*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/y*^{2}

*(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

^{a}

*• 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 ] + y*^{2} *,*
*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*_{i}*D*_{i}*.*

**– D**_{i}*is the modiﬁed duration of the ith asset.*

**– ω***i* 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*_{+}
*P*_{0}*(y*_{+} *− y** _{−}*)

*.*

**– P**_{−}*is the price if the yield is decreased by Δy.*

* – P*+

*is the price if the yield is increased by Δy.*

**– P**_{0} *is the initial price, y is the initial yield.*

**– Δy is small.**

*y* *P*

_{0}

*P*

_{+}

*P*

_{-}

*y*

_{+}*y*

_{-}### Eﬀective Duration (concluded)

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

*• An alternative is to use*

*P*_{0} *− P*_{+}
*P*_{0} *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} *∂*^{2}*P*

*∂y*^{2}
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)

*k*^{2} *.*

### 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

*∂*^{2}*P*

*∂y*^{2}
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 × 2*^{2} = *−17.*

*• C*_{%} equals convexity divided by 100 (prove it!).

### Eﬀective Convexity

*• The eﬀective convexity is deﬁned as*
*P*_{+} *+ P*_{−}*− 2P*_{0}

*P*_{0} *(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.*

**– P**_{0} *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*

^{2}

*f (x)*

^{2}

*/dx*

^{2}

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

^{2}

*• 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?

^{a}

*• 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:

PV_{1} =

*n*1

*i=1*

*C*

*(1 + y*_{1})* ^{i}* +

*F*

*(1 + y*_{1})^{n}^{1} *,*
PV_{2} =

*n*2

*i=1*

*C*

*(1 + y*_{2})* ^{i}* +

*F*

*(1 + y*_{2})^{n}^{2} *.*

### 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 C*1*, C*_{2}*, . . . , C** _{n}* is equivalent to a package of

*zero-coupon bonds with the ith bond paying C*

*dollars*

_{i}*at time i.*

6 6 6 6 -

1 2 3 *n*

*C*1 *C*2 *C*3 *· · ·*

*C**n*

### 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*
*C*_{1}*, C*_{2}*, . . . , C** _{n}* should have a market price of

*P =*

*n*
*i=1*

*C*_{i}*d(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*
*C*_{1}*, C*_{2}*, . . . , 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 y*_{1} *< y*_{2} *<* *· · · (yield curve is normal).*

*• S(k) ≤ y**k* *if y*_{1} *> y*_{2} *>* *· · · (yield curve is inverted).*

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

*• S(k) ≤ y**k* *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))*

*by formula (19) on p. 138.*

^{m−n}*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*
*[ c*_{1}*, c*_{2}*, . . . , 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 C*_{1}*, C*_{2}*, . . . .*

*• Deﬁne duration as*

*−∂P (y)/P (0)*

*∂y*

*y=0*

or *−* *P (Δy)* *− P (−Δy)*
*2P (0)Δy* *.*

*• Modiﬁed duration equals the above when*
*[ c*_{1}*, c*_{2}*, . . . , 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?