August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA
Effective Duration
Another approach to assess the interest rate risk of a bond is to estimate the percentage change in price given a change in a benchmark yield curve. – The effective duration of a bond is the sensitivity of the bond’s price to a change in a benchmark yield curve:
The difference between approximate modified duration and effective duration is in the denominator. Modified duration is a yield duration statistic in that it measures interest rate risk in terms of a change in the bond’s own YTM. Effective duration is a curve duration statistic in that it measures interest rate risk in terms of a change in the benchmark yield curve ( Curve). 0
P
)
Curve
(
2
P
P
Duration
Effective
Interest Rate Risk
Duration and Convexity
5
Modified duration is the approximate price change in a bond‘s price for a 100 basis point change in yield, assuming that the bond‘s expected cash flows do not change when the yield changes (option-free bonds). When calculating P- and P+, the same cash flows used to calculate P0 are used.
Effective (or option-adjusted) duration also estimates a bond‘s price sensitivity to a change in yield, but accounts for how changes in yield will affect cash flows.
Interest Rate Risk
Duration and Convexity
August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA
Modified and Macaulay Duration assume that the cash flows of a bond will not change as yields change. – Thus
effective duration is the appropriate measure of interest rate risk for bonds with embedded options because changes in interest rates may change their future cash flows.
Pricing models are used to determine the prices that would result from a given size change in the benchmark yield curve
Interest Rate Risk
Duration and Convexity
5
How a Bond’s Maturity, Coupon, Embedded Option, and Yield Level Affect its Interest Rate Risk
Holder other factors constant:
– Duration increases (decreases) when maturity increases (decreases).
– Duration decreases (increases) when the coupon rate increases (decreases). – Duration decreases (increases) when YTM increases (decreases).
With a call provision, the value of the call increases as yields fall, so a decrease in yield will have less effect on the price of the callable bond (price callable bond = price straight-bond – price of call)
With a put provision, the bondholder’s option to sell the bond back to the issuer at a set put price reduces the
negative impact of the yield increases on the price of a putable bond (price putable bond = price straight-bond + price of put)
Interest Rate Risk
Duration and Convexity
August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA
Duration of a Portfolio
A portfolio‘s duration is the weighted average duration of the component securities
Example
Compute the duration of the following portfolio:
Bond: Market Value: Duration: % of portfolio:
A USD 3,000,000 3.75 ... B USD 4,000,000 4.25 ... C USD 5,000,000 2.55 ...
Interest Rate Risk
Duration and Convexity
5
A second approach to calculate portfolio duration is to calculate the weighted average number of periods until cash flows will be received using the portfolio’s IRR (its cash flow yield). This method is better theoretically but cannot be used for bonds with embedded options. This approach is also inconsistent with duration capturing the relationship between price and YTM.
Duration will not provide meaningful percentage value change estimates for portfolios unless the yield curve shifts in a parallel manner
Bullet and barbell portfolios with the same durations have the same interest rate risk if the yield curve shifts in a parallel manner
Even if the bullet and barbell portfolios have the same durations, they do not have the same interest rate risk with respect to a non-parallel shift in the yield curve
Interest Rate Risk
Duration and Convexity
August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA
Money Duration and Price Value of a Basis Point (PVBP)
The money duration of a bond (dollar duration) is expressed in currency units and calculated as follows: Money Duration = Annual Modified Duration x Full Price of Bond
Money duration is sometimes expressed as money duration per 100 of bond par value: Money Duration = Annual Modified Duration x Full Price of Bond per 100 of Par Value
The change in bond price can be calculated:
Change in Bond Price = Money Duration x Change in YTM
Interest Rate Risk
Duration and Convexity
5
Example
A life insurance company holds a USD 10 million (par value) position in a 4.50% ArcelorMittal bond that matures
on 25 February 2017. The bond is priced (flat) at 98.125 per 100 of par value to yield 5.2617% on a street-convention semi-annual bond basis for settlement on 27 June 2014. The total market value of the position, including accrued interest, is USD 9,965,000 or 99,65 per 100 par value. The bond’s (annual) Macaulay duration is 2.4988.
1. Calculate the money duration per 100 in par value for the ArcelorMittal Bond?
2. Using the money duration, estimate the loss on the position for each 1 bp increase in the yield-to-maturity for that settlement date?
Interest Rate Risk
Duration and Convexity
August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA
The price value of a basis point is the change in the value of a bond, expressed in currency units, for a change in YTM of one basis point (0.01%):
Example
A newly issued, 10-year, 5% annual coupon paying bond is priced at 92.64. The price value of a basis point for this bond assuming a par value of USD 1 million is closest to:
A. USD 10 B. USD 700 C. USD 1,400
2
P
P
PVBP
Interest Rate Risk
Duration and Convexity
5
Approximate Convexity and Effective Convexity
The divergence from the bond price curve to the straight line (duration) is called convexity
Positive convexity is a larger increase in price than decrease in price, for the same change in interest rates – The upside is greater than the downside
All option-free bonds have positive convexity, but the actual degree of duration and convexity of bonds will vary,
depending on the level of interest rates, coupon, and maturity. A longer maturity, a lower coupon rate, or a lower YTM will all increase convexity, and vice versa. For two bonds with equal duration, the one with cash flows that are more dispersed over time will have the greater convexity.
Interest Rate Risk
Duration and Convexity
August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA
Convexity, is used to approximate the change in price that is not explained by duration. The approximate convexity of a bond can be computed as follows:
2
0
0
)
(
P
P
2
P
P
Convexity
.
approx
Yield
in
change
Example:Compute the convexity of a 6-year, 7% seminannual corporate bond which actually has yield to maturity of 5% when we assume a change in interest rates by 100 basis points
Interest Rate Risk
Duration and Convexity
5
Effective convexity, like effective duration, must be used for bonds with embedded options. The approximate effective convexity of a bond can be computed as follows:
2
0
0
)
(
P
P
2
P
P
Convexity
.
approx
Curve
in
change
effective
Interest Rate Risk
Duration and Convexity
August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA
For callable bonds: The decline in yield will reach the point where the rate of increase in the price of the bond will start slowing down and eventually level off →negative convexity. – This is due to the fact that the issuer has the right to retire the bond prior to maturity at some specified call price.
As long as yields remain below a certain level, callable bonds will exhibit price compression, or negative convexity. As long as yields are above a certain level, those same callable bonds will exhibit all the properties of positive
convexity.
With putable bonds as interest rates move from high to low the duration will increase. Convexity will be positive at all rate levels and convexity will be highest when interest rates are in the area where the put begins to acquire value.
Interest Rate Risk
Duration and Convexity
5
Taylor Approximation
Given values for approximate annual modified duration and approximate annual convexity, the percentage change in the full price of a bond can be calculated as follows:
Estimated price change in % = –Duration ( y) + 0.5 Convexity ( y)2
Example:
Compute the estimated price change in % when the duration is 5.5 and the convexity is 30.5 and assuming an interest shift by 120 basis points!
Interest Rate Risk
Duration and Convexity
August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA
Duration predicts a straight linear-relationship between changes in yield and changes in price. In reality, the price/yield relationship for a bond is convex. Duration therefore underestimates the price increase and overestimates the price decline.