Free Essay

Professor Carpenter

Convexity

Concepts and Buzzwords

• Dollar Convexity • Convexity • Curvature • Taylor series • Barbell, Bullet

Readings

• Veronesi, Chapter 4 • Tuckman, Chapters 5 and 6

Convexity

1

Debt Instruments and Markets

Professor Carpenter

Convexity

• Convexity is a measure of the curvature of the value of a security or portfolio as a function of interest rates. • Duration is related to the slope, i.e., the ﬁrst derivative. • Convexity is related to the curvature, i.e. the second derivative of the price function. • Using convexity together with duration gives a better approximation of the change in value given a change in interest rates than using duration alone.

Price‐Rate Func:on

Example: Security with Positive Convexity

Price

Linear approximation of price function

Approximation error

Interest Rate (in decimal)

Convexity

2

Debt Instruments and Markets

Professor Carpenter

Correc:ng the Dura:on Error

• The price‐rate function is nonlinear. • Duration and dollar duration use a linear approximation to the price rate function to measure the change in price given a change in rates. • The error in the approximation can be substantially reduced by making a convexity correction.

Taylor Series

• The Taylor Theorem from calculus says that the value of a function can be approximated near a given point using its “Taylor series” around that point. • Using only the ﬁrst two derivatives, the Taylor series approximation is:

1 f ''(x 0 ) × (x − x 0 ) 2 2 1 Or, f (x) − f (x 0 ) ≈ f '(x 0 ) × (x − x 0 ) + f ''(x 0 ) × (x − x 0 ) 2 2 f (x) ≈ f (x 0 ) + f '(x 0 ) × (x − x 0 ) +

€

Convexity

3

Debt Instruments and Markets

Professor Carpenter

Dollar Convexity

• Think of bond prices, or bond portfolio values, as functions of interest rates. • The Taylor Theorem says that if we know the ﬁrst and second derivatives of the price function (at current rates), then we can approximate the price impact of a given change in rates.

f (x) − f (x 0 ) ≈ f '(x 0 ) × (x − x 0 ) + 0.5 × f ''(x 0 ) × (x − x 0 ) 2

The ﬁrst derivative is minus dollar duration. Call the second derivative dollar convexity. • Then change in price ≈ ‐$duration x change in rates + 0.5 x $convexity x change in rates squared

€

Dollar Convexity of a PorBolio

If we assume all rates change by the same amount, then the dollar convexity of a portfolio is the sum of the dollar convexities of its securities. Sketch of proof:

∑ f (x) − f (x ) ≈ (∑ f '(x )) × (x − x ) + 0.5 × (∑ f ''(x )) × (x − x ) i i 0 i 0 0 i 0 0

2

I.e., Δ portfolio value ≈ ‐ (sum of dollar durations) x Δr

€

+ 0.5 x (sum of dollar convexities) x (Δrates)2 ⇒ Portfolio dollar duration = sum of dollar durations ⇒ Portfolio dollar convexity = sum of dollar convexities

Convexity

4

Debt Instruments and Markets

Professor Carpenter

Convexity

• Just as dollar duration describes dollar price sensitivity, dollar convexity describes curvature in dollar performance. • To get a scale‐free curvature measure, i.e., curvature per dollar invested, we deﬁne

convexity =

dollar convexity price

⇒ The convexity of a portfolio is the average convexity of its securities, weighted by present value:

€ convexity =

∑ price × convexity ∑ price i i

i

= pv wtd average convexity

• Just like dollar duration and duration, dollar convexities add, convexities average.

€

Dollar Formulas for $1 Par of a Zero

For $1 par of a t‐year zero‐coupon bond price = dt (rt ) = 1 (1+ rt /2) 2t t (1+ rt /2) 2t +1 t 2 + t /2 (1+ rt /2) 2t +2

dollar duration = - dt '(rt ) = dollar convexity = dt ''(rt ) =

For $N par, these would be multiplied by N.

€

Convexity

5

Debt Instruments and Markets

Professor Carpenter

Percent Formulas for Any Amount of a Zero duration = dollar duration N × t /(1+ rt /2) 2t +1 t = = 2t price N ×1/(1+ rt /2) 1+ rt /2 dollar convexity N × (t 2 + t /2) /(1+ rt /2) 2t +2 t 2 + t /2 = = price N ×1/(1+ rt /2) 2t (1+ rt /2) 2

convexity =

€

• These formulas hold for any par amount of the zero – they are scale‐free. • The duration of the t‐year zero is approximately t. • The convexity of the t‐year zero is approximately t2. • (If we deﬁned price as dt = e‐rt, and diﬀerentiated w.r.t. this r, then the duration of the t‐year zero would be exactly t and the convexity of the t‐year zero would be exactly t2.)

Class Problems

Calculate the price, dollar duration, and dollar convexity of $1 par of the 20‐year zero if r20 = 6.50%.

Convexity

6

Debt Instruments and Markets

Professor Carpenter

Class Problems

Suppose r20 rises to 7.50%. 1) Approximate the price change of $1,000,000 par using only dollar duration.

2) Approximate the price change of $1,000,000 using both dollar duration and dollar convexity.

3) What is the exact price change?

Class Problems

Suppose r20 falls to 5.50%. 1) Approximate the price change of $1,000,000 par using only dollar duration.

2) Approximate the price change of $1,000,000 using both dollar duration and dollar convexity.

3) What is the exact price change?

Convexity

7

Debt Instruments and Markets

Professor Carpenter

Sample Risk Measures

Duration and convexity for $1 par of a 10‐year, 20‐year, and 30‐year zero.

Maturity 10 20 30 Rate 6.00% 6.50% 6.40% Price 0.553676 0.278226 0.151084 $Dura/on 5.375493 5.389364 4.391974 Dura/on 9.70874 19.37046 29.06977 $Convexity 54.7987 107.0043 129.8015 Convexity 98.9726 384.5951 859.1356

For zeroes, • duration is roughly equal to maturity, • convexity is roughly equal to maturity squared.

Dollar Convexity of a PorBolio of Zeroes

• Consider a portfolio with ﬁxed cash ﬂows at diﬀerent points in time (K1, K2, … , Kn at times t1, t2, …, tn). • Just as with dollar duration, the dollar convexity of the portfolio is the sum of the dollar convexities of the component zeroes. • The dollar convexity of the portfolio gives the correction to make to the duration approximation of the change in portfolio value given a change in zero rates, assuming all zero rates change by the same amount. n portfolio dollar convexity =

∑K j=1 j

×

t 2 + t j /2 j (1+ rt j /2)

2t j +2

€

Convexity 8

Debt Instruments and Markets

Professor Carpenter

Class Problem: Dollar Convexity of a PorBolio of Zeroes

Consider a portfolio consisting of • $25,174 par value of the 10‐year zero, • $91,898 par value of the 30‐year zero.

Maturity 10 20 30 Rate 6.00% 6.50% 6.40% Price 0.553676 0.278226 0.151084 $Dura/on 5.375493 5.389364 4.391974 Dura/on 9.70874 19.37046 29.06977 $Convexity 54.7987 107.0043 129.8015 Convexity 98.9726 384.5951 859.1356

What is the dollar convexity of the portfolio?

Convexity of a PorBolio of Zeroes n $convexity convexity = = present value n ∑K j=1 j

×

t 2 + t j /2 j (1+ rt j /2) Kj tj 2t j +2

n

∑ (1+ r j=1 /2)

2t j

=

∑ (1+ r j=1 Kj /2)

2t j

×

t 2 + t j /2 j (1+ rt j /2) 2

2t j

tj n

∑ (1+ r j=1 Kj tj /2)

= average convexity weighted by present value ≈ average maturity2 weighted by present value

€

Convexity 9

Debt Instruments and Markets

Professor Carpenter

Class Problem

Consider the portfolio of 10‐ and 30‐year zeroes. • The 10‐year zeroes have market value $25,174 x 0.553676 = $13,938. • The 30‐year zeroes have market value $91,898 x 0.151084 = $13,884. • The market value of the portfolio is $27,822. • What is the convexity of the portfolio?

Maturity 10 20 30

Rate 6.00% 6.50% 6.40%

Price 0.553676 0.278226 0.151084

$Dura/on 5.375493 5.389364 4.391974

Dura/on 9.70874 19.37046 29.06977

$Convexity 54.7987 107.0043 129.8015

Convexity 98.9726 384.5951 859.1356

Barbells and Bullets

• Consider two portfolios with the same duration: • A barbell consisting of a long‐term zero and a short‐term zero • A bullet consisting of an intermediate‐term zero • The barbell will have more convexity.

Convexity

10

Debt Instruments and Markets

Professor Carpenter

Example

• Bullet portfolio: $100,000 par of 20‐year zeroes market value = $100,000 x 0.27822 = 27,822 duration = 19.37 • Barbell portfolio: from previous example $25,174 par value of the 10‐year zero $91,898 par value of the 30‐year zero. market value = 27,822 duration = (13,938 × 9.70874) + (13,884 × 29.06977) = 19.37 13,938 + 13,884

• The convexity of the bullet is 385. • The convexity of the barbell is 478.

€

Securi:es with Fixed Cash Flows: More disperse cash ﬂows, more convexity

• In the previous example, the duration of the bullet is about 20 and the convexity of the bullet is about 202=400. • The duration of the barbell is about 0.5 x 10 + 0.5 x 30 = 20 but the convexity is about 0.5 x 102 + 0.5 x 302 = 500 > 400 = (0.5 x 10 + 0.5 x 30)2 • I.e., the average squared maturity is greater than the average maturity squared. • Indeed, recall Var(X) = E(X2)-(E(X))2 • Think of cash ﬂow maturity t as the variable and pv weights as probabilities. Duration is like E(t) and convexity is like E(t2). • So convexity ≈ duration2 + dispersion (variance) of maturity.

Convexity

11

Debt Instruments and Markets

Professor Carpenter

Value of Barbell and Bullet

Barbell : V2 (s) = 25,174 91,898 + (1+ (0.06 + Δr) /2) 20 (1+ (0.064 + Δr) /2) 60 100,000 (1+ (0.065 + Δr) /2) 40

Bullet : V1 (s) =

€

At current rates, they have the same value and the same slope (duration). But the barbell has more curvature (convexity).

Δr

Does the Barbell Always Outperform the Bullet?

• If there is an immediate parallel shift in interest rates, either up or down, then the barbell will outperform the bullet. • If the shift is not parallel, anything could happen. • If the rates on the bonds stay exactly the same, then as time passes the bullet will actually outperform the barbell: • the bullet will return 6.5% • the barbell will return about 6.2%, the market value‐ weighted average of the 6% and 6.4% on the 10‐ and 30‐year zeroes.

Convexity

12

Debt Instruments and Markets

Professor Carpenter

Value of Barbell and Bullet: One Year Later

If there is a large enough parallel yield curve shift, The barbell will be worth more. If the rates don’t change, the bullet will be worth more.

Convexity and the Shape of the Yield Curve?

• If the yield curve were ﬂat and made parallel shifts, more convex portfolios would always outperform less convex portfolios, and there would be arbitrage. • So to the extent that market movement is described by parallel shifts, bullets must have higher yield to start with, to compensate for lower convexity. • This would explain why the term structure is often hump‐ shaped, dipping down at very long maturities where convexity is greatest relative to duration—investors may give yield to buy convexity. • Some evidence suggests that the yield curve is more curved when volatility is higher and convexity is worth more.

Convexity

13

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...10. BONOS 3 10.1 CONCEPTOS BÁSICOS DE INTRUMENTOS DE DEUDA 3 10.1.1 Definiciones y clasificaciones generales 3 10.1.2 Indicadores Básicos 4 10.1.2.1 Valor residual 4 10.1.2.2 Monto en circulación (millones de $ a Valor nominal) 4 10.1.2.3 Renta anual (coupon yield, %) 4 10.1.2.4 Tasa Interna de RetornoTIR (yield to maturity –YTM- o discounted cash-flow yield -DCFY) 4 10.1.2.5 Intereses corridos ($) 5 10.1.2.6 Precio clean (limpio) o dirty (sucio) 6 10.1.2.7 Valor técnico ($) 6 10.1.2.8 Paridad (%) 6 10.2 TIPOS DE INSTRUMENTOS DE RENTA FIJA 7 10.2.1 Bonos cupón cero (zero coupon bonds): 7 10.2.2 Bonos Amortizables: 8 10.2.3 Bonos con período de gracia 8 10.2.4 Bonos a tasa fija o a tasa variable: 8 10.2.5 Bonos que incluyen contingencias 9 10.3 VALUACIÓN DE UN BONO 11 10.3.1 Flujo de Fondos esperados 11 10.4 LA CURVA DE RENDIMIENTOS Y LA ESTRUCTURA TEMPORAL DE LA TASA DE INTERES (ETTI) 13 10.4.1 Análisis de la curva de los bonos del tesoro americano de contado 14 10.4.2 Tasas de interés implícitas o forwards: 16 10.4.3 ¿Cómo se explica las diferentes formas que puede tomar ala ETTI? 17 10.4.4 La estructura temporal para bonos con riesgo de crédito (soberanos o corporativos) 19 10.5 VALUACIÓN DE UN BONO A TASA VARIABLE 23 10.5.1 Primer Método: utilizar la tasa de interes actual a todos los cupones de renta 23 10.5.2 Segundo método: proyectar una unica tasa de swap para todo el flujo del bono aproximado por el promedio de vida del bono. 23 10.5.3 Tercer...

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... 1 FM: Objec+ves A?er successfully comple+ng this topic, you will be able to: § Apply basic pricing models to evaluate stocks and bonds § Describe the theoreIcal determinants of the level and term structure of interest rates § Explain the concept of “yield” and its rela+on to “interest rate” § Determine the price of coupon and discount bonds § Compute the dura+on and convexity of a bond § Diﬀeren+ate between Macaulay and modiﬁed dura+on § Understand the rela+onship between dura+on and convexity and bond price vola+lity FM2014 5-‐6. Debt Markets: Structure, Par+cipants, Instruments, Interest Rates and Valua+on of Bonds 2 FM: Bond. J. Bond. § Fixed income § Debt instrument § Main instrument...

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...Financial Mathematics for Actuaries Chapter 8 Bond Management Learning Objectives 1. Macaulay duration and modiﬁed duration 2. Duration and interest-rate sensitivity 3. Convexity 4. Some rules for duration calculation 5. Asset-liability matching and immunization strategies 6. Target-date immunization and duration matching 7. Redington immunization and full immunization 8. Cases of nonﬂat term structure 2 8.1 Macaulay Duration and Modiﬁed Duration • Suppose an investor purchases a n-year semiannual coupon bond for P0 at time 0 and holds it until maturity. • As the amounts of the payments she receives are diﬀerent at diﬀerent times, one way to summarize the horizon is to consider the weighted average of the time of the cash ﬂows. • We use the present values of the cash ﬂows (not their nominal values) to compute the weights. • Consider an investment that generates cash ﬂows of amount Ct at time t = 1, · · · , n, measured in payment periods. Suppose the rate of interest is i per payment period and the initial investment is P . 3 • We denote the present value of Ct by PV(Ct ), which is given by Ct . PV(Ct ) = t (1 + i) and we have P = n X (8.1) PV(Ct ). (8.2) t=1 • Using PV(Ct ) as the factor of proportion, we deﬁne the weighted average of the time of the cash ﬂows, denoted by D, as D = = n X t=1 n X t " PV(Ct ) P twt , # (8.3) t=1 where PV(Ct ) wt = . P 4 (8.4) P •...

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