Price Volatility

Bond Price Volatility

“Well, Beethoven, what is this?”

— Attributed to Prince Anton Esterh´azy

Price Volatility

• Volatility measures how bond prices respond to interest rate changes.

• It is key to the risk management of interest rate-sensitive securities.

Price Volatility (concluded)

• What is the sensitivity of the percentage price change to changes in interest rates?

• Define price volatility by

−

∂P∂y

P . (14)

Price Volatility of Bonds

• The price volatility of a level-coupon bond is

−(C/y) n − 

C/y² 

(1 + y)ⁿ⁺¹ − (1 + y)

− nF (C/y) ((1 + y)ⁿ⁺¹ − (1 + y)) + F (1 + y) . – F is the par value.

– C is the coupon payment per period.

– Formula can be simplified a bit with C = F c/m.

• For the above bond,

−

∂P∂y

P > 0.

Macaulay Duration

• The Macaulay duration (MD) is a weighted average of the times to an asset’s cash flows.

• The weights are the cash flows’ PVs divided by the asset’s price.

• Formally,

MD =^Δ 1 P

n i=1

C_i

(1 + y)ⁱ i.

• The Macaulay duration, in periods, is equal to MD = −(1 + y) ∂P

∂y 1

P . (15)

aMacaulay (1938).

MD of Bonds

• The MD of a level-coupon bond is MD = 1

P

 _n



i=1

iC

(1 + y)ⁱ + nF (1 + y)ⁿ



. (16)

• It can be simplified to

MD = c(1 + y) [ (1 + y)ⁿ − 1 ] + ny(y − c) cy [ (1 + y)ⁿ − 1 ] + y² , where c is the period coupon rate.

• The MD of a zero-coupon bond equals n, its term to maturity.

• The MD of a level-coupon bond is less than n.

Remarks

• Formulas (15) on p. 96 and (16) on p. 97 hold only if the coupon C, the par value F , and the maturity n are all independent of the yield y.

– That is, if the cash flow is independent of yields.

• To see this point, suppose the market yield declines.

• The MD will be lengthened.

• But for securities whose maturity actually decreases as a result, the price volatility^a may decrease.

aAs originally defined in formula (14) on p. 94.

How Not To Think about MD

• The MD has its origin in measuring the length of time a bond investment is outstanding.

• But it should be seen mainly as measuring price volatility.

• Duration of a security can be longer than its maturity or negative!

• Neither makes sense under the maturity interpretation.

• Many, if not most, duration-related terminology can only be comprehended as measuring volatility.

Conversion

• For the MD to be year-based, modify formula (16) on p. 97 to

1 P

 _n



i=1

i k

 C

1 + ^y_k_i + n k

 F

1 + ^y_k_n

 ,

where y is the annual yield and k is the compounding frequency per annum.

• Formula (15) on p. 96 also becomes MD = −

1 + y k

 ∂P

∂y 1 P .

• By definition, MD (in years) = MD (in periods)

k .

Modified Duration

• Modified duration is defined as modified duration =^Δ −∂P

∂y 1

P = MD

(1 + y). (17) – Modified duration equals MD under continuous

compounding.

• By the Taylor expansion,

percent price change ≈ −modified duration × yield change.

Example

• Consider a bond whose modified duration is 11.54 with a yield of 10%.

• If the yield increases instantaneously from 10% to

10.1%, the approximate percentage price change will be

−11.54 × 0.001 = −0.01154 = −1.154%.

Modified Duration of a Portfolio

• By calculus, the modified duration of a portfolio equals



i

ω_iD_i.

– D_i is the modified duration of the ith asset.

– ωi is the market value of that asset expressed as a percentage of the market value of the portfolio.

Effective Duration

• Yield changes may alter the cash flow or the cash flow may be too complex for simple formulas.

• We need a general numerical formula for volatility.

• The effective duration is defined as P₋ − P₊ P₀(y₊ − y₋).

– P₋ is the price if the yield is decreased by Δy.

– P+ is the price if the yield is increased by Δy.

– P₀ is the initial price, y is the initial yield.

– Δy is small.

y P

P

P

y

y

Effective Duration (concluded)

• One can compute the effective duration of just about any financial instrument.

• An alternative is to use

P₀ − P₊ P₀ Δy .

– More economical but theoretically less accurate.

The Practices

• Duration is usually expressed in percentage terms — call it D_% — for quick mental calculation.^a

• The percentage price change expressed in percentage terms is then approximated by

−D_% × Δr

when the yield increases instantaneously by Δr%.

– Price will drop by 20% if D_% = 10 and Δr = 2 because 10 × 2 = 20.

• D_% in fact equals modified duration (prove it!).

aNeftci (2008), “Market professionals do not like to use decimal points.”

Hedging

• Hedging offsets the price fluctuations of the position to be hedged by the hedging instrument in the opposite direction, leaving the total wealth unchanged.

• Define dollar duration as

modified duration × price = −∂P

∂y .

• The approximate dollar price change is

price change ≈ −dollar duration × yield change.

• One can hedge a bond portfolio with a dollar duration D by bonds with a dollar duration −D.

Convexity

• Convexity is defined as

convexity (in periods) =^Δ ∂²P

∂y² 1 P .

• The convexity of a level-coupon bond is positive (prove it!).

• For a bond with positive convexity, the price rises more for a rate decline than it falls for a rate increase of equal magnitude (see plot next page).

• So between two bonds with the same price and duration, the one with a higher convexity is more valuable.^a

aDo you spot a problem here (Christensen & Sørensen, 1994)?

0.02 0.04 0.06 0.08Yield 50

100 150 200 250

Price

Convexity (concluded)

• Suppose there are k periods per annum.

• Convexity measured in periods and convexity measured in years are related by

convexity (in years) = convexity (in periods)

k² .

Use of Convexity

• The approximation ΔP /P ≈ − duration × yield change works for small yield changes.

• For larger yield changes, use ΔP

P ≈ ∂P

∂y 1

P Δy + 1 2

∂²P

∂y² 1

P (Δy)²

= −duration × Δy + 1

2 × convexity × (Δy)².

• Recall the figure on p. 110.

The Practices

• Convexity is usually expressed in percentage terms — call it C_% — for quick mental calculation.

• The percentage price change expressed in percentage terms is approximated by

−D_% × Δr + C_% × (Δr)²/2

when the yield increases instantaneously by Δr%.

– Price will drop by 17% if D_% = 10, C_% = 1.5, and Δr = 2 because

−10 × 2 + 1

2 × 1.5 × 2² = −17.

• C_% equals convexity divided by 100 (prove it!).

Effective Convexity

• The effective convexity is defined as P₊ + P₋ − 2P₀

P₀ (0.5 × (y+ − y−))² ,

– P₋ is the price if the yield is decreased by Δy.

– P+ is the price if the yield is increased by Δy.

– P₀ is the initial price, y is the initial yield.

– Δy is small.

• Effective convexity is most relevant when a bond’s cash flow is interest rate sensitive.

• How to choose the right Δy is a delicate matter.

Approximate d

f (x)

/dx

at x = 1, Where f (x) = x

• The difference of [ (1 + Δx)² + (1 − Δx)² − 2 ]/(Δx)² and 2:

• This numerical issue is common in financial engineering but does not admit general solutions yet (see pp. 869ff).

Interest Rates and Bond Prices: Which Determines Which?

• If you have one, you have the other.

• So they are just two names given to the same thing: cost of fund.

• Traders most likely work with prices.

• Banks most likely work with interest rates.

aContributed by Mr. Wang, Cheng (R01741064) on March 5, 2014.

Term Structure of Interest Rates

Why is it that the interest of money is lower, when money is plentiful?

— Samuel Johnson (1709–1784) If you have money, don’t lend it at interest.

Rather, give [it] to someone from whom you won’t get it back.

— Thomas Gospel 95

Term Structure of Interest Rates

• Concerned with how interest rates change with maturity.

• The set of yields to maturity for bonds form the term structure.

– The bonds must be of equal quality.

– They differ solely in their terms to maturity.

• The term structure is fundamental to the valuation of fixed-income securities.

Term Structure of Interest Rates (concluded)

• The term “term structure” often refers exclusively to the yields of zero-coupon bonds.

• A yield curve plots the yields to maturity of coupon bonds against maturity.

• A par yield curve is constructed from bonds trading near par.

Yield Curve of U.S. Treasuries as of July 24, 2015

Four Typical Shapes

• A normal yield curve is upward sloping.

• An inverted yield curve is downward sloping.

• A flat yield curve is flat.

• A humped yield curve is upward sloping at first but then turns downward sloping.

Spot Rates

• The i-period spot rate S(i) is the yield to maturity of an i-period zero-coupon bond.

• The PV of one dollar i periods from now is by definition [ 1 + S(i) ]⁻ⁱ.

– It is the price of an i-period zero-coupon bond.^a

• The one-period spot rate is called the short rate.

• Spot rate curve:^b Plot of spot rates against maturity:

S(1), S(2), . . . , S(n).

aRecall Eq. (9) on p. 69.

bThat is, term structure.

Problems with the PV Formula

• In the bond price formula (4) on p. 41,

n i=1

C

(1 + y)ⁱ + F

(1 + y)ⁿ,

every cash flow is discounted at the same yield y.

• Consider two riskless bonds with different yields to maturity because of their different cash flows:

PV₁ =

n1



i=1

C

(1 + y₁)ⁱ + F

(1 + y₁)ⁿ¹ , PV₂ =

n2



i=1

C

(1 + y₂)ⁱ + F

(1 + y₂)ⁿ² .

Problems with the PV Formula (concluded)

• The yield-to-maturity methodology discounts their contemporaneous cash flows with different rates.

• But shouldn’t they be discounted at the same rate?

Spot Rate Discount Methodology

• A cash flow C1, C₂, . . . , C_n is equivalent to a package of zero-coupon bonds with the ith bond paying C_i dollars at time i.

6 6 6 6 -

1 2 3 n

C1 C2 C3 · · ·

Cn

Spot Rate Discount Methodology (concluded)

• So a level-coupon bond has the price P =

n i=1

C

[ 1 + S(i) ]ⁱ + F

[ 1 + S(n) ]ⁿ . (18)

• This pricing method incorporates information from the term structure.

• It discounts each cash flow at the matching spot rate.

Discount Factors

• In general, any riskless security having a cash flow C₁, C₂, . . . , C_n should have a market price of

P =

n i=1

C_id(i).

– Above, d(i) = [ 1 + S(i) ]^{Δ −i}, i = 1, 2, . . . , n, are called the discount factors.

– d(i) is the PV of one dollar i periods from now.

– The above formula will be justified on p. 222.

• The discount factors are often interpolated to form a continuous function called the discount function.

Extracting Spot Rates from Yield Curve

• Start with the short rate S(1).

– Note that short-term Treasuries are zero-coupon bonds.

• Compute S(2) from the two-period coupon bond price P by solving

P = C

1 + S(1) + C + 100 [ 1 + S(2) ]².

Extracting Spot Rates from Yield Curve (concluded)

• Inductively, we are given the market price P of the n-period coupon bond and

S(1), S(2), . . . , S(n − 1).

• Then S(n) can be computed from Eq. (18) on p. 127, repeated below,

P =

n i=1

C

[ 1 + S(i) ]ⁱ + F

[ 1 + S(n) ]ⁿ .

• The running time can be made to be O(n) (see text).

• The procedure is called bootstrapping.

Some Problems

• Treasuries of the same maturity might be selling at different yields (the multiple cash flow problem).

• Some maturities might be missing from the data points (the incompleteness problem).

• Treasuries might not be of the same quality.

• Interpolation and fitting techniques are needed in practice to create a smooth spot rate curve.^a

aOften without economic justifications.

Which One (from P. 121)?

Yield Spread

• Consider a risky bond with the cash flow C₁, C₂, . . . , C_n and selling for P .

• Calculate the IRR of the risky bond.

• Calculate the IRR of a riskless bond with comparable maturity.

• Yield spread is their difference.

Static Spread

• Were the risky bond riskless, it would fetch P^∗ =

n t=1

C_t

[ 1 + S(t) ]^t.

• But as risk must be compensated, in reality P < P^∗.

• The static spread is the amount s by which the spot rate curve has to shift in parallel to price the risky bond:

P =

n t=1

C_t

[ 1 + s + S(t) ]^t.

• Unlike the yield spread, the static spread explicitly incorporates information from the term structure.

Of Spot Rate Curve and Yield Curve

• y_k: yield to maturity for the k-period coupon bond.

• S(k) ≥ y_k if y₁ < y₂ < · · · (yield curve is normal).

• S(k) ≤ yk if y₁ > y₂ > · · · (yield curve is inverted).

• S(k) ≥ yk if S(1) < S(2) < · · · (spot rate curve is normal).

• S(k) ≤ yk if S(1) > S(2) > · · · (spot rate curve is inverted).

• If the yield curve is flat, the spot rate curve coincides with the yield curve.

Shapes

• The spot rate curve often has the same shape as the yield curve.

– If the spot rate curve is inverted (normal, resp.), then the yield curve is inverted (normal, resp.).

• But this is only a trend not a mathematical truth.^a

aSee a counterexample in the text.

Forward Rates

• The yield curve contains information regarding future interest rates currently “expected” by the market.

• Invest $1 for j periods to end up with [ 1 + S(j) ]^j dollars at time j.

– The maturity strategy.

• Invest $1 in bonds for i periods and at time i invest the proceeds in bonds for another j − i periods where j > i.

• Will have [ 1 + S(i) ]ⁱ[ 1 + S(i, j) ]^j−i dollars at time j.

– S(i, j): (j − i)-period spot rate i periods from now.

– The rollover strategy.

Forward Rates (concluded)

• When S(i, j) equals

f (i, j) =^Δ

(1 + S(j))^j (1 + S(i))ⁱ

_1/(j−i)

− 1, (19)

we will end up with [ 1 + S(j) ]^j dollars again.

• As expected,

f (0, j) = S(j).

• The f(i, j) are the (implied) forward (interest) rates.

– More precisely, the (j − i)-period forward rate i periods from now.

Time Line

f(0, 1) f(1, 2) f(2, 3) f(3, 4) -

Time 0

-^S(1)

-^S(2)

-^S(3)

-^S(4)

Forward Rates and Future Spot Rates

• We did not assume any a priori relation between f(i, j) and future spot rate S(i, j).

– This is the subject of the term structure theories.

• We merely looked for the future spot rate that, if realized, will equate the two investment strategies.

• The f(i, i + 1) are the instantaneous forward rates or one-period forward rates.

Spot Rates and Forward Rates

• When the spot rate curve is normal, the forward rate dominates the spot rates,

f (i, j) > S(j) > · · · > S(i).

• When the spot rate curve is inverted, the forward rate is dominated by the spot rates,

f (i, j) < S(j) < · · · < S(i).

spot rate curve forward rate curve yield curve

(a)

spot rate curve forward rate curve yield curve

(b)

Forward Rates ≡ Spot Rates ≡ Yield Curve

• The FV of $1 at time n can be derived in two ways.

• Buy n-period zero-coupon bonds and receive [ 1 + S(n) ]ⁿ.

• Buy one-period zero-coupon bonds today and a series of such bonds at the forward rates as they mature.

• The FV is

[ 1 + S(1) ][ 1 + f (1, 2) ]· · · [ 1 + f(n − 1, n) ].

Forward Rates ≡ Spot Rates ≡ Yield Curves (concluded)

• Since they are identical,

S(n) = {[ 1 + S(1) ][ 1 + f(1, 2) ]

· · · [ 1 + f(n − 1, n) ]}^1/n − 1. (20)

• Hence, the forward rates (specifically the one-period forward rates) determine the spot rate curve.

• Other equivalencies can be derived similarly, such as f (T, T + 1) = d(T )

d(T + 1) − 1. (21)

Locking in the Forward Rate f (n, m)

• Buy one n-period zero-coupon bond for 1/(1 + S(n))ⁿ dollars.

• Sell (1 + S(m))^m/(1 + S(n))ⁿ m-period zero-coupon bonds.^a

• No net initial investment because the cash inflow equals the cash outflow: 1/(1 + S(n))ⁿ.

• At time n there will be a cash inflow of $1.

• At time m there will be a cash outflow of (1 + S(m))^m/(1 + S(n))ⁿ dollars.

aNote that (1 +S(m))^m/(1 + S(n))ⁿ = (1 +f(n, m))^m−n by formula (19) on p. 138.

Locking in the Forward Rate f (n, m) (concluded)

• This implies the interest rate between times n and m equals f (n, m) by formula (19) on p. 138.

6 -

?

n m

1

(1 + S(m))^m/(1 + S(n))ⁿ

Forward Loans

• We had generated the cash flow of a type of forward contract called the forward loan.

• Agreed upon today, it enables one to

– Borrow money at time n in the future, and

– Repay the loan at time m > n with an interest rate equal to the forward rate

f (n, m).

• Can the spot rate curve be arbitrarily drawn?^a

aContributed by Mr. Dai, Tian-Shyr (B82506025, R86526008, D88526006) in 1998.

Synthetic Bonds

• We had seen that forward loan

= n-period zero − [ 1 + f(n, m) ]^m−n × m-period zero.

• Thus

n-period zero

= forward loan + [ 1 + f (n, m) ]^m−n × m-period zero.

• We have created a synthetic zero-coupon bond with forward loans and other zero-coupon bonds.

• Useful if the n-period zero is unavailable or illiquid.

Spot and Forward Rates under Continuous Compounding

• The pricing formula:

P =

n i=1

Ce^−iS(i) + F e^−nS(n).

• The market discount function:

d(n) = e^−nS(n).

• The spot rate is an arithmetic average of forward rates,^a S(n) = f (0, 1) + f (1, 2) + · · · + f(n − 1, n)

n .

aCompare it with formula (20) on p. 144.

Spot and Forward Rates under Continuous Compounding (continued)

• The formula for the forward rate:

f (i, j) = jS(j) − iS(i)

j − i . (22)

– Compare the above formula with (19) on p. 138.

• The one-period forward rate:^a

f (j, j + 1) = − ln d(j + 1) d(j) .

aCompare it with formula (21) on p. 144.

Spot and Forward Rates under Continuous Compounding (concluded)

• Now, the (instantaneous) forward rate curve is:

f (T ) =^Δ lim

ΔT →0 f (T, T + ΔT )

= S(T ) + T ∂S

∂T . (23)

• So f(T ) > S(T ) if and only if ∂S/∂T > 0 (i.e., a normal spot rate curve).

• If S(T ) < −T (∂S/∂T ), then f(T ) < 0.^a

aConsistent with the plot on p. 142. Contributed by Mr. Huang, Hsien-Chun (R03922103) on March 11, 2015.

An Example

• Let the interest rates be continuously compounded.

• Suppose the spot rate curve is^a

S(T ) = 0.08^Δ − 0.05 e^−0.18T.

• Then by Eq. (23) on p. 151, the forward rate curve is f (T )

= S(T ) + T S^(T )

= 0.08 − 0.05 e^−0.18T + 0.009T e^−0.18T.

aHull & White (1994).

Unbiased Expectations Theory

• Forward rate equals the average future spot rate,

f (a, b) = E[ S(a, b) ]. (24)

• It does not imply that the forward rate is an accurate predictor for the future spot rate.

• It implies the maturity strategy and the rollover strategy produce the same result at the horizon “on average.”

Unbiased Expectations Theory and Spot Rate Curve

• It implies that a normal spot rate curve is due to the fact that the market expects the future spot rate to rise.

– f(j, j + 1) > S(j + 1) if and only if S(j + 1) > S(j) from formula (19) on p. 138.

– So

E[ S(j, j + 1) ] > S(j + 1) > · · · > S(1) if and only if S(j + 1) > · · · > S(1).

• Conversely, the spot rate is expected to fall if and only if the spot rate curve is inverted.

A “Bad” Expectations Theory

• The expected returns^a on all possible riskless bond strategies are equal for all holding periods.

• So

(1 + S(2))² = (1 + S(1)) E[ 1 + S(1, 2) ] (25) because of the equivalency between buying a two-period bond and rolling over one-period bonds.

• After rearrangement, 1

E[ 1 + S(1, 2) ] = 1 + S(1) (1 + S(2))².

aMore precisely, the one-plus returns.

A “Bad” Expectations Theory (continued)

• Now consider two one-period strategies.

– Strategy one buys a two-period bond for (1 + S(2))⁻² dollars and sells it after one period.

– The expected return is

E[ (1 + S(1, 2))⁻¹ ]/(1 + S(2))⁻².

– Strategy two buys a one-period bond with a return of 1 + S(1).

A “Bad” Expectations Theory (continued)

• The theory says the returns are equal:

1 + S(1)

(1 + S(2))² = E

1

1 + S(1, 2)

.

• Combine this with Eq. (25) on p. 155 to obtain E

1

1 + S(1, 2)

= 1

E[ 1 + S(1, 2) ].

A “Bad” Expectations Theory (concluded)

• But this is impossible save for a certain economy.

– Jensen’s inequality states that E[ g(X) ] > g(E[ X ]) for any nondegenerate random variable X and

strictly convex function g (i.e., g^(x) > 0).

– Use

g(x) = (1 + x)^{Δ −1} to prove our point.

Local Expectations Theory

• The expected rate of return of any bond over a single period equals the prevailing one-period spot rate:

E 

(1 + S(1, n))⁻⁽ⁿ⁻¹⁾ 

(1 + S(n))⁻ⁿ = 1 + S(1) for all n > 1.

• This theory is the basis of many interest rate models.

Duration, in Practice

• We had assumed parallel shifts in the spot rate curve.

• To handle more general shifts, define a vector [ c₁, c₂, . . . , c_n ] that characterizes the shift.

– Parallel shift: [ 1, 1, . . . , 1 ].

– Twist: [ 1, 1, . . . , 1, −1, . . . , −1 ], [ 1.8, 1.6, 1.4, 1, 0,−1, −1.4, . . . ], etc.

– . . . .

• At least one c_i should be 1 as the reference point.

Duration in Practice (concluded)

• Let

P (y) =^Δ 

i

C_i/(1 + S(i) + yc_i)ⁱ

be the price associated with the cash flow C₁, C₂, . . . .

• Define duration as

−∂P (y)/P (0)

∂y

y=0

or − P (Δy) − P (−Δy) 2P (0)Δy .

• Modified duration equals the above when [ c₁, c₂, . . . , c_n ] = [ 1, 1, . . . , 1 ],

S(1) = S(2) = · · · = S(n).

Some Loose Ends on Dates

• Holidays.

• Weekends.

• Business days (T + 2, etc.).

• Shall we treat a year as 1 year whether it has 365 or 366 days?