Of Spot Rate Curve and Yield Curve

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) ≡

 (1 + S(j))^j (1 + S(i))ⁱ

1/(j−i)

− 1, (15)

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

• By definition, f(0, j) = S(j).

• f(i, j) is called the (implied) forward 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.

• f(i, i + 1) are called 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. (16)

• 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.

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.

• 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.

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

• This implies the rate f(n, m) between times n and m.

6 -

?

n m

1

(1 + S(m))^m/(1 + S(n))ⁿ

Forward Contracts

• We had generated the cash flow of a financial instrument called forward contract.

• 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 an arbitrary curve?^a

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

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 Eq. (16) on p. 129.

Spot and Forward Rates under Continuous Compounding (continued)

• The formula for the forward rate:

f (i, j) = jS(j) − iS(i) j − i .

• The one-period forward rate:

f (j, j + 1) = − ln d(j + 1) d(j) .

Spot and Forward Rates under Continuous Compounding (concluded)

• Now,

f (T ) ≡ lim

ΔT →0 f (T, T + ΔT )

= S(T ) + T ∂S

∂T .

• 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

aContributed by Mr. Huang, Hsien-Chun (R03922103) on March 11, 2015.

Unbiased Expectations Theory

• Forward rate equals the average future spot rate,

f (a, b) = E[ S(a, b) ]. (17)

• 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 the 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 Eq. (15) on p. 123.

– 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.

More Implications

• The theory has been rejected by most empirical studies with the possible exception of the period prior to 1915.

• 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 bonds, regardless of their different maturities, are expected to earn the same return on the average.

• That would mean investors are indifferent to risk.

A “Bad” Expectations Theory

• The expected returns 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) ] (18) 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.

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^a 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).

aMore precisely, the one-plus return.

A “Bad” Expectations Theory (concluded)

• The theory says the returns are equal:

1 + S(1)

(1 + S(2))² = E

 1

1 + S(1, 2)

 .

• Combine this with Eq. (18) on p. 139 to obtain E

 1

1 + S(1, 2)



= 1

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

• 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)⁻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

• To handle more general types of spot rate curve changes, define a vector [ c₁, c₂, . . . , c_n ] that characterizes the

perceived type of change.

– 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) ≡ 

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) P (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?

Fundamental Statistical Concepts

There are three kinds of lies:

lies, damn lies, and statistics.

— Misattributed to Benjamin Disraeli (1804–1881) If 50 million people believe a foolish thing, it’s still a foolish thing.

— George Bernard Shaw (1856–1950) One death is a tragedy, but a million deaths are a statistic.

— Josef Stalin (1879–1953)

Moments

• The variance of a random variable X is defined as Var[ X ] ≡ E 

(X − E[ X ])²  .

• The covariance between random variables X and Y is Cov[ X, Y ] ≡ E [ (X − μ_X)(Y − μ_Y ) ] ,

where μ_X and μ_Y are the means of X and Y , respectively.

• Random variables X and Y are uncorrelated if Cov[ X, Y ] = 0.

Correlation

• The standard deviation of X is the square root of the variance,

σ_X ≡ 

Var[ X ] .

• The correlation (or correlation coefficient) between X and Y is

ρ_X,Y ≡ Cov[ X, Y ] σ_Xσ_Y ,

provided both have nonzero standard deviations.^a

aPaul Wilmott (2009), “the correlations between financial quantities are notoriously unstable.”

Variance of Sum

• Variance of a weighted sum of random variables equals Var

_n



i=1

a_iX_i

=

n i=1

n j=1

a_ia_j Cov[ X_i, X_j ].

• It becomes

n i=1

a²_i Var[ X_i ] when X_i are uncorrelated.

Conditional Expectation

• “X | I” denotes X conditional on the information set I.

• The information set can be another random variable’s value or the past values of X, say.

• The conditional expectation E[ X | I ] is the expected value of X conditional on I; it is a random variable.

• The law of iterated conditional expectations:

E[ X ] = E[ E[ X | I ] ].

• If I2 contains at least as much information as I₁, then E[ X | I1 ] = E[ E[ X | I2 ]| I1 ]. (19)

The Normal Distribution

• A random variable X has the normal distribution with mean μ and variance σ² if its probability density

function is

1 σ√

2π e^{−(x−μ)2/(2σ2)}.

• This is expressed by X ∼ N(μ, σ²).

• The standard normal distribution has zero mean, unit variance, and the following distribution function

Prob[ X ≤ z ] = N(z) ≡ 1

√2π

 _z

−∞

e^−x2/2 dx.

Moment Generating Function

• The moment generating function of random variable X is

θ_X(t) ≡ E[ e^tX ].

• The moment generating function of X ∼ N(μ, σ²) is θ_X(t) = exp



μt + σ²t² 2



. (20)

The Multivariate Normal Distribution

• If X_i ∼ N(μ_i, σ_i²) are independent, then



i

X_i ∼ N



i

μ_i,

i

σ_i²

.

• Let Xi ∼ N(μi, σ_i²), which may not be independent.

• Suppose

n i=1

t_iX_i ∼ N

⎛

⎝ⁿ

i=1

t_i μ_i,

n i=1

n j=1

t_it_j Cov[ X_i, X_j ]

⎞

⎠

for every linear combination _n

i=1 t_iX_i.^a

• X_i are said to have a multivariate normal distribution.

Generation of Univariate Normal Distributions

• Let X be uniformly distributed over (0, 1 ] so that Prob[ X ≤ x ] = x, 0 < x ≤ 1.

• Repeatedly draw two samples x₁ and x₂ from X until ω ≡ (2x1 − 1)² + (2x₂ − 1)² < 1.

• Then c(2x₁ − 1) and c(2x₂ − 1) are independent standard normal variables where^a

c ≡ 

−2(ln ω)/ω .

aAs they are normally distributed, to prove independence, it suffices to prove that they are uncorrelated, which is easy. Thanks to a lively class discussion on March 5, 2014.

A Dirty Trick and a Right Attitude

• Let ξ_i are independent and uniformly distributed over (0, 1).

• A simple method to generate the standard normal variable is to calculate^a

 ₁₂



i=1

ξ_i

− 6.

• But why use 12?

• Recall the mean and variance of ξi are 1/2 and 1/12, respectively.

aJ¨ackel (2002), “this is not a highly accurate approximation and

A Dirty Trick and a Right Attitude (concluded)

• So the general formula is (_n

i=1ξ_i) − (n/2)

n/12 .

• Choosing n = 12 yields a formula without the need of division and square-root operations.^a

• Always blame your random number generator last.^b

• Instead, check your programs first.

aContributed by Mr. Chen, Shih-Hang (R02723031) on March 5, 2014.

b“The fault, dear Brutus, lies not in the stars but in ourselves that we are underlings.” William Shakespeare (1564–1616), Julius Caesar.

Generation of Bivariate Normal Distributions

• Pairs of normally distributed variables with correlation ρ can be generated.

• Let X1 and X₂ be independent standard normal variables.

• Set

U ≡ aX₁,

V ≡ ρU + 

1 − ρ² aX₂.

Generation of Bivariate Normal Distributions (concluded)

• U and V are the desired random variables with Var[ U ] = Var[ V ] = a²,

Cov[ U, V ] = ρa².

• Note that the mapping between (X₁, X₂) and (U, V ) is one-to-one.

The Lognormal Distribution

• A random variable Y is said to have a lognormal distribution if ln Y has a normal distribution.

• Let X ∼ N(μ, σ²) and Y ≡ e^X.

• The mean and variance of Y are

μ_Y = e^μ+σ2/2 and σ_Y² = e^2μ+σ2



e^σ2 − 1 ,

(21) respectively.

– They follow from E[ Y ⁿ ] = e^nμ+n2σ2/2.

Option Basics

The shift toward options as the center of gravity of finance [ . . . ]

— Merton H. Miller (1923–2000)

Calls and Puts

• A call gives its holder the right to buy a number of the underlying asset by paying a strike price.

- 6

? ?

option primium

stock

strike price

Calls and Puts (continued)

• A put gives its holder the right to sell a number of the underlying asset for the strike price.

- 6

? ?

option primium

strike price

stock

Calls and Puts (concluded)

• An embedded option has to be traded along with the underlying asset.

• How to price options?

– It can be traced to Aristotle’s (384 B.C.–322 B.C.) Politics, if not earlier.

Exercise

• When a call is exercised, the holder pays the strike price in exchange for the stock.

• When a put is exercised, the holder receives from the writer the strike price in exchange for the stock.

• An option can be exercised prior to the expiration date:

early exercise.

American and European

• American options can be exercised at any time up to the expiration date.

• European options can only be exercised at expiration.

• An American option is worth at least as much as an otherwise identical European option.