Tyco Spins Off Two Units

On Friday Tyco International spun-off to its shareholders two of its five units -- Tyco Healthcare (which has been renamed Covidien) and Tyco Electronics.

About 500 million Tyco Electronics shares and 500 million Covidien shares will be distributed to Tyco International shareholders. Immediately following the distributions, Tyco International's shareholders will own 100% of the common shares of Covidien and Tyco Electronics.

Tyco International also said that its board approved a one-for-four reverse split which will become effectively immediately following the completion of the dividend distribution. As a result, Tyco International will have about 500 million shares outstanding after the reverse stock split.

Tyco Electronics and Covidien have applied for listing their shares on the New York Stock Exchange and the Bermuda Stock Exchange under the trading symbols "TEL" and "COV" respectively.

Blackstone Down Another 9.8%

Blackstone closed at $29.27 today, down another 9.8% from its $32.44 closing price a week ago. It's dropping even faster than I thought.

Beazer Homes cans chief accounting officer for ethics violations

Beazer Homes, already under a Securities and Exchange Commission informal investigation, has fired Chief Accounting Officer Michael Rand "due to violations of the company's ethics policy stemming from attempts to destroy documents." The move comes four months after the company fired General Counsel Kenneth Gary "for a pattern of personal conduct which includes violations of company policies." In a filing with the SEC, the Atlanta-based homebuilder said Rand's termination followed a briefing by independent legal counsel retained by the company's audit committee, which is conducting an internal investigation of the company's mortgage origination business and related matters. Beazer said it recently became aware of Rand's actions during that investigation.

You have to think that the news in the housing sector is even worse than we think if accounting officers are destroying documents to hide the facts.

Blackstone Down 7.5%

Shares of Blackstone Group fell in their second day of trading as doubts set in about the valuation of the private equity firm. Blackstone shares were down 7.5% to $32.44 on the NYSE.

It could be that Friday was the high point of the market for the foreseeable future. A lot of smart people are wondering about that when the smart money (i.e., Blackstone) starts selling.

Bear Stearns Bails Out Hedge Fund

Bear Stearns said it would provide up to $3.2 billion in financing for a struggling hedge fund it manages, raising concern about other funds that invested in bonds linked to subprime mortgages. The biggest bailout since Wall Street's 1998 rescue of Long-Term Capital Management signaled that the funds' main investments -- a type of bond known as a collateralized debt obligation (CDO) -- may be riskier than previously reckoned. Bear Stearns, the fifth-largest U.S. investment bank, said it would provide secured financing to its High-Grade Structured Credit Strategies Fund so the fund can sell assets in an orderly fashion. Bear also said a second fund (High-Grade Structured Credit Strategies Enhanced Leverage Fund) that took greater risk is still working out a restructuring plan with creditors. The two funds melted down after rising U.S. subprime mortgage defaults earlier this year depressed prices of CDOs, which were essentially repackaged portfolios of subprime home loans and were among the funds' main investments. Bear Stearns' High-Grade Structured Credit Strategies Fund was down about 5% so far this year through the end of April. The High-Grade Structured Credit Strategies Enhanced Leverage Fund, meanwhile, which borrowed more to magnify potential returns and potential risk, was down 23% over the same period.

Another story that provides more of the numbers and paints a much clearer -- and much bleaker -- picture than above...

Merrill Lynch seized $850 million of bonds held as collateral for loans it had made to the funds. Lehman Brothers, JPMorgan Chase and Cantor Fitzgerald also pulled out, leaving Bear Stearns to sort through the wreckage of bad bets on subprime mortgage bonds and collateralized debt obligations. Without assistance from his Wall Street peers, Bear Stearns was forced to salvage the healthier of the two funds, putting $3.2 billion of the firm's capital at risk in the biggest bailout since LTCM. Bear Stearns may dissolve the second fund after more than $600 million of investors' money dwindled to less than $200 million.

First Lucent, Then Agere, And Now Finally Avaya

Avaya announced that it has entered into a definitive merger agreement with Silver Lake, the leading investment firm focused on large-scale investments in technology-driven growth industries, and TPG Capital, a leading private investment partnership, for approximately $8.2 billion or $17.50 (cash!) per common share.

2006 Fortune 500

Top Fifteen by Revenue

1 Wal-Mart Stores $351.1B
2 Exxon Mobil $347.3B
3 General Motors $207.3B
4 Chevron $200.6B
5 ConocoPhillips $172.5B
6 General Electric $168.3B
7 Ford Motor $160.1B
8 Citigroup $146.8B
9 Bank of America $117.0B
10 AIG $113.2B
11 JP Morgan Chase $100.0B
12 Berkshire Hathaway $98.5B
13 Verizon Communications $93.2B
14 Hewlett-Packard $91.7B
15 IBM $91.4B

Top Fifteen by Profit

1 Exxon Mobil $39.5B
2 UAL $22.9B (due to bankruptcy accounting)
3 Citigroup $21.5B
4 Bank of America $21.1B
5 General Electric $20.8B
6 Pfizer $19.3B
7 Chevron $17.1B
8 ConocoPhillips $15.6B
9 JP Morgan Chase $14.4B
10 AIG $14.0B
11 Microsoft $12.6B
12 Altria Group $12.0B
13 Wal-Mart Stores $11.3B
14 Johnson & Johnson $11.1B
15 Berkshire Hathaway $11.0B

Top Fifteen by Market Cap

1 Exxon Mobil $426B
2 General Electric $368B
3 Microsoft $274B
4 Citigroup $256B
5 AT&T $243B
6 Bank of America $231B
7 Procter & Gamble $201B
8 Wal-Mart Stores $198B
9 Pfizer $182B
10 Altria Group $179B
11 AIG $177B
12 Johnson & Johnson $175B
13 JP Morgan Chase $169B
14 Berkshire Hathaway $167B
15 Chevron $159B

You can slice and dice the data anyway you want at
http://money.cnn.com/magazines/fortune/fortune500/2007/

Housing Index

http://www.speculativebubble.com/videos/real-estate-roller-coaster.php

Structured Products

Just got my hands on Structured Products by Satyajit Das. It consists of two volumes, 2600 pages total. Volume 1 covers applications of derivatives, synthetic assets, exotic options, and interest rate and currency structured products. Volume 2 covers equity linked structures, commodity linked structures, credit derivatives and new markets (e.g., inflation, insurance, weather, etc).

You'd think that such massive tomes, covering such an encyclopedic list of topics, contain a wealth of knowledge. You'd be wrong. The explanations are at such a basic level that they are essentially useless to anybody with even an elementary understanding of structured products. And to top it off, the material is incredibly dated (even though the third "revised" edition was published in 2006). For example, the author dedicates pages to employee stock option plans and how they are not reflected in the financial statements, with references to articles written in the 1990s. He appears completely unaware that IASB and FASB both now require employee stock options to be accounted for in company's financial statements.

A total waste of money.

WHD Problem 5.9

Calculate the theta, gamma, vega and rho for European call and put options.

Note: I have assumed a non-dividend stock.

Θ = ∂Π/∂t

Θ_c = - ½S₀N'(d₁)σ/√T - rKe^-rTN(d₂)

Θ_p = - ½S₀N'(d₁)σ/√T + rKe^-rTN(-d₂)

Γ = ∂₂Π/∂S₂

Γ_c = Γ_p = N'(d₁)/S₀σ√T

v = ∂Π/∂σ

v_c = v_p = S₀N'(d₁)√T

ρ = ∂Π/∂r

ρ_c = KTN(d₂)e^-rT

ρ_p = -KTN(-d₂)e^-rT

Transforming the Black-Scholes equation into the Heat equation

(∂V/∂t) + ½ σ² S² (∂²V/∂S²) + r S (∂V/∂S) - rV = 0

First substitution: u = V e^-rt

V = u e^rt

(∂u/∂t) = (∂V/∂t) e^-rt - V r e^-rt

(∂u/∂t) + V r e^-rt = (∂V/∂t) e^-rt

(∂V/∂t) = e^rt (∂u/∂t) + V r = e^rt (∂u/∂t) + r u e^rt

(∂u/∂S) = e^-rt (∂V/∂S)

(∂V/∂S) = e^rt (∂u/∂S)

(∂²V/∂S²) = e^rt (∂²u/∂S²)

And the equation changes to...

e^rt (∂u/∂t) + r u e^rt + ½ σ² S² e^rt (∂²u/∂S²) + r S e^rt (∂u/∂S) - r e^rt u = 0

(∂u/∂t) + ½ σ² S² (∂²u/∂S²) + r S (∂u/∂S) = 0

Second substitution: S = e^x

x = ln S

(∂S/∂x) = e^x
(∂x/∂S) = 1 / S

(∂u/∂x) = (∂u/∂S) (∂S/∂x) = (∂u/∂S) e^x = S (∂u/∂S)

S² (∂²u/∂S²) = S² ∂/∂S (∂u/∂S)

= S² (∂x/∂S) ∂/∂x (∂u/∂S)

= S² (1/S) ∂/∂x (1/S ∂u/∂x)

= S [ 1/S ∂²u/∂x² + ∂u/∂x ∂/∂x (1/S) ]

= ∂²u/∂x² + S (∂u/∂x) (-1/S² ∂S/∂x)

= ∂²u/∂x² - (1/S) e^x (∂u/∂x)

= ∂²u/∂x² - ∂u/∂x

And the equation changes to...

(∂u/∂t) + ½ σ² (∂²u/∂x² - ∂u/∂x) + r (∂u/∂x) = 0

(∂u/∂t) + ½ σ² ∂²u/∂x² + (r - ½σ²) ∂u/∂x = 0

Third substitution: z = x - (r - ½σ²)t to cancel the first derivative term, and t' = - t to conform to the usual sign convention.

(∂u/∂t) = (∂u/∂z)(∂z/∂t) + (∂u/∂t')(∂t'/∂t) = (∂u/∂z)[-(r - ½σ²)] + (∂u/∂t')(-1)

∂u/∂x = ∂u/∂z

∂²u/∂x² = ∂²u/∂z²
And the equation changes to (dropping the ' on the t variable) ...

- (∂u/∂t) - (r - ½σ²) (∂u/∂z) + ½σ² (∂²u/∂z²) + (r - ½σ²) (∂u/∂z) = 0

- (∂u/∂t) + ½σ² (∂²u/∂z²) = 0

And finally, voila, we have the heat equation...

∂u/∂t = ½σ² (∂²u/∂z²)

WHD Problem 3.6b

Find the most general solution of the Black-Scholes equation that has the special form V=A(t)B(S).

(∂V/∂t) + ½ σ² S² (∂²V/∂S²) + r S (∂V/∂S) - rV = 0

V = A(t) B(s)

(∂V/∂t) = B (dA/dt)

(∂V/∂S) = A (dB/dS)

(∂²V/∂S²) = A (d²B/dS²)

B (dA/dt) + ½ σ² S² A (d²B/dS²) + r S A (dB/dS) - r A B = 0

(1/A) (dA/dt) + ½ σ² S² (1/B) (d²B/dS²) + r S (1/B) (dB/dS) - r = 0

½ σ² S² (1/B) (d²B/dS²) + r S (1/B) (dB/dS) - r = - (1/A) (dA/dt)

Since the left-hand side depends only on S and the right-hand side depends only on t, the only way the equality can hold is if both sides are equal to a constant K.

- (1/A) (dA/dt) = K

A(t) = c e^-Kt

½ σ² S² (1/B) (d²B/dS²) + r S (1/B) (dB/dS) - r = K

½ σ² S² (d²B/dS²) + r S (dB/dS) - (r + K) B = 0

S² (d²B/dS²) + (2r/σ²) S (dB/dS) - (2/σ²) (r+K) B = 0

This equation proceeds as in the previous exercise.

λ² + (2r/σ² -1)λ + (-2/σ²)(r+K) = 0

λ = { (1 - 2r/σ²) ± [ (2r/σ² - 1)² + (8/σ²)(r+K) ]^½ } / 2

λ = { (1 - 2r/σ²) ± [ (2r/σ² - 1)² + (8/σ²)(r+K) ]^½ } / 2

λ = { (1 - 2r/σ²) ± [ (2r/σ² + 1)² + (8K/σ²) ]^½ } / 2

There are three cases depending on the roots: two real roots, one real root, two complex roots. And to be explicit, we should note that the case of two real roots can come about either with a real value of K (which leaves the form of A(t) above unchanged) or with a complex value of K (which would require rewriting A as we'll see in a minute).

Case 1: λ₁ and λ₂ are distinct real roots.

B(S) = c₁ S^λ₁ + c₂ S^λ₂

V(S,t) = (c₁ S^λ₁ + c₂ S^λ₂) e^-Kt

Case 1A: K = 0
(Special scenario under case 1)

This reduces to the special case where there is no time dependence V = B(S)

V = c₁ S + c₂ S ^(-2r/σ2)

Case 2: λ₁ = λ₂ = λ is the only real root.

B(S) = S^λ (c₁ + c₂ ln S)

V(S,t) = S^λ (c₁ + c₂ ln S) e^-Kt

Case 3: λ₁ = a+ib and λ₂ = a-ib are complex roots (with K real).

B(S) = S^a [ c₁ cos (b ln S) + c₂ sin (b ln S) ]

V(S,t) = S^a [ c₁ cos (b ln S) + c₂ sin (b ln S) ] e^-Kt

Case 3A: λ₁ = a+ib and λ₂ = a-ib are complex roots (with complex K = c + id).
(This is not strictly separate from case 3; it's simply a matter of expanding e^-Kt for complex K.)

B(S) = S^a [ c₁ cos (b ln S) + c₂ sin (b ln S) ]

V(S,t) = S^a [ c₁ cos (b ln S) + c₂ sin (b ln S) ] e^-ct (cos dt - i sin dt)

Note that all of these solutions (except Case 1A) have too many degrees of freedom, due to the freedom of choice in choosing K. Normally boundary conditions would restrict the universe of valid values for K, but the problem as stated did not provide any such conditions.

WHD Problem 3.6a

Find the most general solution of the Black-Scholes equation that has the special form V=V(S).

(∂V/∂t) + ½ σ² S² (∂²V/∂S²) + r S (∂V/∂S) - rV = 0

Substituting V=V(S) reduces the partial differential equation to an ordinary differential equation.

S² (d²V/dS²) + (2r/σ²) S (dV/dS) - (2r/σ²) V = 0

This is an Euler differential equation x²y'' + axy' - by = 0, the solutions of which can be found with the aid of the characteristic equation.

λ² + (a-1)λ + b = 0

λ² + (2r/σ² -1)λ + (-2r/σ²) = 0

Applying the quadratic equation, one obtains (after a bit of algebra) two roots

λ = 1 and λ = (-2r/σ²)

Using these roots, we determine that the most general solution of the Euler equation is
V(S) = c₁ S + c₂ S ^(-2r/σ2)

WHD Problem 2.5

There are n assets S_i,...,S_n satisfying the following stochastic differential equations dS_i = σ_i S_i dX_i + μ_i S_i dt, for i = 1, ..., n

The Wiener processes dX _i satisfy E[dX_i] = 0 and E[dX_i²] = dt, as usual, but the asset price changes are correlated with E[dX_idX_j] = ρ_ijdt where -1 ≤ ρ_ij ≤ 1.

Derive Ito's Lemma for a function f(S_i,...,S_n) of the n assets.

Actually, I already know most or all of the content of The Mathematics of Financial Derivatives; I am reading it as a review in preparation for more advanced materials I am hoping to get to in the near future.

This exercise is pretty trivial, mostly a matter of keeping track of all the variables.

The Taylor series expansion is
df = (∂f/∂t)dt + Σ(∂f/∂S_i)dS_i + ½Σ(∂²f/∂S_i∂S_j)dS_idS_j

dS_idS_j = (σ_i S_i dX_i + μ_i S_i dt)(σ_j Sj dXj + μj Sj dt) = σ_i σ_j S_i Sj dX_i dXj = ρ_ij σ_i σ_j S_i Sj dt
dropping all higher order terms and then substituting in the correlation assumption.

So finally we obtain
df = [ (∂f/∂t) + Σ μ_i S_i (∂f/∂S_i) + ½ Σ ρ_ij σ_i σ_j S_i Sj (∂²f/∂S_i∂S_j) ] dt + Σ σ_i S_i (∂f/∂S_i) dX_i

Here we go...

Senator Charles Schumer and other members of the Senate Banking Committee said the federal government should spend "hundreds of millions of dollars'' to bail out subprime mortgage borrowers facing foreclosure. Non-profit groups would distribute the money to help homeowners refinance loans they can't repay, Democratic Senators Schumer of New York, Robert Menendez of New Jersey and Sherrod Brown of Ohio.

So, let me get this straight those of us who did the RESPONSIBLE thing and did not buy more house than we could afford (while watching prices skyrocket out of our reach due to these irresponsible buyers) are now being asked to bend over and take it once more as our tax money goes to subsidize those who did buy more house than they could afford? Why?

And I'm tired of hearing that folks didn't understand what they were getting as a mortgage. Everyone understands adjustable interest. And everyone understands whether or not a payment is more than they can afford. And last of all everyone should understand the consequences of lying about his/her income on a mortgage application. I don't have any sympathy.

These folks took a risk. If it had worked out for them, the profit was all for them. They weren't planning to share it with me. It didn't work out, so I have to share the loss? WTF?

Job Cuts at Citigroup

Citigroup will eliminate about 17,000 jobs as part of a companywide restructuring to reduce costs and improve profit. Overall, the cuts amount to about 5% of the bank's 327,000-strong work force. Citigroup said its plans include "shrinking the size of corporate centers," several of which are in New York. It also expects to move some 9,500 jobs to lower-cost locations. Still, the elimination of the jobs won't reduce the bank's work force, but merely slow its growth, Citi executives said.

Ravings Brands Selling Moe's Southwestern Grill

Moe's Southwest Grill, the crown jewel of Atlanta-based parent Raving Brands, is being sold to Focus Brands, the Atlanta-based owner of Carvel Ice Cream, Schlotzsky's, Cinnabon and international stores of Seattle's Best Coffee. The announcement Wednesday comes on the heels of a lawsuit by franchisees, who among other things said the chain's management was secretly considering a sale. Raving Brands founder Martin Sprock denied that claim as recently as last week.

BearingPoint

The Company's continuing failure to timely file certain required periodic reports with the SEC imposes significant risks to the Company's business, including the possible loss of business, delisting from the New York Stock Exchange and defaults under the Company's credit facility. The Company has identified material weaknesses in its internal control over financial reporting, which could materially and adversely affect its business and financial condition. The Company's current cash resources might not be sufficient to meet its expected near-term cash needs, especially to fund intra-quarter operating cash requirements and non-recurring cash requirements (e.g., to settle lawsuits).

Apparently, BearingPoint didn't file its annual reports for fiscal 2004 and 2005 on time, nor will it file its 2006 report on time. It also has failed to file quarterly reports on time for the past six quarters in a row.

MBA Salaries

Came across the following in the WSJ...

The following business schools reported the highest average annual base salaries for full-time graduates, according to "The Wall Street Journal Guide to the Top Business Schools 2006".

Stanford University: $100,400

Harvard University: $99,848

Massachusetts Institute of Technology: $94,131

University of Pennsylvania: $92,986

Dartmouth College: $91,900

Northwestern University: $91,390

New Century Cuts Over Half Its Workforce

New Century Financial will immediately cut 3200 jobs (54% percent of its workforce) as part of its Chapter 11 bankruptcy reorganization. The company also said it agreed to sell its servicing assets and platform to Carrington Capital Management for $139 million, subject to bankruptcy approval. Separately, New Century said CIT Group and Greenwich Capital Financial Products have agreed to provide up to $150 million of debtor-in-possession financing to keep the company in business as it reorganizes.