1. Duration

1.1 Introduction

Duration(久期): The sensitivity of bond’s full price to changes in the bond’s YTM or in benchmark interest rates.

Yield duration(收益率久期): sensitivity of bond price to the bond’s own YTM

Curve duration(曲线久期): sensitivity of bond price to a benchmark yield curve.

1.2 Yield Duration

Macaulay duration: (现金流现值的平均回流时间) the average time the bond holder has to wait before receiving the present value.

$MacDur=∑t=1nt×PVCFt∑PVCFt\text{MacDur}=\frac{\sum^n_{t=1}t\times PVCF_t}{\sum PVCF_t}$

$∑PVCFt\sum PVCF_t$ is bond price.
MacDur(5-year zero-coupon bond) = 5
MacDur(5-year coupon bond) < 5

Example: Consider a two-year bond that provides annual coupons at the rate of $66\ %$ . The YTM of the bond is $5$ .

$N = 2$ , $1/Y=5%1/Y=5\%$ , $P M T = 6$ , $F V = 100$ $→CPTPV=−101.8594\to CPT \; PV=-101.8594$

$MacDur=1×61+5%101.8594+2×106(1+5%)2101.8594=1.9439\text{MacDur} = 1\times \frac{\frac{6}{1+5\%}}{101.8594}+2\times \frac{\frac{106}{(1+5\%)^2}}{101.8594}=1.9439$

Modified duration: provides a linear estimate of the percentage price change for a bond given a change in yield.

$ModDur=−Δ%PΔy=MacDur1+y/m\text{ModDur}=-\frac{\Delta\%P}{\Delta y}=\frac{\text{MacDur}}{1+y/m}$

$Δ%P=ΔP/P≈−ModDur×Δy\Delta\%P = \Delta P/P \approx -\text{ModDur}\times \Delta y$

The Macaulay duration applies in the situation where $y$ is measured with continuous compounding.

$Δ%P≈−MacDur×Δy(continuouscompounding)\Delta\%P \approx -\text{MacDur} \times \Delta y(\text{continuous\; compounding})$

Example: If the annual yield on a $5.5%5.5\%$ semiannual coupon payment bond which is an option-free(不含权债券) noncallable bond increase from $6%6\%$ to $7%7\%$ , and the bond’s modified duration is $4.32$ , what should be the bond’s percentage price change?

$Δ%P≈−4.32×(7%−6%)=−4.32%\Delta \%P\approx-4.32\times(7\%-6\%)=-4.32\%$

Dollar duration: a measure of the dollar change in a bond’s value to a change in the yield.

$DollarDur(D)=−ΔPΔy=ModDur×pFull\text{DollarDur(D)}=-\frac{\Delta P}{\Delta y}=\text{ModDur}\times p^{Full}$
$ΔP≈−Ddollar×Δy\Delta P\approx -D_{dollar}\times \Delta y$

DV01(基点价值): describes the impact of a one-basis-point( $0.0001$ )change in interest rates on the value of a portfolio.
$DV01=−ΔP10,000×Δy=DollarDur10,000\text{DV01}=-\frac{\Delta P}{10,000\times\Delta y}=\frac{\text{DollarDur}}{10,000}$

Example: Suppose a 10-year, $8%8\%$ annual-pay straight bond priced at $105$ . Calculate the $D V 01$ if it has a par value of $10.000$ .

Step 1: $N = 10$ , $P V = - 105$ , $P M T = 8$ , $F V = 100$ ; $C P T : 1 / Y = 7.28$
Step 2: calculate bond price with YTM of $7.27$ and $7.29$

$N = 10$ , $F V = 10, 000$ , $P M T = 800$ , $F V = 10, 000$ , $1 / Y = 7.27$ ; $C P T : P V = - 10506.3$
$N = 10$ , $F V = 10, 000$ , $P M T = 800$ , $F V = 10, 000$ , $1 / Y = 7.29$ ; $C P T : P V = - 10492.1$

Step 3: $D V 01 = (10506.3 - 10492.1) / 2 = 7.1$

Summary
Sensitivity of percentage changes in bond price:

Macaulay duration: continuously compounded
Modified duration: other compounding frequencies

Sensitivity of actual changes in bond price:

Dollar duration
DV01: one-basis-point change

Macaulay duration, modified duration and money duration cannot be used for bonds with embedded option due to uncertain future cash flow.

1.3 Curve Duration

The One Factor Assumption: assumes that all interest rates move by the same amount, which means the shape of the term structure never changes(parallel shift(平行移动)).

Curve Duration: used for bonds with embedded option due to uncertain future cash flow.

Effective Duration: describes the percentage change in the price of a bond, due to a small change in all rates.

$EffectiveDuration=P−Δy−P+ΔyP02×ΔCurve\text{Effective\;Duration}=\frac{\frac{P_{-\Delta y}-{P_{+\Delta y}}}{P_0}}{2\times\Delta \text{Curve}}$

Yield-based DV01: The change in price from a one-basis-point increase in a bond’s yield.

DVDZ(or DPDZ): The change in price from a one-basis-point increase in all spot rates.

DVDF or DPDF: The change in price from a one-basis-point increase in forward rates.

Example: Consider a portfolio consists of a Treasury bond with a face value of USD 1 million paying a $10%10\%$ per annum coupon semi-annually. The tenor(到期日) of this bond is one year. Suppose that spot rates are as shown in the table below. Calculate the DV01 and effective duration, if spot rates move $5$ bp.

Maturity(Years)	Rate(%)	+5bp Rate(%)	-5bp Rate(%)
0.5	7.0	7.05	6.95
1.0	7.5	7.55	7.45

The value of the bond is:
$50,0001.035+1,050,0001.03752=1,023,777.32\frac{50,000}{1.035}+\frac{1,050,000}{1.0375^2}=1,023,777.32$

The rates increase by five basis points, the value of the bond:
$50,0001.03525+1,050,0001.037752=1,023,295.72\frac{50,000}{1.03525}+\frac{1,050,000}{1.03775^2}=1,023,295.72$
$DV01′=1,023,777.32−1,023,295.725=96.32DV01'=\frac{1,023,777.32-1,023,295.72}{5}=96.32$

The rates decrease by five basis points, the value of the bond:
$50,0001.03475+1,050,0001.037252=1,024,259.26\frac{50,000}{1.03475}+\frac{1,050,000}{1.03725^2}=1,024,259.26$
$DV01′′=1,024,259.26−1,023,777.325=96.39DV01''=\frac{1,024,259.26-1,023,777.32}{5}=96.39$

The two estimates of DV01 differ slightly because the bond’s price is not exactly a linear function of interest rates. We can get a good estimate of DV01 by averaging the two estimates:

$D V 01 = (96.32 + 96.39) / 2 = 96.355$

$EffectiveDuration=P−Δy−P+ΔyP02×ΔCurve=(1,024,259.26−1,023,295.72)/1,023,777.322×0.05%=0.9412\text{Effective\;Duration}=\frac{\frac{P_{-\Delta y}-{P_{+\Delta y}}}{P_0}}{2\times\Delta Curve}=\frac{(1,024,259.26-1,023,295.72)/1,023,777.32}{2\times0.05\%}=0.9412$

1.4 Limitations of Duration

Duration provides a good approximation of the effect of a small parallel shift in the interest rate term structure.

But these equations cannot be relied upon, if the change in the bond yield arises from a non-parallel shift in the interest rate term structure or the change is large.

1.5 Properties of Bond Duration

Longer time-to-maturity usually leads to higher duration.

Higher coupon rate leads to lower duration.

Higher yield-to-maturity leads to lower duration.

2. Convexity

2.1 Convexity

Convexity(凸度): measures the non-linear relationship of bond prices to changes in interest rates and the curvature in the relationship between bond prices and bond yields that demonstrates how the duration of a bond changes as the interest rate changes.

The duration plus convexity approximation fits a quadratic function and captures some of the curvature, which provides a better approximation.（长多跌少）

Duration overestimates(高估) the magnitude of price decreases
Duration underestimates(低估) the magnitude of price increases

2.2 Macaulay Convexity

$MacConvexity=∑t=1nt2×PVCFt∑PVCFt\text{MacConvexity}=\frac{\sum^n_{t=1}t^2\times {PVCF}_t}{\sum {PVCF}_t}$

Example: Consider a bond that provides annual coupons at the rate of $6%6\%$ . The maturity is 2 years. The YTM is $5%5\%$ . Calculate the Macaulay convexity of it.

$N = 1$ , $1 / Y = 5$ , $P M T = 6$ , $\to CPT \; PV=-101.8594$

$MacCovexity=12×6(1+5%)101.8594+22×106(1+5%)2101.8594=3.8317\text{MacCovexity}=1^2\times\frac{\frac{6}{(1+5\%)}}{101.8594}+2^2\times\frac{\frac{106}{(1+5\%)^2}}{101.8594}=3.8317$

Year	Time Squared	CF	PV(CF)	Weight	Time Squared ☓ Weight
1	1	6	5.7143	5.61%	0.0561
2	4	106	96.1451	94.39%	3.7756

2.3 Modified Convexity

$ModifiedConvexity=MacConvexity(1+y/m)2\text{Modified\;Convexity}=\frac{\text{MacConvexity}}{(1+y/m)^2}$

Example: Consider a bond a bond’s Macaulay Convexity is $8.13904$ . If the bond is compounded semi-annually and YTM is $5.2455%5.2455\%$ , calculate the modified convexity of it.
$ModifiedConvexity=8.13904(1+5.2455%/2)2=7.7283\text{Modified\;Convexity}=\frac{8.13904}{(1+5.2455\%/2)^2}=7.7283$

With continuous compounding
$Δ%P≈−MacDur∗Δy+12∗MacConvexity∗(Δy)2\Delta\%P\approx-MacDur*\Delta y+\frac{1}{2}*MacConvexity*(\Delta y)^2$

$ΔP≈−MacDur∗P∗Δy+12∗MacConvexity∗P∗(Δy)2\Delta P\approx-MacDur*P*\Delta y+\frac{1}{2}*MacConvexity*P*(\Delta y)^2$

With discrete compounding frequencies
$Δ%P≈−ModDur∗Δy+12∗ModConvexity∗(Δy)2\Delta\%P\approx-\text{ModDur}*\Delta y+\frac{1}{2}*\text{ModConvexity}*(\Delta y)^2$

$ΔP≈−ModDur∗P∗Δy+12∗ModConvexity∗P∗(Δy)2\Delta P\approx-\text{ModDur}*P*\Delta y+\frac{1}{2}*\text{ModConvexity}*P*(\Delta y)^2$

Example: Suppose a bond with modified duration of $31.32$ and modified convexity of $667$ , when its yield is expected to fall by $50bps50\;bps$ , what should be the expected percentage price change?

$Δ%P=−31.32∗(−0.0050)+12∗667∗(−0.0050)2=16.49%\Delta\%P=-31.32*(-0.0050)+\frac{1}{2}*667*(-0.0050)^2=16.49\%$

2.4 Effective Convexity

Effective convexity measures the sensitivity of the duration measure to changes in interest rates. The effective convexity( C ) of a position worth $P$ can be estimated as

$C=1p[P++P−−2P(ΔCurve)2]C=\frac{1}{p}\left[\frac{P^++P^--2P}{(\Delta Curve)^2}\right]$

Example: A $ $1, 000$ par bond with $17$ years to maturity and a 4% semiannual coupon. The original price of the bond is $€ 867, 4808$ . When all the interest rates rise $5 b p s$ , the price of the bond would be $€ 861.4837$ . When all the interest rates decrease $5 b p s$ , the price of the bond would be $€ 873.5338$ . Please calculate the effective convexity of this bond.

$C=1p[P++P−−2P(ΔCurve)2]=873.5338+861.4837−2×867,4808867,4808×0.00052=257.7579C=\frac{1}{p}\left[\frac{P^++P^--2P}{(\Delta Curve)^2}\right]=\frac{873.5338+861.4837-2\times 867,4808}{867,4808\times 0.0005^2}=257.7579$

2.5 Negative Convexity

Callable bonds often have negative convexity(concavity 凹度 ), especially when interest rates are low.

Putable bonds often have higher positive convexity, especially when interest rates are high.

The security with more convexity outperforms the less convex security in both bull (rising price) and bear(failing price) markets.

The bigger the volatility of the interest rate, the greater the gains from the positive convexity.

Long volatility of the interest rate $→\to$ choosing a security with positive convexity
Short volatility of the interest rate $→\to$ choosing a security with negative convexity

2.6 Portfolio Calculations

Consider the situation where a portfolio consists of a number of bonds.

Duration of portfolio: $∑i=1nwi×Di\sum^n_{i=1}w_i\times D_i$

$w_i$ : the bond’s market value to the whole portfolio value

Convexity of portfolio: $∑i=1nwi×Ci\sum^n_{i=1}w_i\times C_i$

$w_i$ : the bond’s market value to the whole portfolio value

DV01: The DV01 for a portfolio is simply the sum of the DV01s of the components of the portfolio.

Example: A portfolio consists of three bonds worth (in USD millions) 10,15 and 25. Based on the information shown in the table, calculate the duration and the convexity of the portfolio.

Bonds	Value	Duration	Convexity
Bond A	10	6	200
Bond B	15	8	215
Bond C	25	11	245

Bonds	Value	Weight	DurationXWeight	ConvexityXWeight
Bond A	10	0.2	1.2	40
Bond B	15	0.3	2.4	64.5
Bond C	25	0.5	5.5	122.5
Portfolio	50		9.1	227

2.7 Barbell vs. Bullet Portfolio

Barbell portfolio(杠铃组合): securities in this portfolio concentrate in short and long maturities but less intermediate maturities.

Bullet portfolio(子弹组合): has more exposure at intermediate maturities.

For bonds with same duration, the one that has the greater dispersion(离散) of cash flows has the greater convexity.

Example: A manager purchase $1 million Bond B. The coupon payments are semi-annual. Using A and C to construct a portfolio with the same cost and duration.

Bond	Coupon-Semi	Maturity	Price	Yield	Duration	Convexity
A	2%	5	95.3889	3%	4.7060	25.16
B	4%	10	100	4%	8.1755	79
C	6%	30	115.4543	5%	14.9120	331.73

$V_A+V_C=1\; million$

$VA1million×4.7060+VC1million×14.9120=8.1755\frac{V_A}{1\;million} \times4.7060+\frac{V_C}{1\;million}\times 14.9120=8.1755$

$V_A=0.66\;million,\;V_C=0.34\;million$

$ConvexityA+C=0.66×25.16+0.34×331.73=129.39Convexity_{A+C}=0.66\times25.16+0.34\times331.73=129.39$

Advantage for barbell portfolio:

These two strategies will have the same duration and different convexity.
The barbell strategies produces a better result when there is a parallel shift in the yield curve.

Disadvantage for barbell portfolio:

The bullet investment would perform better than the barbell investment for many non-parallel shifts.

Example: Jackson, portfolio manager of Venus Fund, is managing a fixed income portfolio. He wants to measure the duration of the portfolio in order to determine yield curve strategy. There are three bonds within this portfolio. What is the modified duration of this portfolio?

Bond	Par Value	Price	Modified Duration
Bond A	2 million	$95.7 per $100	7
Bond B	3.5 million	$112 per $100	5
Bond C	1 million	$90.79 per $100	4

Bond	Market Value	Weight	Modified Duration
Bond A	1.914 million	0.2839	1.9873
Bond B	3.92 million	0.5814	2.9072
Bond C	0.9079 million	0.1347	0.5387

Portfolio duration = 1.9873+2.9072+0.5387=5.4332

3. Hedging

3.1 Duration Hedging

To construct a portfolio that can hedge a small change in interest rates.

$Δ\Delta$ (Price change of underlying asset) + $Δ\Delta$ (Price change of hedging instrument) = 0

Example: a fund manager takes a long position in a 10-year treasury with DV01 of 0.165. She is quite worried about the uncertainty of the interest rate in next three months. After analyzing the potential changes in term structure of interest rates, she decides to hedge this position with a T-note futures which has DV01 of 0.123. What kind of action should she undertake to provide an appropriate hedge for small changes in yield?

DV01 underlying asset + N $×\times$ DV01 hedging instrument $→n=−1.3415\to n=-1.3415$

Short 1.3514 T-note futures

3.2 Duration and Convexity Hedging

We can make both duration and convexity zero by choosing $P_1$ and $P_2$ so that:

$V×DV+P1D1+P2D2=0V\times D_V+P_1D_1+P_2D_2=0$

$V×CV+P1C1+P2C2=0V\times C_V+P_1C_1+P_2C_2=0$

The position is hedged against relatively large parallel shifts in the term structure. However, it will still have exposure to non-parallel shifts.

Example: An investor has a bond position worth USD $20, 000$ with a duration of 7 and a convexity of 33. Two bonds are available for hedging. Bond A has a duration of 10 and a convexity of 80. Bond B has a duration of six and a convexity of 25. How can a duration plus convexity hedge be set up?

$10PA+6PB+20,000×7=010P_A+6P_B+20,000\times7 =0$
$80PA+25PB+20,000×33=080P_A+25P_B+20,000\times33=0$
$→PA=−2,000,PB=−20,000\to P_A=-2,000,\;P_B=-20,000$

A short position A of USD 2,000 and a short position B of USD 20,000 are required.

Example: A market maker sell call options on underlying bonds when the interest rate is $6%6\%$ . The value of the call option is $ $3$ million and its duration is $6$ . At the same $6%6\%$ rate, the underlying bond has a duration of $7$ and pays $6%6\%$ coupon rate with par value $100. How should market maker do to hedge the risk of short call options？

$Δ\Delta$ (Price change of underlying asset) + $Δ\Delta$ (Price change of hedging instrument)=0

$3million×6+P×7=0→P=−2.57million3million\times6+P\times7=0\to P=-2.57million$

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4.2.1 Duration and Convexity