Combination Sum question – no solution yet

Given a set of candidate numbers (C) and a target number (T), find all unique combinations in C where the candidate numbers sums to T. The same repeated number may be chosen from C unlimited number of times.

All numbers (including target) will be positive integers. Elements in a combination (a1, a2, … , ak) must be in
non-descending order. (ie, a1 ≤ a2 ≤ … ≤ ak). The solution set must not contain duplicate combinations.

For example, given candidate set 2,3,6,7 and target 7, A solution set is:
[2, 2, 3]

book value of leverage

A simple analog is the leverage of a property bought on an (unsecured) commercial loan

Suppose the house was bought for $600k with $480k loan. After a few years, loan stays at $480 (to be paid off at maturity), but house doubles to $1.2m.

Book value of EQ is still 600-480 = $120k, but current EQ would be 1.2m – 480k = 720k.

The book value of leverage was and is still 600/120 = 5.0

The current value of leverage would be (1200k)/720k, which is lower and safer.

Now the bleak picture — suppose AS value drops from 600k to 500k. Book leverage remains 600/120 = 5.0
Current value of leverage is 500/(500 – 480) = 25.0. Dangerously high leverage. Further decline in asset valuation would wipe out equity and the entire account is under water. Some say the property is under-water but i feel really we are talking about the borrower and owner of the property — i call it the account.
(Book value of) Leverage in “literature” is defined as

(book value of) ASset / EQuity (book value)


   (LIability + EQ) / EQ …. (all book values)

The denominator is much lower as book value than the current value. For a listed company, Current value of total equity is total market cap == current share price * total shares issued so far. In contrast, Book value is the initial capital of the founder + actual dollars raised through the IPO, ignoring the increase in value of each share. Why is this book value less useful? We need to understand the term “shareholder equity”.  This term logically means the “value” of the shares held by the shareholders (say a private club of …. 500 teachers). Like the value of your house, this “value” increases over time.

OLS ^ AutoRegressive models

Given some observed data Y, you first pick some explanatory variables X_1, X_2 etc

If you pick a linear model to explain the observed Y, then OLS is the best, linear, unbiased and efficient (BLUE) solution using a computer. It will give you all the parameters of your linear model – the b_0, b_1, b_2 etc.

If you feel the relationship isn’t linear, you still can use OLS. As an alternative to a linear model, you could use AR(1) models to explain Y using the X1 X2 etc. You use AR models when you believe there’s strong serial correlation or autocorrelation.

I believe AR models use additional parameters beside the b1, b2 etc. The computation is more efficient than OLS.

asset^liability on a bank’s bal sheet

When a bank issues a financial statement, the meaning of AS (asset) and LI (liability) tend to confuse me.

Suppose JPMC bank has client IBM…

Liability – Deposits (incl. CD) at the bank
Liability – overnight borrowing. This interest rate could surge like in 2008.
Liability – Commercial papers issued by the bank
Liability – Bonds issued by the bank
Asset – securities owned by the bank (treasury department?), including stocks, govt bonds and corp bonds etc. Securities could devalue like bad loan!
Asset – Loans to corporations like IBM — on the balance sheet treated like a govt bond!
Asset – Loans/mtg to retail — on the balance sheet treated like a govt bond!
Asset – spare cash

AS = LI + share holders’ equity

If the bank issues 600M shares in an IPO, the $600mil collected is considered share holders’ equity capital or simply “capital” or simply “equity”.

Chronologically, the balance sheet starts with the initial share holders’ equity. Then Deposits come in and sitting there as spare cash. Similarly the bank can issue bonds.
Then the bank could use the spare cash to buy securities — without change on the LI side.
The bank can also use the spare cash to give loans — without change on the LI side.

Each type of transaction above affects the balance sheet only in a “realized” sense i.e. book values —

Big warning – all the AS numbers and LI numbers and equity values are book values.
* Latest share price doesn’t enter the equation. Those 600M shares will always be recorded as worth $600M on the balance sheet.
* market value (m2m) of the loans lent out doesn’t matter
* market value (m2m) of the securities owned by the bank doesn’t matter.

Fair Value accounting tries to change that. Mark-to-market is a big effort in GS and many investment banks.