TOC
1. Motivation
Interest rate risk refers to the adverse price movement of a bond resulting from the change in market interest rate. For bond investors, it is typically the risk that the interest rate will rise. The inverse convex relationship between price and yield are shown as below
There are two methods measuring the interest rate risk: Duration and Convexity.
2. Duration
2.1 Definition
Duration of a debt security is its discounted-cash-flow-weighted time to receipt of all promised cash flows divided by its price. Consider a coupon bond price expressed in continuous compounding, , duration is expressed as
Duration is the price elasticity of interest rate, which measures a fixed income security’s exposure to interest rate risk.
which can be expressed as
MaCaulay Duration & Modified Duration
Doing the same exercise in discrete compounding leads to
which is referred to as MaCaulay duration.
An alternative measure is the modified duration
which coincides with duration in continuous compounding.
2.2 PVBP / DV01
The price value of a basis point (PVBP) or dollar value of 01 (DV01) is the change in a bond’s price due to a change in the interest rate. PVBP is typically expressed as the change in value of a $1 million (par value) position in reaction to a change in the yield of one basis point, that is, .
The relation between PVBP and duration: Note that tells us how much a bond price will change when increases by one, that is, 10,000 basis points. Dividing by 10,000, we obtain the change in P due to a one basis point change in .
is equal to the change in the price of a $100-face-value bond for a one-basis-point change in the yield. A $1 million position involves 10,000 bonds, so we have
Suppose the initial yield is 9% so that
Now
If the yield increases or decreases by 1 bp:
- ⇒
- ⇒
The PVBP-implied change is 646.90. The errors are about 0.04% of actual change or 0.000031% of the initial bond value.
If the yield changes more significantly, say 100 bps:
- ⇒
- ⇒
The PVBP-implied change is 64,690. The errors are about 4% of the realized change or 0.32% of the initial bond value.
2.3 Properties of Duration
A zero-coupon bond’s duration equals its maturity, implying that the zero-coupon bond is the most interest-rate-elastic instrument for a given maturity.
A coupon bond’s duration is less than its maturity.
- Duration increases with maturity
- Duration decreases with coupon rate and yield to maturity
- Duration is time-varying
As a linear approximation of the pirce-yield relationship, duration may fail miserably if the term structure tilts while shifting parallelly.
Term Structure and Yield Curve
Term structure of interest rates refers to the relationship between spots rates (yields to maturity of default-free zero-coupon bonds) and their maturities.
Yield curve is the plot of yield to maturity against time-to-maturity or a risk measure (e.g., duration). The shape of yield curve can be normal (upward), inverted (downward) or humped. Components of yield curve include the level factor, the slope factor and the curvature factor.
2.4 Hedging and Trading
Hedging interest rate risk with Treasury futures contract
The objective function is
thus
we should short 94 such Treasury futures contracts.
Hedging interest rate risk with interest rate swap
The objective function is
thus
the firm should enter into 575 contracts, paying fixed rate and receiving floating rate.
Spread Trading
Spread trading bets on changing slop of the term structure. We construct a trading srtategy that pay off if the yield curve steepens or flattens sufficiently but is insensitive to parallel shifts in the term structure.
There are two scenarios where the yield curve can steepen:
- Short rates increase less than long rates (in a bearish market)
- Short rates fall more than long rates (in a bullish market)
To gain exposure to the yield curve slop, we need to trade (at least) two bonds of different maturities: short the long bond and long the short bond.
Suppose we long 100 short bonds, to calculate the bonds we need to short in establishing duration eutrality,we have the objective function
The result shows that we need to short 33.57 long bonds. This duration-neutral portfolio is worth $6,132, and its value will increase if the yield curve slope rises substantially.
A more realistic example is shown as below
The current spread between the yields of the bonds is 7.96%-5.30%=266bps. The investor is considering a strategy that pays off if the spread rises above 266 bps. There are two scenarios might do the trick
- in a bullish market: the long yield falls less than the short yield
- in a bearish market: the long yield rises more than the short yield
The investor will long the short and short the long. The position will be neutral to parallel shifts in the yield curve, that is, duration-neutral
Suppose we take a $100 million (par value) short position in the long bond, then
The trades will typically be implemented involving repo and reverse repo:
- to take a long position in the short bond, a repo will be used to finance the transaction
- to take a short position in the long bond, a reverse repo will be used to borrow the security
Operations and Cash flows are shown as below
The final profit is -$2995. The main reason for this loss is bid-ask spread — Ignoring the bid-ask spread yields a profit of roughly $400,000. Other potential factors include haircut and special repo rates.
3. Convexity
3.1 Definition
Convexity is the change in the slope of the price-yield curve for a slight change in the yield.
A bond’s convexity is the rate of change of its durations and is measured as the second derivative of the bond’s price to its yield.
the first term is PVBP and the second term is dollar convexity in continuous time
dividing both sides of above euqation by P yields
which are Duration and relative convexity.
3.2 Properties of convexity
Positive and negative convexity
Security exhibtis positive convexity when its price rises more due to a downward movement in yield than its price falling due to an equal upward movement in yield. Bullet securities typically exhibit positive convexity.
Callable bonds usually exhibit negative convexity at lower yields. The price of a callable bond might drop as the likelihood that the bond will be called increases, and the shape of a callable bond’s curve of price to yield is concave or negatively convex.
when interest rate fall, .
Most mortgage securities exhibit negative convexity due to the mortgage borrower’s incentive to refinance at a lower rate when market yields decline.
Remarks:
- Positive convexity is desirable. Convexity bias explains the curvature of the yield curve.
- Bonds with greater convexity tend to gain more in price when yields fall than they lose when yields rise.
- The more convex a bond price is, the more valuable the bond is.
- If interest rates are volatile, this asymmetric response of price to yield is more desirable.
Butterfly trade
A trading strategy to take advantage of convenxity by establishing a duration-neutral trade. The is replicate a bullet position in an intermediate maturity by a barbell position in straddling maturities.
For example, a long position in 5-year T-note (zero coupon bond) is a bullet position. A long position in a portfolio of a 2-year T-note and a 10-year T-note is a barbell position, reflecting the two balloon payments. Let be the par value of the 2-year T-note and let be the par value of the 10-year T-note needed to replace $100 million par value of a 5-year T-note. We require that the cash proceeds from the sale of a 5-year T-note to be sufficient to buy the requisite numbers of 2-year and 10-year T-notes. This is the self-financing condition
We further require that the DV01 of the 5-year T-note that is sold is equal to the PVBP of the portfolio that is purchased, that is
By construction, at the prevailing market yields, the market value of a 5-year T-note and its PVBP are exactly matched by those of the barbell portfolio. When the yields drop, the PVBP of the barbell portfolio exceeds the PVBP of the bullet security. This indicates that the barbell portfolio will benefit more from the reduction in yields. On the other hand, as the yields go up, the PVBP of the barbell portfolio is always lower than that of the 5-year T-note. Consequently, the barbell portfolio will loss less value compared to the bullet position.
Actually,
We have hedged off the first term (Duration) by constructing this portfolio and the second term is positive. Therefore, whenever there is a deviation off the inital price (), which means we are making profits.
We can make money when yield curve move parallelly. We gets a lot of money when the yield curve flattens (tails down) and lose a lot of money when the yield curve steepens. Explanations:
- When yield curve gets flattening:
- for pair (short 5, long 10): short the SHORT, long the LONG ⇒ we are gonna make money
- for pair (long 2, short 5) : short the LONG, long the SHORT ⇒ we are gonna loss money
- since bond5 and bond10 have larger duration, we can overall make money
- When yield curve gets steepening:
- for pair (short 5, long 10): short the SHORT, long the LONG ⇒ we are gonna loss money
- for pair (long 2, short 5) : short the LONG, long the SHORT ⇒ we are gonna make money
- since bond 5 and bond 10 have larger duration, we overall loss money
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