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# Price Volatility

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(1)

## Bond Price Volatility

(2)

“Well, Beethoven, what is this?”

— Attributed to Prince Anton Esterh´azy

(3)

### Price Volatility

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

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

(4)

### 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)

(5)

### 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.

(6)

### 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

Ci

(1 + y)i i.

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

∂y 1

P . (15)

aMacaulay (1938).

(7)

### 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.

(8)

### 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 volatilitya may decrease.

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

(9)

### 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.

(10)

### Conversion

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

1 P

 n



i=1

i k

 C

1 + yki + n k

 F

1 + yk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 .

(11)

### 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.

(12)

### 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%.

(13)

### Modiﬁed Duration of a Portfolio

• By calculus, the modiﬁed duration of a portfolio equals



i

ωiDi.

– Di 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.

(14)

### 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.

(15)

0

+

-

+

-

(16)

### 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.

(17)

### 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.”

(18)

### 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.

(19)

### 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)?

(20)

0.02 0.04 0.06 0.08Yield 50

100 150 200 250

Price

(21)

### 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 .

(22)

### 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.

(23)

### 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!).

(24)

### 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.

(25)

2

2

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).

(26)

### 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.

(27)

## Term Structure of Interest Rates

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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

(29)

### 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.

(30)

### 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.

(31)

(32)

### 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.

(33)

### 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.

(34)

### 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 .

(35)

### 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?

(36)

### Spot Rate Discount Methodology

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

6 6 6 6 -

1 2 3 n

C1 C2 C3 · · ·

Cn

(37)

### 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.

(38)

### Discount Factors

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

P =

n i=1

Cid(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.

(39)

### Extracting Spot Rates from Yield Curve

– 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.

(40)

### 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.

(41)

### 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.

(42)

### Which One (from P. 121)?

(43)

• Consider a risky bond with the cash ﬂow C1, C2, . . . , Cn 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.

(44)

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

n t=1

Ct

[ 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

Ct

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

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

(45)

### Of Spot Rate Curve and Yield Curve

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

• S(k) ≥ yk 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.

(46)

### 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.

(47)

### 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.

(48)

### 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.

(49)

### Time Line

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

Time 0

-S(1)

-S(2)

-S(3)

-S(4)

(50)

### 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.

(51)

### 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).

(52)

spot rate curve forward rate curve yield curve

(a)

spot rate curve forward rate curve yield curve

(b)

(53)

### 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) ].

(54)

### 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)

(55)

### 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.

(56)

### 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

(57)

### 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.

(58)

### 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.

(59)

### 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.

(60)

### 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.

(61)

### 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.

(62)

### An Example

• Let the interest rates be continuously compounded.

• Suppose the spot rate curve isa

S(T ) = 0.08Δ − 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).

(63)

### 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.”

(64)

### 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.

(65)

• The expected returnsa 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.

(66)

### 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).

(67)

### 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) ].

(68)

### 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.

(69)

### 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.

(70)

### Duration, in Practice

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

• To handle more general shifts, deﬁne a vector [ c1, c2, . . . , cn ] 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 ci should be 1 as the reference point.

(71)

### Duration in Practice (concluded)

• Let

P (y) =Δ 

i

Ci/(1 + S(i) + yci)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, . . . , cn ] = [ 1, 1, . . . , 1 ],

S(1) = S(2) = · · · = S(n).

(72)

### 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?

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• The XYZ.com bonds are equivalent to a default-free zero-coupon bond with \$X par value plus n written European puts on Merck at a strike price of \$30.. – By the

• P u is the price of the i-period zero-coupon bond one period from now if the short rate makes an up move. • P d is the price of the i-period zero-coupon bond one period from now

• The XYZ.com bonds are equivalent to a default-free zero-coupon bond with \$X par value plus n written European puts on Merck at a strike price of \$30.. – By the

• P u is the price of the i-period zero-coupon bond one period from now if the short rate makes an up move. • P d is the price of the i-period zero-coupon bond one period from now

• P u is the price of the i-period zero-coupon bond one period from now if the short rate makes an up move. • P d is the price of the i-period zero-coupon bond one period from now

• Goal is to construct a no-arbitrage interest rate tree consistent with the yields and/or yield volatilities of zero-coupon bonds of all maturities.. – This procedure is

• The XYZ.com bonds are equivalent to a default-free zero-coupon bond with \$X par value plus n written European puts on Merck at a strike price of \$30. – By the

• The binomial interest rate tree can be used to calculate the yield volatility of zero-coupon bonds.. • Consider an n-period

• The XYZ.com bonds are equivalent to a default-free zero-coupon bond with \$X par value plus n written European puts on Merck at a strike price of \$30. – By the

• The XYZ.com bonds are equivalent to a default-free zero-coupon bond with \$X par value plus n written European puts on Merck at a strike price of \$30.. – By the

• Now suppose the settlement date for a bond selling at par (i.e., the quoted price is equal to the par value) falls between two coupon payment dates. • Then its yield to maturity

– at a premium (above its par value) when its coupon rate c is above the market interest rate r;. – at par (at its par value) when its coupon rate is equal to the market

– at a premium (above its par value) when its coupon rate c is above the market interest rate r;. – at par (at its par value) when its coupon rate is equal to the market

•In a stable structure the total strength of the bonds reaching an anion from all surrounding cations should be equal to the charge of the anion.. Pauling’ s rule-

• Since the term structure has been upward sloping about 80% of the time, the theory would imply that investors have expected interest rates to rise 80% of the time.. • Riskless

However, the SRAS curve is upward sloping, which indicates that an increase in the overall price level tends to raise the quantity of goods and services supplied and a decrease in

However, the SRAS curve is upward sloping, which indicates that an increase in the overall price level tends to raise the quantity of goods and services supplied and a decrease in

The current yield does not consider the time value of money since it does not consider the present value of the coupon payments the investor will receive in the future.. A more