Lecture 1 – Financial Intermediation
Functions and products of financial intermediates
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Financial intermediation: The process of transforming financial assets from one form into
another.
The six functions of a financial system as described by Merton (1993):
o A financial system provides a payments system for the exchange of goods and services
o A financial system provides a mechanism for the pooling of funds to undertake largescale indivisible enterprise.
o A financial system provides a way to transfer economic resources through time and
across geographic regions and industries
o A financial system provides a way to manage uncertainty and control risk
o A financial system provides price information which helps coordinate decentralized
decision-making in various sectors of the economy.
o A financial system provides a way to deal with the asymmetric information problems
when one party to a financial transaction has information that the other party does not.
The most efficient institutional structure for fulfilling the functions of the financial system
generally changes over time and differs across geopolitical subdivision. In contrast, the basic
functions of the financial system are essentially the same in all economies.
Financial intermediates improve the economy in three different ways:
o Meeting investor or issuer demands to “complete the markets” with new securities or
products that offer expanded opportunities for risk sharing, risk-pooling, hedging, and
inter-temporal or spatial transfers of resources
o Lowering transactions costs or increasing liquidity; and
o Reducing agency costs that arise from either information asymmetries between trading
parties or incomplete monitoring of their agents’ performance.
The process of ‘securitization’ is essentially the removal of (non-traded) assets from a financial
intermediary’s balance sheet by packaging them in a convenient form and selling the packaged
securities in a financial market.
Summary – Chapter 2 (M10)
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The basic function of financial markets is to channel funds from savers who have an excess of
funds to spenders who have a shortage of funds. Financial markets can do this either through
direct finance, in which borrowers borrow funds directly from lenders by selling them securities,
or through indirect finance, which involves a financial intermediary that stands between the
lender-savers and the borrower-spenders and helps transfer funds from one to the other. This
channeling of funds improves the economic welfare of everyone in society. Because they allow
funds to move from people who have no productive investment opportunities to those who
have such opportunities, financial markets contribute to economic efficiency. In addition,
channeling of funds directly benefits consumers by allowing them to make purchases when they
need them most.
Financial markets can be classified as debt and equity markets, primary and secondary markets,
exchanges and over-the-counter markets, and money and capital markets.
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The principal money market instruments (debt instruments with maturities of less than one
year) are U.S. Treasury bills, negotiable bank certificates of deposit, commercial paper,
repurchase agreements, and federal funds. The principal capital market instruments (debt and
equity investments with maturities greater than one year) are stock, mortgages, corporate
bonds, U.S. government securities, U.S. government agency securities, state and local
government bonds, and consumer and bank commercial loans.
An important trends in recent years is the growing internationalization of financial markets.
Eurobonds, which are denominated in a currency other than that of the country in which they
are sold, are now the dominant security in the international bond market and have surpassed
U.S. corporate bonds as a source of new funds. Eurodollars, which are U.S. dollars deposited in
foreign banks, are an important source of funds for American banks.
Financial intermediaries are financial institutions that acquire funds by issuing liabilities and, in
turn, use those funds to acquire assets by purchasing securities or making loans. Financial
intermediaries play an important role in the financial system because they reduce transaction
costs, allow risk sharing, and solve problems created by adverse selection and moral hazard.
As a result, financial intermediaries allow small savers and borrowers to benefit from the
existence of financial markets, thereby increasing the efficiency of the economy. However, the
economics of scope that help make financial intermediaries successful can lead to conflicts of
interest that make the financial system less efficient.
The principal financial intermediaries fall into three categories: (a) banks – commercial banks,
savings and loan associations, mutual savings banks, and credit unions; (b) contractual savings
institutions – life insurance companies, fire and casualty insurance companies, and pension
funds; and (c) investment intermediaries – finance companies, mutual funds, and money market
mutual funds.
The government regulates financial markets and financial intermediaries for two main reasons:
to increase the information available to investors and to ensure the soundness of the financial
system. Regulations include requiring disclosure of information to the public, restriction on who
can set up a financial intermediary, restrictions on what assets financial intermediaries can hold,
the provision of deposit insurance, limits on competition, and restrictions on interest rates.
Financial structure
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Stocks are not the most important source of external financing for businesses
Issuing marketable debt and equity securities is not the primary way in which businesses finance
their operations.
Indirect finance, which involves the activities of financial intermediaries, is many times more
important than direct finance, in which businesses raise funds directly from lenders in financial
markets.
Banks are the most important source of external funds used to finance businesses
The financial system is among the most heavily regulated sectors of the economy.
Only large, well-established corporations have easy access to securities markets to finance their
activities.
Collateral is a prevalent feature of debt contracts for both households and businesses.
Debt contracts typically are extremely complicated legal documents that place substantial
restrictions on the behavior of the borrower.
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Transaction costs freeze many small savers and borrowers out of direct involvement with
financial markets. Financial intermediaries can take advantage of economies of scale and are
better able to develop expertise to lower transaction costs, thus enabling their savers and
borrowers to benefit from the existence of financial markets.
Asymmetric information: Adverse selection and moral hazard (summary chapter 8 M10)
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Asymmetric information results in two problems: adverse selection, which occurs before the
transaction, and moral hazard, which occurs after the transaction. Adverse selection refers to
the fact that bad credit risks are the ones most likely to seek loans, and moral hazard refers to
the risk of the borrower’s engaging in activities that are undesirable from the lender’s point of
view.
Adverse selection interferers with the efficient functioning of financial markets. Tools to help
reduce the adverse selection problem include private production and sale of information,
government regulation to increase information, financial intermediation, and collateral and net
worth. The free-rider problem occurs when people who do not pay for information take
advantage of information that other people have paid for. The problem explains why financial
intermediaries, particularly banks, play a more important role in financing the activities of
businesses than securities markets do.
Moral hazard in equity contracts is known as the principal-agent problem, because managers
(the agents) have less incentive to maximize profits than stockholders (the principals). The
principal-agent problem explains why debt contracts are so much more prevalent in financial
markets than equity contracts. Tools to help reduce the principal-agent problem include
monitoring, government regulation to increase information, and financial intermediation.
Tools to reduce the moral hazard problem in debt contracts include net worth, monitoring and
enforcement of restrictive covenants, and financial intermediaries.
Financial crises are major disruption of financial markets. They are caused by increases in
adverse selection and moral hazard problems that prevent financial markets from channeling
funds to people with productive investment opportunities, leading to a sharp contraction in the
economic activity. The five types of factors that lead to financial crises are increases in interest
rates, increases in uncertainty, asset market effects on balance sheets, problems in the banking
sector, and government fiscal imbalances.
Structured models
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Structural models for corporate bonds are models that take as starting point a model of the
firm’s asset value or the firm’s earnings Equity and bonds are priced as derivative securities
whose cash flows depend on either the firms’ asset values of the firms’ earnings.
In the standard Merton model, we assume that the firm is financed by equity and a single zero
coupon debt issue with principal D and maturity T. The pay-off at maturity to equity ST and to
debt BT at maturity are given as:
o ST = max(VT – D, 0)
o BT = min(VT, D) = D – min(D – VT, 0)
Equity is a call option on the value of the firm
Debt corresponds to riskless debt minus a put option
Firm value follows a Geometric Brownian motion:
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o Vt = V0 exp((µ − 0.5σ2)t + σWt)
Taking the logarithm of the underlying asset value we note the below which means that the
logarithm of the price increment is normally distributed with mean (µ − 0.5σ2)t and variance σ2t
The price of a put option is:
o PBS( Vt , t) = CBS( Vt , t) − Vt + D exp(− r( T − t))
The price of a corporate bond in this model can be expressed as at time t either using the put or
the call as:
o Bt = D exp(−r(T − t)) − PBS(Vt , t)
o Bt = Vt − CBS(Vt , t)
The thing to note in the above formula is that the difference between a bond with default risk
and a riskless bond is given by the value of a put option. The difference is large when the value
of the put option is large. Where the most notable quantities that increase the put value are
leverage and volatility.
Since prices may be difficult to compare across maturities we look at spreads. The yield at t on a
credit risky zero coupon bond with expiration T is
o
, i.e. the quantity satisfying
Doing some tedious computations of the probability that the total value of the firm will be lower
than the debt we can compute the probability of default using the Black-Scholes model (that is,
dependent on the normal distribution).
Note that the expected return on assets (the drift µ) affects the default probability but not the
credit spread. This highlights the importance of distinguishing between credit spreads on
corporate bonds and the expected excess return. Credit spreads increase with asset volatility,
but expected returns only increase with the systematic part of σ.
Regulation of banks involves looking at the default probabilities of loans. Adjustments can
therefore be made to the models to see how banks can take on more systematic risk without
changing regulatory capital and how the risk in a loan portfolio depends on correlation.
We can keep volatility constant in the model but let a larger fraction of the volatility be due to
systematic risk. This can be done by increasing beta and simultaneously decreasing the
contribution from the non-systematic part. Because the drift increase when beta increases, this
decreases the actual probability of default. Since volatility is kept constant, the credit spread is
unchanged. Stated differently, we can have the same credit spreads and very different actual
default probabilities (and vice versa).
Leverage effect: In the Merton model (ignoring the effect of shortening maturity of the option) a
loss in equity value is due to a fall in the firm’s asset value. With debt fixed, this implies higher
leverage, hence higher equity beta and therefore a higher expected return on equity.
Several debt classes can easily be handled in the option-based framework:
o
o
Lecture 2 - Banks
Banks
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The balance sheet includes:
o Liabilities side
 Deposits (demand deposits, time deposits)
 Other interest-bearing liabilities (bond issues, interbank loans)
 Non-interest bearing liabilities (negative fair-value derivatives positions,
deferred taxes)
 Equity
o Asset side
 Reserves (vault cash, deposits in central bank)
 Cash items in process of collection (uncleared cheques)
 Deposits at other banks (interbank lending)
 Securities (bonds, derivatives)
 Loans (households, firms)
 Other assets (buildings, foreclosed assets)
 Intangible assets (goodwill, brands and trademarks)
Income statement
o For retail banks the key source of income is net interest income and income from
commission and fees. Key expenses are personnel and loans loss provisions.
 Interest income on loans: Interest and commissions received on loans, advances
and leasing
 Other interest income: Interest income from the trading book, short-term funds
and investment securities
 Interest expense on customer deposits: Fees, commission and transaction costs
 Other interest expense: Interest paid on debt securities and other borrowed
funds excluding insurance related interest expense if separately identified.
Risk of banks:
o Loss of asset value (leading to insolvency)
 Loss on loans (credit risk)
 Loss from trading (derivatives, counterparty default)
 Non-profitable business lowering PV of business
 Mitigation: Diversification, asset selection, capital, hedging, monitoring,
covenants
o Liquidity risk (the risk of a “run” on deposits)
 Depositors withdraw loans suddenly and in large numbers
 Liquid assets are insufficient
 Illiquid assets may have to be sold at “fire sale” prices
 Mitigation: Reserves, liquid assets, stable funding
o Interest rate risk due to asset-liability mismatch
 Assets and liabilities have different maturity (the essence of “maturity
transformation” is to lend long term and borrow short term)
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Fixed-rate loans with different maturity have different sensitivity to movements
in interest rates (duration)
 The mix of variable-rate and fixed-rate instruments may differ on the two side
of the balance sheet.
 Mitigation: Hedging (swaps), duration matching
o Currency mismatch in assets and liabilities
 European banks had used dollar funding to fund non-dollar assets before crisis
where the dollar funding dried out.
Why not just fund the bank with a lot of capital to prevent it from going bust when the value of
the assets decreases?
o Banking without leverage is not that profitable. I.e. the ROA is small and therefore
leverage is needed to allow equity to deliver high expected returns.
The balance sheet of commercial banks can be thought of as a list of the sources and uses of
bank funds. A bank’s liabilities are its source of funds, which include checkable deposits, time
deposits, discount loans from the Fed, borrowings from other banks and corporations, and bank
capital. A bank’s assets are its uses of funds, which include reserves, cash items in process of
collection, deposits at other banks, securities, loans, and other assets (most physical capital).
Banks make profits through the process of asset transformation: They borrow short (accept
short-term deposits) and lend long (make long-term loans). When a bank takes in additional
deposits, it gains an equal amount of reserves; when it pays out deposits, it loses an equal
amount of reserves.
Although more-liquid assets tend to earn lower returns, banks still desire to hold them.
Specifically, banks hold excess and secondary reserves because they provide insurance against
the costs of a deposit outflow. Banks manage their assets to maximize profits by seeking the
highest returns possible on loans and securities while at the same time trying to lower risk and
making adequate provisions for liquidity. Although liability management was once a staid affair,
large (money center) banks now actively seek out sources of funds by issuing liabilities such as
negotiable CDs or by actively borrowing from other banks and corporations. Banks manage the
amount of capital they hold to prevent bank failure and to meet bank capital requirements set
by the regulatory authorities. However, they do not want to hold too much capital because by
doing so doing they will lower the returns to equity holders.
The concepts of adverse selection and moral hazard explain many credit risk management
principles and involving loan activities: screening and monitoring, establishment of long-term
customer relationships and loan commitments, collateral and compensating balances, and credit
rationing.
With the increased volatility of interest rates that occurred in the 1980s, financial institutions
became more concerned about their exposure to interest-rate risk. Gap and duration analyses
tell a financial institution if it has more rate-sensitive liabilities than assets (in which case a rise
in interest rates will reduce profits and a fall in interest rates will raise profits). Financial
institutions manage their interest-rate risk by modifying their balance sheets but can also use
strategies involving financial derivatives.
Off-Balance sheet activities
o Another way to boost returns on assets (and therefore equity) is through “off-balance
sheet” activities.
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Guarantees: Includes any guarantees that the bank makes against the default of
third party debt.
 Committed credit lines: Includes any undrawn credit facilities made available by
the bank
 Other contingent liabilities: Includes any reported off balance sheet liabilities
not input into the above lines, i.e. security obligations, reverse repurchases,
security lending agreements.
 Derivatives? Maybe?
Off-balance sheet activities consist of trading financial instruments and generating
income from fees and loan sales, all of which affect bank profits but are not visible on
bank balance sheets. Because these off-balance-sheet activities expose banks to
increased risk, bank management must pay particular attention to risk assessment
procedures and internal controls to restrict employees from taking on too much risk.
Lecture 3 & 4 – Banking regulation & A model of a bank run
Reasons for regulation
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Deposit insurance gives moral hazard problem
Too-big-to-fail problem
Importance of banks for the payments system and the economy (lending)
Level playing field (internationally)
The experience
Major financial legislations in the US
Summary chapter 12 – M10
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The concepts of asymmetric information, adverse selection, and moral hazard help explain the
ten types of financial regulation that we see in the United States and other countries: the
government safety net, restrictions on financial institutions’ asset holdings, capital
requirements, prompt corrective action, financial institution supervision, assessment of risk
management, disclosure requirements, consumer protection, restrictions on competition, and
macroprudential policies.
Financial innovation and deregulation increased adverse selection and moral hazard problems in
the 1980s and resulted in huge losses for U.S. S&Ls, banks, and taxpayers. Similar crises occurred
in other countries.
The parallels between the banking crisis episodes that have occurred in other countries
throughout the world are striking, indicating that similar forces are at work.
The Dodd-Frank Act of 2010 is the most comprehensive financial reform legislation since the
Great Depression. It has provisions in five areas: (1) consumer protection, (2) resolution
authority, (3) systemic risk regulation, (4) Volcker rule, and (5) derivatives. Future regulation
needs to address five areas: (1) capital requirements, (2) compensation, (3) GSEs, (4) credit
rating agencies, and (5) the dangers of overregulation.
Basel committee
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Started in 1974 after breakdown of the Bretton Woods system of managed exchange rates had
broken down
Financial turmoil – many banks suffered large losses
G10 countries established what is now the Basel Committee on Banking Supervision
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Initially, purpose was merely to have a forum for cooperation on banking supervisory issues but
has expended its role since then
Sets minimum standards for regulation and supervision of banks
Decisions have no legal force, but the committee strongly encourages full implementation of its
standards by members. In EU countries, it is the European Commission that proposes the
directives that implement the Basel Committee’s standards.
Basel I: 1988. Calls for minimum capital ratio of capital to Risk Weighted Assets of 8%. In 1997
expanded to also include charges for market risk arising from foreign exchange, traded
securities, and derivatives.
Basel II: 1999. Introduces three pillars (more fine-grained risk weights and possibility of using
internal models):
o Minimum capital requirements
o Supervisory review of capital adequacy and assessment process
o Enhancing market discipline through effective use of disclosure
Basel III: 2010: Proposal and still under implementation. Key features:
o Higher capital requirements (and more layers)
o Liquidity regulation
Three key elements of regulation
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Capital
o Risk-weighted assets
o Leverage ratio
Liquidity Coverage Ratio (LCR)
o The objective of the LCR is to promote the short-term resilience of the liquidity risk
profile of banks.
o Ensures that banks have an adequate stock of unencumbered (i.e. not pledged as
collateral) high-quality liquid assets (HQLA) that can be converted easily and
immediately in private markets into cash to meet their liquidity needs for a 30 calendar
day liquidity stress scenario
o
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𝑆𝑡𝑜𝑐𝑘 𝑜𝑓 ℎ𝑖𝑔ℎ 𝑞𝑢𝑎𝑙𝑖𝑡𝑦 𝑙𝑖𝑞𝑢𝑖𝑑 𝑎𝑠𝑠𝑒𝑡𝑠
𝑁𝑒𝑡 𝑐𝑎𝑠ℎ 𝑜𝑢𝑡𝑓𝑙𝑜𝑤𝑠 𝑜𝑣𝑒𝑟 𝑎 30 𝑑𝑎𝑦 𝑡𝑖𝑚𝑒 𝑝𝑒𝑟𝑖𝑜𝑑
≥ 100
Net Stable Funding Ratio (NSFR)
o The objective of NSFR is to promote resilience over a longer time horizon by creating
additional incentives for banks to fund their activities with more stable sources of
funding on an ongoing basis.
o The NSFR supplements the LCR and has a time horizon of one year
o It has been developed to provide sustainable maturity structure of assets and liabilities.
The NSFR limits overreliance on short-term wholesale funding, encourages better
assessment of funding risk across all on- and off-balance sheet items, and promotes
funding stability (and thus mitigate the risk of future funding stress)
o The NSFR is defined as the amount of available stable funding (ASF) relative to the
amount of required stable funding (RSF) which is required to be larger than 100% on an
ongoing basis
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ASF is the amount of capital and liabilities that are expected to be reliable over
the time horizon considered by the NSFR, which extends to one year. It is based
on the relative stability of an institution’s funding sources including the
contractual maturity of its liabilities and differences in the propensity of
different types of funding provides to withdraw their assets.
ASF is calculated as a weighted sum of an institutions capital and liability, where
the weights/factors reflects the reliability of the capital/liabilities
RSF is calculated as a weighted sum of an institution’s assets. The
weights/factors are intended to approximate the amount of a particular asset
that would have to be funded, either because it will be rolled over, or because it
could not be monetized through sale or used as collateral in a secured
borrowing transaction over the course of one year without significant expense.
The Diamond and Dybvig model
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The Diamond and Dybvig model is a classic in banking theory as it explains the liquidity creation
of banks and also explains bank runs.
The goal is not to be realistic in terms of institutional setup but to capture a central insight in the
simplest possible model.
In the model there are three dates, T = 0, T = 1, and T = 2 and each investor has to consume at
either of those dates. At date 0 the investor does not know at which date he will need to
consume. Type 1 investors need to consume at date 1 and vice versa. Investors start off with
one unit at date 0. The probability of being type 1 is denoted t.
The illiquid assets will be worth r1 at date 1 and r2 at date 2. The smaller r1/r2 is (assuming a
constant discount rate) the less liquid is the asset.
Assume the investor invests in an asset that offers the choice of delivering r1 at date 1 or r2 at
date 2. The expected utility from this asset is then:
o tU(r1) + (1-t)U(r2)
o Where utility is assumed to be U(c) = 1 – 1/c, where the important feature is that U is
concave.
The Diamond and Dybvig model – example
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t = 0.25
There is an illiquid asset in the economy (e.g. investment in a project) that yields r1 = 1 but r2 = 2
The expected utility of holding the illiquid asset for the investor is:
o 0.25U(1) + 0.75U(2) = 0.375
An asset offers an investor the choice between r1 = 1.28 and r2 = 1.813, the expected utility is
then:
o 0.25U(1.28) + 0.75U(1.813) = 0.391 > 0.375
The point is that a bank taking deposits from, say, 100 investors can create such an asset. The
bank can invest 100 in the ‘project’ initially using the deposits of the investors. When 25 which
to consume at date 1, the bank gives them each 1.28 which requires liquidation of 25 * 1.28 = 32
assets. The remaining 68 invested in the ‘project’ yield 136 at date 2 which is enough to give
136/75 = 1.813 to each investor
The goal is to maximize tU(r1) + (1-t)U(r2) subject to the constrains:
o
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Where 1 – tr1 is the value of assets left after Type 1 investors have received their
pay-off
The remaining amount is worth (1-tr1)R in period 2
This amount is to be shared among the remaining fraction of investors, 1 – t
The Diamond and Dybvig model – example cont. (Runs)
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It is important that the bank knows that a fourth of the investors will need liquidity at date 1
If there is uncertainty as to how many investors need liquidity at date 1, the types who need to
consume at date 2 may start worrying that there is not enough assets in the bank.
If more than 78 liquidate receiving 1.28 each, all projects have to be sold, and therefore is
nothing left for the second type! A run may occur!
In other words, the model has two equilibriums: One in which the agents do ‘as they are
supposed to’ (i.e. type 1 investors get their money at time 1, and type 2 wait until time 2).
Another equilibrium is one in which all agents withdraw money at time 1 (Type 2 store with low
return until time 2) and the bank goes under. To prevent his the bank would have to implement
deposit insurance and suspension of convertibility.
Lecture 5 – VaR, RAROC, Capital Allocation
Value-at-Risk (VaR)
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Value-at-risk (VaR) is heavily used in capital regulation for market risk, i.e. fluctuations in the
value of the bank’s assets due to fluctuations in interest rates, exchange rates, and commodity
prices. Regulatory capital depends strongly on VaR.
VaR is really just a quantile.
o Imagine that the distribution of profit/loss x in money terms (say USD) is through a
density function f(x). Negative x is a loss whereas positive x is a profit.
∞
o The 99% VaR is the solution to 𝑐 = ∫−𝑉𝑎𝑅 𝑓(𝑥)𝑑𝑥 where c = 1-0.99 = 0.01
o We write a minus sign in front of VaR because the number is reported as a positive
number.
Empirical VaR
o VaR can be computed using a parametric distribution. In this case you use the time of
observed profits/losses to estimate the parameters in a family of distributions.
o For example:
 Estimate mean and variance in a sequence of profits/losses
 Plug in the estimates in the quantile function for the normal to get the VaR at
the required confidence level.
o It is critical that the distributional assumption is reasonable. I.e. for the same mean and
standard deviation, the extreme quantiles (far left tail) are very different for (say) a
normal and a t-distribution with a low number of degrees of freedom. Thus if the ”true”
profits and losses are t-distributed but you use an assumption of normality, you
underestimate VaR.
o VaR could also be based on an empirical distribution function. The procedure is then (for
daily VaR):
 Observe N days of profits and losses
 Decide your confidence level (say 95%)
 Order the observations and cut out the lowest 5%.
 The lowest remaining observation is the 95% VaR
o The larger the N, the more accurate is your estimate – if we assume it is reasonable to
think of the data as coming from the same distribution.
o In practice, VaR must be computed both using a long-enough period (1 year) – but there
is also a multiplicative factor to allow for stress scenarios.
Note that VaR does not say anything about what the worst possible loss is. In the VaR
computation based on the empirical distribution function, you can move the most extreme
losses further to the right without changing the estimated VaR.
VaR is often transformed from one day to T-days by multiplying with the square root of T. This
works with IID data, but does not hold in general.
Risk-Adjusted Return On Capital (RAROC)
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A bank needs to prioritize its activities.
Many different groups in the bank are competing for funds to trade, extend credit, etc.
A bank often wishes to reward the units that perform well (performance-based pay)
How can this be done in a fashion consistent with maximizing shareholder value?
Risk-adjusted Return on Capital is an attempt to addressing these issues.
Consider an example, in which
o A trader in one business unit (say fixed income) has invested 200 USDm, makes a 10
USDm profit and has a 99%-VaR of 19 USDm.
o A trader in another unit has invested 100 USDm, also made 10 USDm in profits, but has
had a 99%-VaR of 28 USDm.
o If regulatory capital (or the competition for internal funds) is proportional to the VaR,
then even if the fixed income trader has invested more, he has used a small amount of
capital.
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𝑅𝐴𝑅𝑂𝐶 =
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𝑃𝑟𝑜𝑓𝑖𝑡−(𝐶𝑎𝑝𝑖𝑡𝑎𝑙×𝑘)
𝐶𝑎𝑝𝑖𝑡𝑎𝑙
(se slides—may differ)
o Where k is the required rate of return on capital
Whether a project has RAROC positive, depends on the capital it consumes and the “hurdle
rate”. In practice, many banks use Capital = Regulatory Capital since this is their binding
constraint.
Froot and Stein mention three potential pitfalls in the implementation of the RAROC:
o Inadequate separation of priced and non-priced risks. One of the key implications of
our two-factor model is that a bank should evaluate investments according to both their
correlation with any priced market factors and their correlation with the bank’s existing
portfolio. Taken literally, the one-factor RAROC approach does not allow for these two
degrees of freedom.
o Capital allocations based on measures of variance rather than covariance. The amount
of capital allocated to an investment is typically related to some measure of that
investment’s risk. But in some applications of RAROC, the risk measure used in the
investment’s total volatility. IT makes more sense for the capital allocation to be driven
by the investment’s covariance with the rest of the bank’s portfolio.
o Using the incorrect cost of capital. Even if the RAROC capital allocation is based on the
appropriate covariance measure, one still has to come up with the right values of
required return.
Lecture 6 – Risk capital, the risk in loan portfolios – modeling correlation, simulating nonhomogeneous portfolios
Modeling distributions of loan portfolios



Loans to firms are a major part of a bank’s assets.
Have a pay-off which make traditional stock portfolio analysis unsuitable. Maximum pay-off is
the face value (+ coupons). Except for distressed loans, the default probability is small.
Distribution has concentration around expected pay-off and a “left tail”.
Remember Merton model
1 2
𝜎 )
2
o
Underlying asset value (and writing ∝ = 𝜇 −
o
o
Taking logarithm, we note that log 𝑉𝑡 − log 𝑉0 = ∝ 𝑡 + 𝜎𝑊𝑡
Which means that the logarithm of the price increment is normally distributed with a
mean αt and a variance σ2t. is this correct??
For a fixed time horizon T, we may therefore write log 𝑉𝑇 = log 𝑉0 + ∝ 𝑇 + 𝜎√𝑇𝜀
where ε is a standard normal random variable with a mean 0 and a variance of 1.
We can rewrite the condition log 𝑉𝑇 = log 𝑉0 + ∝ 𝑇 + 𝜎√𝑇𝜀 < log 𝐷 as:
o
o

𝜀<
is 𝑉𝑡 = 𝑉0 exp(∝ 𝑡 + 𝜎𝑊𝑡 )
log 𝐷−log 𝑉0 −𝛼𝑇
𝜎 √𝑇
o

Hence we can think of default as happening when a normally distributed random
variable (log asset value of borrow) is under a barrier D at the maturity date of the loan.
Default probability
o From this we recover the result from lecture 1 that the default probability
corresponding to this assumption as

𝑃(𝑉𝑇 < 𝐷) = Φ (−
𝑉
log( 0 )+ 𝛼𝑇
𝐷
𝜎 √𝑇
)
o

In applications of this framework to loans, we will assume that if the firm survives, it
pays the face value of the loan – if it defaults it pays a fraction (the recovery value). The
recovery value is often taken as a constant (which of course has to be estimated based
on the type of loans under consideration).
When generalizing to portfolios, the idea is simply to let the normal disturbances corresponding
to each firm (denoted εi) be correlated. The normally distributed random variable εi is affected
both by ‘systematic risk’ (which hits all issuers) and by borrower-specific risk. To be precise, the
default of a firm i occurs if 𝜀 𝑖 = 𝜌𝑀 + √1 − 𝜌2 𝑧 𝑖 < 𝐾𝑖
o Where M is a N(0,1) distributed ‘market factor’ and zi is a N(0,1) distributed
‘idiosyncratic risk’. Note that zi’s are independent of each other and of M.
log 𝐷 𝑖 −log 𝑉0𝑖 −𝛼𝑖 𝑇
𝜎 𝑖 √𝑇
o
𝐾𝑖 =
o
o
Also note that εi is still normally distributed with a mean 0 and variance 1.
ρ is a correlation parameter. The larger ρ, the larger the correlation between the
borrowing firms’ assets. For ρ = 0 they are independent.
In applications, we assume we know (from some empirical exercise) that i’s default
probability pi and we pick Ki so that P(εi < Ki) = pi
o

Homogeneous case
o If all firms in a portfolio have the same default probability 𝑝̅ . The barrier K must then be
chosen as Φ−1 (𝑝̅ ). For a given level of the market factor M we can now compute the
probability that firm i defaults:

o
o
o
o
Φ−1 (𝑝̅ )− 𝜌𝑚
√1−𝜌2
)
This gives the probability of default if M is known and the default events therefore are
independent. Hence, for a known M = m the probability of k defaults is binomially
distributed with this default probability, which we denote ρ(m). We can therefore
express the probability that k out of N firms default as:
𝑁−𝑘
𝑁
 𝑃(𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑒𝑓𝑎𝑢𝑙𝑡𝑠 = 𝑘) = ∫ ( ) 𝑝(𝑚)𝑘 (1 − 𝑝(𝑚))
𝜙(𝑚)𝑑𝑚
𝑘
 Where 𝜙 is the density function of a standard normal distribution
As N becomes larger (i.e., a large portfolio of loans with the same default probability)
and the same pairwise correlations), the probability that there are less than a fraction θ
of the loans that default as given by:

o
𝑃(𝜀 𝑖 < 𝐾|𝑀 = 𝑚) = 𝑃 (𝜌𝑚 + √1 − 𝜌2 𝑧 𝑖 < Φ−1 (𝑝̅ )) = Φ (
1
𝜌
𝐹(𝜃) = Φ ( (√1 − 𝜌2 Φ−1 (𝜃) − Φ−1 (𝑝̅ )))
In the following graph the distribution is called the mixture distribution. Because of the
asymptotic result mentioned earlier, it is this distribution, which describes the loss
frequency in a large portfolio, where firms are modeled as here.
There is a typo in the graph as the top right box should say ρ2

Inhomogeneity
o Note that we may change the model to handle individual default probabilities. We just
need to choose the default barriers Ki separately to match 𝑝̅ i. We can also handle
individual correlations with the market factor.
o With a simulation we get:
 For each firm, determine the level of the barrier Ki which produces the desired
default probability.
 Simulate M and the zi’s, compute the firm specific factor εi, and check if there is
a default for each firm.
 For each simulation count the number of defaults, repeat the simulation, count
again etc.
 The probability of k defaults is then the number of times we see k defaults
divided by the number of times we simulated the vector of N + 1 variables
 This is a Monte Carlo simulation!
Risk capital - Lecture 18. March 2015








Risk capital does not depend on face value of debt (when liabilities are “non-random” as in
Merton model).
Let A0 be current asset value and D be current face value of debt.
Risk capital is defined as the cost of making sure that AT-D are at least as large as [A0-De-rt]ert
Note that this means we must have: A0ert-D < AT-D, i.e. A0erT < AT
Need a put that pays max(A0ert-AT, 0) at T.
How is risk capital computed? (p. 31 middle). Why a call?
Remember the put/call parity for European options, i.e. C – P = S – Ke-rt which can also be
written as P = C – S + Ke-rt. Because the underlying is A0 and the strike is A0ert the S and K cancels
out so we are left with P = C. When you plug everything into the BS formula you should end up
with a result that implies that the strike = 1 and r = 0 or A0c(1,1,0,T,sigma). This is supposed to
show how to get a formula in the appendix of Merton & Perlod (section 6).
Risk capital is not equity. Difference between accounting statement and risk capital.
Lecture 7 – Securitization – pricing CDOs
What is securitization?




The typical setup is:
o Pooling of assets (typically debt contracts)
o Selling the assets to a ‘special purpose vehicle’ (SPV)
o Financing the SPV’s purchase by issuing notes that are claims to the cash flows from the
pool
The idea of the SPV is to create an entity that is ‘default remote’ from the original owner of the
assets. Then the buyers of the notes can concentrate on the risks of the underlying pool of
contracts and not worry about the credit quality of the original owner of the assets
We have seen banking presented as a way of permitting liquid deposits to finance illiquid
projects. Banking practice and regulation (LCR) limit the extent to which assets can be invested
in illiquid assets. Regulation also sets capital standards that limit the risk taking.
Securitization offers a way for a bank to shed the risk of loans on its balance sheet while
maintaining the benefit of a sustained customer relationship. Securitization is also a way for
borrowers to pool their collateral and borrow against the pool thereby collectively guaranteeing
each other’s loans and mitigating asymmetric information problems. Securitization permits
‘market funding’ of activities outside of the normal banking system.
Pricing the tranches


We want to understand the pricing of the tranches when the underlying assets in the pool are
simple loans.
o How does the default probability affect prices?
o The main issue: How does correlation affect prices?
o How are prices affected by systematic risk?
The mixed binomial model
o A very useful model for building intuition
o The most important models used in practice fall within the class
o We recall the relevant definitions of mean, variance, and covariance for random
variables that have only finitely many outcomes
o

Binomial distribution
 Consider N independent issuers each having a default probability of p over a
certain period. The probability of k defaults among the N issuers is given by the
binomial distribution. Let D denote the (random) number of defaults. Then:
𝑁
 𝑃(𝐷 = 𝑘) = ( ) 𝑝𝑘 (1 − 𝑝)𝑁−𝑘
𝑘
 When D is binomially distributed (N, p), we have:
o
 E(D) = Np
and
V(D) = Np(1-p) variance?
 A special case is the ‘Bernoulli distribution’ where N = 1.
 The Binomial distribution arises because the individual default events are
independent. It is a sum of independent, identically distributed Bernoulli
variables. Note the homogeneity assumption that all firms have the same
default probability.
The mixed binomial model is a way of introducing dependence (‘correlation’) and still
maintain the computational simplicity of the binomial model. Note that homogeneity is
still assumed and hence, to begin with, all issuers are alike. In the beginning our main
goal is just to understand the importance of correlation which the model captures just
fine.
The world’s simples CDO





We will consider risk in loan portfolios, but we will also analyze what happens to prices of
tranches of a CDO with a loan portfolio as collateral. For that purpose we introduce the world’s
simples CDO. The CDO has three so-called tranches, which are prioritized claims to the cash
flows of a very simple loan portfolio. The loan portfolio consists of 50 loans, each with a
principal of 1 and maturing in one year. We also assume 0 recovery in case of default.
The three tranches are a senior tranche, a junior tranche and an equity tranche. The principals
are 35, 10, and 5 respectively (i.e. sum to 50).
If no defaults in the underlying loan portfolio (the collateral pool) then all tranches get the full
principal.
o If 1 default occurs, then equity gets 4, the others (still) get the full principal.
o With 5 defaults, equity gets nothing, but junior and senior get full pay
o With 6 defaults, equity is (still) wiped out, junior gets 9, senior is paid in full
o With 16 defaults equity and junior are wiped out and senior gets only 34, etc.
Pooling and tranching was/is common
o Real contracts are of course more complicated, but the basic idea is important.
o Subprime mortgages were typically sold as RMBS (Residential Mortgage Back Securities)
to investors after pooling and tranching. Some of the tranches were further put as
collateral into new CDOs (and even CDOs on CDOs).
o Banks used CDO structures to offload loans from balance sheets – in principle getting rid
of the risk. But reputation concerns were important when the crisis struck and some
banks ended ‘rescuing’ troubled structures.
o Today we have a revival of CLOs (CDOs where bank loans or other debt instruments are
the collateral).
o Prioritization is not unique to CDOs. Think of the liability side of companies: Many types
of debt, equity in various forms, etc.
Expected tranche pay-offs
o We will illustrate the price effects by just looking at expected pay-off. We could discount
as well using a one-year rate to get the price. The expected pay-off of the trances can be
computed as follows:
 Let Ti(D) be the pay-off to tranche i (i=equity, junior, senior) as a function of the
number of defaults:




 𝐸𝑇𝑖 (𝐷) = ∑50
𝑘=0 𝑇𝑖 (𝑘)𝑃(𝐷 = 𝑘)
The index in the sum for the equity case only needs to run from 0 to 5, and for
the junior tranche from 0 to 15, as the remaining terms are zero.
Note that his expression is always true (for zero recovery) whether there is
correlation or not.
We could plug in binomial probabilities for P(D=k) if default probabilities are the
same for all issuers and defaults are independent.
In reality we need to take into account common dependence of, say, the
business cycle.
The mixed binomial distribution




The mixed binomial is obtained by making the mixture parameter stochastic. Intuitively, this
corresponds to making the ‘state of the economy’ a common random variable which determines
all default probabilities at once.
Let 𝑝̃ be the stochastic default probability which we assume has a density f. Conditional on the
value of 𝑝̃ the number of defaults follows a binomial distribution with probability parameter 𝑝̃.
The precise expression for the probability of k defaults is:
1 𝑁
o 𝑃(𝐷 = 𝑘) = ∫0 ( ) 𝑝𝑘 (1 − 𝑝)𝑁−𝑘 𝑓(𝑝)𝑑𝑝
𝑘
In the discrete case where 𝑝̃ takes on only finite many values the integral is replaced by a sum.
If, for example, 𝑝̃ can take on only two different values, p1 and p2, with probabilities f(p1) and
f(p2), then:
𝑁
𝑁
o 𝑃(𝐷 = 𝑘) = ( ) 𝑝1𝑘 (1 − 𝑝1 )𝑁−𝑘 𝑓(𝑝1 ) + ( ) 𝑝2𝑘 (1 − 𝑝2 )𝑁−𝑘 𝑓(𝑝2 )
𝑘
𝑘
We will now derive analytically how properties of the mixture distribution affects the properties
of the distribution of the number of defaults:
o Before we begin with the mixed binomial model, let us recall two useful rules for
computing the mean and variance through conditioning:
 If X, Y are stochastic variables, then:
 E(X) = E(E(X|Y))
 V(X) = E(V(X|Y)) + V(E(X|Y))
 We will use this with X = D and Y = 𝑝̃
o Let Xi denote the default indicator of issuer i, i.e. 1 if i defaults and 0 otherwise.
Let 𝑝̅ = 𝐸𝑝̃. From the conditioning rules we obtain:
 EXi = E(E(Xi|𝑝̃)) = E(𝑝̃) = 𝑝̅
o One can also show that:
 VXi = 𝑝̅ (1-𝑝̅ )
 Cov(Xi,Xj) = E(𝑝̃2) - 𝑝̅ 2 = V(𝑝̃)
when i≠j
 Note that when 𝑝̃ is a constant, we get covariance = 0.
o The correlation between default events is:
𝐸𝑝̃2 −𝑝̅ 2
𝑝̅ (1−𝑝̅ )
𝑉(𝑝̃)
𝑝̅ (1−𝑝̅ )

𝑃(𝑋𝑖 , 𝑋, ) =

This number is not easy to interpret in terms of magnitude but note that for a
given 𝑝̅ , increasing the variance of 𝑝̃ will increase the correlation
=
o




Let us now consider the more interesting case with N borrowers. Let D denote the
number of defaults (a random variable). Again we can use the conditioning rules to get
ED = N𝑝̃ and (after a bit of algebra):
 V(D) = N𝑝̅ (1-𝑝̅ ) + N(N-1)(E𝑝̃2-𝑝̃2)
 If 𝑝̃ = 𝑝̅ , there is no variation in p and we are back to the old binomial
distribution.
 The opposite extreme with the same default probability occurs when 𝑝̃ is 1 with
probability 𝑝̅ and 0 otherwise.
 The critical insight: Higher variability in 𝑝̃ gives larger correlation between
default events and larger variance in the number of defaults.
Default frequency of large portfolios:
o If we compute the variance of the loss fraction, then we find that:
𝐷𝑁
)
𝑁
𝑝̅ (1−𝑝̅ )
𝑁(𝑁−1)
(𝐸𝑝̃2
+
𝑁
𝑁2
2
2
− 𝑝̅ 2 )

𝑉(


→ 𝐸𝑝̃ − 𝑝̅
𝑓𝑜𝑟 𝑁 → ∞
Hence when N becomes large the variance of the loss frequency is equal to the
variance of 𝑝̃. For large N we have a conditional version of the Law of Large
Numbers. In fact, it is not only the variance which approaches that of 𝑝̃ but the
entire distribution does! Therefore, for large portfolios (i.e. portfolios with many
names), it is the distribution of 𝑝̃ which determines the loss fraction.
=
Choosing 𝑝̃
o We will study the effect on correlation and the distribution of the number of defaults
when we vary the mixture distribution. The beta-distribution is a flexible class of
distributions on (0,1) (with no real economic motivation). The density of this is:
 f(x) = Be(α,β)xα-1(1-x)β-1, 0 < x < 1
o Be(α,β) is a constant which ensures that the integral is 1.
The Beta-distribution
o The Beta distribution is a distribution on [0,1], hence we may indeed interpret the
outcomes as a probability. The beta distribution has two parameters which we denote
α,β. We have:
∝
 𝐸𝑝̃ = ∝ + 𝛽
∝𝛽
(∝ + 𝛽)2 (𝛼+ 𝛽+1)

𝑉𝑝̃ =

If we look at the combination of the two parameters for which 𝑝̅ =
∝
∝+𝛽
for a
given default probability 𝑝̅ , then the variance of the distribution will fall when
we increase α.
o We use this to understand the effect of correlation on the expected loss of different
tranches in the world’s simplest CDO. The moral is: Higher correlation (for a given
default probability) gives ‘fatter tails’.
Merton’s model
o The Beta distribution is not satisfactory as there is no clear interpretation. Of course,
one could say that the α, β could be chosen to fit a given 𝑝̅ and a given ρ, that is, a given
default indicator correlation. But the default indicator correlations are not nice to work

with. Further, the parameter identification comes from first and second order moment
conditions.
o There is no economic reason why the beta should be the relevant underlying
distribution. The Merton has an economic rationale and there are way of estimating
parameters.
Homogeneous case
o If all firms in a portfolio have the same default probability 𝑝̅ the barrier K must then be
chosen as Φ−1 (𝑝̅ ). For a given level of the market factor M we can now compute the
probability that firm i defaults:


𝑃(𝜀 𝑖 < 𝐾|𝑀 = 𝑚) = 𝑃 (𝜌𝑚 + √1 − 𝑝2 𝑧 𝑖 < Φ−1 (𝑝̅ )) 𝑎
= Φ(
Φ−1 (𝑝̅ )− 𝜌𝑚
√1−𝜌2
)


This gives the probability of default if M is known and the default events
therefore are independent.
o Hence for a known M = m the probability of k defaults is binomially distributed with this
default probability, which we denote p(m)
o We can therefore express the probability that k out of N firms default as
𝑁−𝑘
𝑁
 𝑃(𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑒𝑓𝑎𝑢𝑙𝑡𝑠 = 𝑘) = ∫ ( ) 𝑝(𝑚)𝑘 (1 − 𝑝(𝑚))
𝜙(𝑚)𝑑𝑚
𝑘
 Where 𝜙 is the density function of a standard normal distribution
How would we express using mixed binomial?
o Asking what the chance is that the probability of default becomes θ is like asking what
the probability is that p(M) is θ.
o We get a slightly better result if we ask what the probability is that p(M) ≤ θ where the
result is:

o
o
1
𝑃𝑟𝑜𝑏(𝑝(𝑀) ≤ 𝜃) ≡ 𝐹(𝜃) = Φ (𝑝 (√1 − 𝑝2 Φ−1 (𝜃) − Φ−1 (𝑝̅ )))
Which results in the following density function (Note that there is a typo in the graph as
it should say p2 in the box in the top right corner):

With the mixture distribution at hand, we may apply it to a finite number of firms just as
in the definition of the mixed binomial or use the asymptotic approximation and say,
that the mixture distribution approximates the default frequency of a large loan
portfolio. This is Vasicek’s large homogeneous portfolio argument.
Lecture 8 – Securitization – rating and tranching. Securitization without risk transfer. Shadow banking.
Rating and tranching





The goal of the paper is to quantify the gains that can be obtained from tranching when
investors price tranches according to their ratings.
Two rating systems: Moody’s system rates according to expected loss, i.e. both (actual) default
probability and recovery are taken into account. S&P system rates according to (actual) default
probability. Note that both systems deal with actual default probabilities.
o For example, a bond whose default probability (or expected loss) is 2% and whose
default (or loss) always occurs in ‘the best of times’ has the same rating as one with the
same characteristics except that default occurs during ‘the worst of times’. Yet their
prices should be very different according to finance theory.
Finance 101 says that systematic risk is prices but non-systematic risk is not. You are willing to
pay extra for not defaulting in states where you will need the money. If investors forget this,
then there is a possibility to extract a profit by securitizing debt. A rough indication is given in
the figure below that it was indeed the case, up to the beginning of the crisis, that investors
were pricing tranches of CDOs with a given rating at roughly the same level as a corporate bond
with the same rating.
o
The starting point of the analysis is that there exists a firm, whose bond issues are correctly
priced for all choices of principal. They refer to this firm as the reference firm. We know the Pdynamics of the firm’s assets, and hence we can compute physical default probabilities and
expected loss under this measure. By choosing the face value appropriately, we can create
bonds with designated default probabilities/expected losses corresponding to certain ratings.
o For example, the paper works with default probability of 0.219% for AA-rated securities
in the S&P system. The expected loss in the Moody’s system is 0.037% for an AA-rating.
When setting up the model, we mark all variables related to the reference firm with an asterisk.
We think of the total risk of the reference firm as stemming from both a systematic, marketwide component, as well as a firm-specific component. To model this, suppose there is an
aggregate market which can be modelled by a process (Mt)t ≥ 0 with dynamics:
o
𝑀𝑡 = 𝑀0 𝑒
1
2
((𝑟𝑚 − 𝜎 2 )𝑡+𝜎𝑚 𝑊𝑡 )

o Where (Wt) is a standard Brownian motion, and σm describes the market-wide risk.
Suppose that the asset value process of the reference firm has the dynamics:
1
((𝜇∗ − (𝜎 ∗ )2 )𝑡+𝛽 ∗ 𝜎 𝑊 +𝜎∗ 𝑍∗ )
𝑚 𝑡
𝜀 𝑡
2
𝑉𝑡∗ = 𝑉0∗ 𝑒
Under the physical measure P, where µ* = rf + β*(rm-rf), rf is the risk-free rate, β* is the
CAPM beta coefficient, and 𝜎𝜀∗ denotes the idiosyncratic risk; W is the same Brownian
motion as the one driving the market, and Z* is another standard Brownian motion,
independent of W.
Now we introduce another issuer into the model whose asset value V also follows a Geometric
Brownian Motion with parameters β, σε. To begin with, think of this issuer as another firm with
possibly different risk characteristics than the reference firm. Suppose this new firm is able to
issue tranches of subordinated debt. Each tranche will have a different rating, k. The key
assumption in the model is the following: Investment bankers are able to sell each tranche with
rating k at exactly the same yield as a bond issued by the reference firm with an identical rating
of k. Formally:
o Let 𝐵𝑘∗ and 𝑊𝑘∗ denote the face value and market value of a bond with rating k and
maturity T issued by the reference firm, respectively. Let 𝑦𝑘∗ be the yield to maturity of
∗
this reference bond such that 𝑊𝑘∗ = 𝑒 −𝑦𝑘 𝑇 𝐵𝑘∗ .
o All variables relating to the issuer of tranched debt will be written without asterisk.
o Let 𝐵𝑖,𝑘𝑖 denote the face value of tranche i with rating ki, where i = 1,…,6 if there are 6
tranches. Now, the key assumption is that the sales price of the ith tranche, 𝑆𝑖,𝑘𝑖 is:
 𝑆𝑖,𝑘𝑖 = 𝑒 −𝑦∗𝑘𝑖𝑇 𝐵𝑖,𝑘𝑖
 Note that the reference yield is calculated based on a single debt issue and
applied to equivalently rated tranches.
o The marketing gain will then be as follows. Let 𝑊𝑖,𝑘𝑖 denote the market value of tranche
i with rating ki. The marketing gain from selling tranche i is then:
 Ω𝑖 = 𝑆𝑖,𝑘𝑖 − 𝑊𝑖,𝑘𝑖 and the total marketing gain becomes Ω𝑖 = Σ𝑖 Ω𝑖
How do we compute the face- and market values?
o In the S&P system, each rating k corresponds to an exogenously given default
probability Пk. The face value of a bond with rating k issued by the reference firm is
defined implicitly be the equation:
 𝑃(𝑉𝑇∗ ≤ 𝐵𝑘∗ ) = Π𝑘
o The market value of the reference bond with rating k can now be found by Merton’s
formula:
 𝑊𝑘∗ = 𝐵𝑘∗ 𝑒 −𝑟𝑓𝑇 𝑁(𝑑2 ) + 𝑉0∗ 𝑁(−𝑑1 )
o
o




𝑤ℎ𝑒𝑟𝑒 𝑑1 =
ln(𝑉0∗ / 𝐵𝑘∗ ) +(𝑟𝑓 +0.5𝜎 ∗2 )𝑇
𝜎 ∗ √𝑇
𝑎𝑛𝑑 𝑑2 = 𝑑1 − 𝜎 ∗ √𝑇
Moody’s expected loss approach
o We define the expected default loss as:
 𝐿∗𝑘 = 𝐸 𝑝 1(𝑉𝑇∗ ≤ 𝐵𝑘∗ ) (𝐵𝑘∗ − 𝑉𝑇∗ )
o
This is again an analytical formula in our parametric model. A given rating k corresponds
to an exogenously given expected default loss rate Λk. We can then bind 𝐵𝑘∗ as the
solution to:

Λ𝑘 =
𝐿∗𝑘
𝐵𝑘∗
CDO tranches


Let us continue with the S&P system. Consider a portfolio of 125 B-rated underlying bonds.
Suppose all issuing firms have the same idiosyncratic risk, σε, and the same β. Assume that each
firm i has asset value dynamics:
o 𝑑𝑉𝑡𝑖 = 𝜇𝑉𝑡𝑖 𝑑(𝛽𝜎𝑚 𝑊 + 𝜎𝜀 𝑍 𝑖 )𝑡
o where µ = rf + β(rm – rf), and Zi is a standard Brownian motion independent of W and Zj
whenever j ≠ i.
Simulation procedure
o We choose the principal 𝐵̂𝑘 of each bond such that it corresponds to a B-rating, i.e. we
use Пk = 0.24460. The SPV issues 6 tranches. We choose 100,000 simulations runs. We
simulate the asset value at time T under both the physical and the risk neutral measure:

𝑉𝑗,𝑛 (𝑇) = 𝑉𝑗 (0)𝑒 (𝜇−0.5𝜎
2
𝑄
(𝑇) = 𝑉𝑗 (0)𝑒 (𝑟𝑓−0.5𝜎 )𝑇+𝛽𝜎𝑚 √𝑇𝑍0 +𝜎𝜀 √𝑇𝑍𝑗
𝑉𝑗,𝑛
Where j denotes the jth bond and n denotes the nth simulation run. Also, Z0 and
Zj are IIDN(0,1) variables for j = 1,…, 125
We compute the cash flow from bond j for each simulation run n as
 𝐶𝐹𝑗,𝑛 = min(𝑉𝑗,𝑛 (𝑇), 𝐵̂𝑘 )
The total portfolio cash flow under the physical measure is
 𝐶𝐹𝑆𝑃𝑉,𝑛 = ∑125
𝑗=1 𝐶𝐹𝑗,𝑛
And under the risk-neutral measure,
𝑄
𝑄
̂
 𝐶𝐹𝑆𝑃𝑉,𝑛
= ∑125
𝑗=1 min(𝑉𝑗,𝑛 (𝑇), 𝐵𝑘


o
o
o
o
o
𝑄
We note that (CFSPV,n)n=1,…,100000 and (𝐶𝐹𝑆𝑃𝑉,𝑛
)n=1,…,100000 are distributions of the portfolio
value at time T under the physical and risk-neutral measure, respectively.
The market value of the entire SPV at time t = 0 is

o
o
o
o
𝑊𝑆𝑃𝑉 = 𝑒 −𝑟𝑓𝑇
1
100000
𝑄
∑100000
𝐶𝐹𝑆𝑃𝑉,𝑛
𝑛=1
The SPV issues tranches with ratings ki for i =1,…, 6. We are looking for an aggregate
face value Bki for the SPV portfolio such that:
 P(CFSPV ≤ Bki) = Пki
 Hence we find Bki as the Пk-quantile of the distribution of the SPV value under
the physical measure, CFSPV.
Computationally, this can be done by sorting the vector (CFSPV,n)n=1,…,100000 in ascending
order and then choosing Bki as entry number Пki * 100000 in the sorted vector.
Knowing Bki for each i, the market value of the aggregate bond written on the SPV is
then:

o
2 )𝑇+𝛽𝜎
𝑚 √𝑇𝑍0 +𝜎𝜀 √𝑇𝑍𝑗
1
𝑄
𝑊𝑘𝑖 = 𝑒 −𝑟𝑓𝑇 100000 ∑100000
𝑛=1 (𝐶𝐹𝑆𝑃𝑉,𝑛 , 𝐵𝑘𝑖 )
The face and market value of the tranches are then Bi,ki = Bki – Bki-1, Wi,ki = Wki – Wki-1, i ≥
2, and B1,k1 = Bk1, W1,k1 = Wk1. The market value of the equity tranche is then:
 𝑊𝑒𝑞𝑢𝑖𝑡𝑦 = 𝑊𝑆𝑃𝑉 − ∑6𝑖=1 𝑊𝑖,𝑘𝑖
As a final step, all the numbers are normalized and reported as a percentage of WSPV =
7802.88.



∗
What should be noted is that the 𝑦𝑘𝑖
’s from Rating and Tranching section are called the ratingbased yields. For each ki the rating-based yield is lower than the equilibrium yield of the CDO
with an identical rating.
In other words, our model implies that the equilibrium yields on tranches ought to be much
higher than the yields on corporate bonds with an identical rating. Yet, Brennan et al. showcase
a sample in which the average between e.g. BBB CDO tranches and BBB corporate yields over
the period mid 2005 until mid 2007 was close to zero. It is this lack of spread which gave rise to
the arbitrage opportunities from tranching and securitization.
A final point: Brennan et al. use the model above to show quantitatively that the higher the beta
of the corporate bonds in the SPV pool, the higher is the marketing gain from tranching.
Securitization without risk transfer





We discuss paper by Acharya, Scnabl, and Suarez (2012)
Example of securitization that played an important role in the financial crisis. Indicates that
structure of securitization did not really rid the institution of the risk. The study involves ABCP
(Asset Backed Commercial Paper) conduits (a conduit is an SPV set up by a financial institution)
ABCP conduits
o The conduit purchases and holds financial assets from different asset sellers. The
conduit finances its purchases of assets by issuing commercial paper. While the assets
are commonly long term (3-5 years, say) the funding is short term (30 days) and debt is
rolled over.
Sponsor guarantees
o Sponsors provide guarantees to investors who buy the commercial paper. There are
four types of guarantees:
 Credit guarantee: Sponsor pays of maturing ABCP regardless of the conduit’s
asset values
 Liquidity guarantee: Sponsor pays off maturing ABCP only if conduits assets are
not in default
 Extendible note: Conduit issuer has the right to extend the maturity of the CP
(usually 60 days)
 SIV (Structured Investment Vehicle) guarantees: A form of liquidity guarantee,
but funding is also involved non-guaranteed liabilities
o Key points: Liquidity guarantees offer capital relief, credit guarantees do not.
Regulatory arbitrage
o Costs of bank distress are in part borne by agents different from debt and equity
holders. This may give banks an incentive to take on more risk, especially in high
aggregate risk states where the entire sector is in trouble. Capital requirements are
imposed because of these externalities associated with bank risk taking. But to the
extent that capital requirements are imperfect, banks may not try to circumvent capital
requirements
o The paper argues that ABCP conduits were used to minimize regulatory capital and that
in effect the risk of the loans that were securitized remained on the sponsoring banks’
balance sheet.





A key observation
o Liquidity guarantees implied a reduction by a factor 10 of the capital requirement of the
sponsoring bank. In effect the liquidity guarantees would shield outside investors (i.e.
those who bought the ABCP) from losses.
o If there were no buyers to help roll over maturing ABCP, the sponsor had to step in.
o If there is a slow decline in credit quality (or doubts about the credit quality), but not a
default, the sponsor buys back the assets at par value before default occurs.
o This prevents losses from reaching the outside investors.
There was a large drop in ACBP issuance around the crisis and the spread widened.
Most guarantees were for liquidity and most guarantees were issued by commercial banks
Important findings
o It is mostly commercial banks that had a regulatory motive – and they are the main
issuers.
o It is the most capital constrained who issue liquidity guarantees
o Measure constraint through leverage ratio. This is better because Tier 1 compared to
risk weighted assets is improved using conduits. Other empirical studies have confirmed
that leverage ratio is a good predictor of distress.
o Strikingly, conduits with credit guarantees (that do not buy capital relief), are not used
more by constrained banks. This strengthens the evidence that the motive was
regulatory arbitrage.
Capital ‘saved’
o The authors estimate for the 30 largest banks the capital saving that followed from not
having the assets fully on the balance sheet. For some it was a large saving – the largest
was for Sachsen Landesbank – a bank that was an early casualty of the crisis.
o The probability of the conduits does seem like ‘picking up nickels in front of a
steamroller’.
o Carry of 10 bps. With stricter capital requirements, a negative carry would probably
have prevented the construction.
Shadow banking




The example of conduits is an example of what broadly is referred to as ‘shadow banking’. There
is not a universally agreed upon definition of the term shadow banking.
Shadow banking is often associated with provision of credit outside the traditional banking
systems.
The traditional banking system is the system that benefits from deposits insurance and from
access to the central banks’ lending facilities. To the extent that conduits were outside of
banking regulation, they were part of the shadow banking system. Other examples include
money market funds, mortgage institutions (in the US), securities firms (investment banks), and
finance companies.
Adrian and Shin (2009) have interesting material on the nature and growth of the shadow
banking system. Lots of leverage is maintained in this system and this is a source of systemic
risk. Main points from the article:
o Mortgages went from assets in banks to being sold by creators of MBS
o Home mortgages evolved from being held by banks to being held by the market
o
o
o
o
o
o
o
o
o

More financial assets were held by the market than by banks in Q2 of 2007
Many securitizations involved complicated chains of intermediaries
Securities brokers/dealers grew rapidly from 1980-2009 (much higher than assets of
banks/households/etc.)
Lehman’s balance sheet was mainly composed of Collateralized lending (assets) and
collateralized borrowing (liabilities) at the end of 2007.
Household leverage decreased as assets increased in value while the corporate sector
leverage was relatively insensitive to asset growth. Broker-dealer leverage increased
with asset value.
The credit crunch hit hard in the ABCP
REPO was hit. Higher haircuts meant less borrowing possible for same amount of
collateral
o Commercial banks did not actually shrink in the crisis while Broker-Dealers decreased
significantly as well as ABS issuers.
Concluding remarks
o Up to the crisis we saw a huge increase in the shadow banking system
o Many of the intermediaries here relied on high leverage and even increasing leverage
o Dryouts in ABCP and repo financing affected the shadow banking sector
o Assets of commercial banks were more resilient
Lecture 9 – Bank capital structure and lending
n/a
Lecture 10 – Central banking and banks

A simplified balance sheet for a central bank looks like:
o
o




Assets
 Securities are mostly bonds, but the crisis has expanded the types of assets
 Loans can be unsecured loans or they can be loans against collateral (repo). In
U.S. these loans mostly consist of U.S. Treasury securities and MBS.
 Assets earn higher rates than paid on liabilities
o Liabilities
 Currency in circulation (on the liability side) pays no interest (increases steadily
over the years) and is not held by banks (vault cash). Currency held by banks is
counted as reserves:
 Reserves are commercial bank deposits at a central bank (main component of
the banks liabilities) + currency physically held by the banks (vault cash) which
earn lower rates than lending rate (even negative)
Monetary base
o We focus on changing the monetary base through open market operations
o Monetary base = currency in circulation + total reserves in the banking system
Open market operations
o Open market operations means buying and selling of securities (bonds) by the central
bank. Monetary base can also be changes through lending, i.e. offering “discount loans”
to banks.
Open market purchase of bond
o Assume The Fed buys 100 million USD worth of bonds from a bank
o It either credits the bank’s account in the Fed with 100 million USD or pays using cash
o
o Banks have less securities, more reserves – unchanged total assets (and liabilities)
o Assets have become more liquid
o Balance sheet of The Fed has increased by 100 million USD
Open market purchase from non-bank public
o
o
Imagine now that the bonds are purchased from, say, a pension fund
Distinguish between two cases:
 The central bank pays by crediting the pension fund’s account in a local bank (or
paying by check, with the fund then deposits in its bank)
 In this case both reserves and the monetary base increase by 100 million



The pension fund receives cash which it hides away in a big box at its
headquarters
In this case only the monetary base increases (because of increased currency in
circulation). Because cash is held by the fund, reserves are unchanged.



Purchase always increases monetary base. The degree to which the public holds
cash affects the increase in reserves
Open market sale
o It is easy to reverse the steps for a sale
o Just do one case where pension fund has 100 million in currency and pays for the bonds
o
o
o

…and another (more realistic) where it draws on the account in a local bank
The conclusion is again that effect on monetary base is clear, whereas the effect on
reserves are less certain
Shifts from deposit to currency
o Shifts from deposits into currency can change reserves. Imagine a case where bank
customers withdraw cash from deposits and store in their wallets. The monetary base is
unaffected but reserves are!
o



Remember: Currency in circulation refers to currency outside of the banking system
which explains the change in the Fed balance sheet.
o
Loans to financial institutions
o Monetary base is also affected by loans made to financial institutions. Imagine a cash
where the Fed makes a loan of 100 million to a financial institution which is credited to
the bank’s account at the Fed (or given to the bank in cash).
o Monetary base increases with the amount of the loan and so does the reserve
o
Monetary base split
o Open market operations are controlled by the Fed while lending to financial institutions
is less so.
o The non-borrowed monetary base is the part of the monetary base under control by the
Fed. MBn = MB – BR where BR denotes the borrowed reserves from the Fed
Multiple deposit creation
o The monetary base is not a sufficient measure of money supply. An important money
creation occurs through the banking system because deposits are lent out, deposited,
etc.
o This is captured through the monetary aggregate denoted M1 which contains money in
circulation outside of banks’ vaults and demand deposits (may contain other ultra-liquid
forms of money, but these are the main two components).
o

o
Creation of deposits
o When creating deposits it is assumed that the bank uses all its excess reserves to make
loans, i.e. only retains the required reserve fraction rr. The total amount of deposits thus
equals:

o
𝑅
𝑟𝑟
If instead the bank keeps excess reserves in a ratio equal to e, and consumers retain
currency as a fraction c of D, then it is shown in the book that the money supply (M1
really) is:

𝐷 + 𝐶 ≡ 𝑀1 =
1+𝑐
𝑀𝐵
𝑟𝑟+𝑒+𝑐


Monetory Base = Currency + Reserves
e = currency ratio = C/D

𝑀1
𝑀𝐵
𝐶+𝐷
= 𝐶+𝑅 =
𝐶
+1
𝐷
𝐶 𝑅
+
𝐷 𝐷
=
𝐶+1
𝐶+𝑟𝑟+𝑒
(𝑎𝑠 𝐷 =
𝑅
), 𝑖. 𝑒. 𝑀1
𝑟𝑟+𝑒
=
𝐶+1
𝑀𝐵
𝐶+𝑟𝑟+𝑒
 𝑖𝑓 𝑟𝑟 + 𝑒 < 1, 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒𝑟 > 1
The creation of liquid money is lower when more is retained as cash and as excess
reserves
An expanded balance sheet
o Two (three) other major items are important for understanding central bank policy:
 The government’s account (liability)
 The foreign currency reserve (asset)
o

𝑘
𝐷 = 𝑅 ∑∞
𝑘=0(1 − 𝑟𝑟) =

o
A small additional item is gold (asset) which is not significant in value but
perhaps symbolically
Foreign currency interventions have implications for domestic banking system
o

o
Paying for deficits issuing money
o Note that a troubled government could finance activities through its central bank by
borrowing (in which case its account becomes an asset to the bank) and financing it
through printing money
o This could be done for example by having the central bank to buy treasury bonds and
“print” the money
o It could also force banks to hold large fractions of government debt through regulation
(and example of “financial repression”)



o
Summary
o
Was a filed monetary policy the leading cause of the Great Depression?
o A big discussion in monetary economics to what extent the money supply affects
economic activity/output
o Failing banks lead to customers withdrawing deposits (there was no deposit insurance).
o Banks had to retain large reserves to resist runs
o In our notation, e and c were changing
o This caused a decline in M1
o
The 2007 – 2009 crisis
o We have seen that the monetary base and M1 did not move in sync during the Great
Depression.
o
o
2007-2009 is an extreme example where there was huge growth in MB because if asset
purchases and new lending facilities but much less growth in M1.
Interest on reserves played a big role
o

o
Fed funds equilibrium
o Goal of having lower Fed funds target than lending rate is to encourage banks to lend
and borrow liquidity from each other.
o Remember: Fed funds rate is a rate at which banks actually borrow and lend to each
other. The target is a target set by the Fed and the Fed can influence the market for Fed
funds through its open market operations.
o
o
o
o
o

o
Unconventional monetary tools
o The recent financial crisis has called for a number of new monetary tools
o Has been introduced to battle illiquidity in the system
o And because the short term interest rate became so low (reaching the zero lower
bound) that other tools had to be invented to bring down longer term rates, namely
o Liquidity provision
 Lower rate for borrowing from the “discount window”
 Term Auction Facility – alternative to discount window
o Asset purchases
 (Fed) Purchases of Mortgage Backed Securities
 Large monthly purchases of Treasury securities
o Commitment to future policy actions
 Commitment to keep the Fed funds target rate between 0 and 0.25% (!)
Summary chapter 14 – M10

The Federal Reserve System was created in 1913 to lessen the frequency of bank panics.
Because of public hostility to central banks and the centralization of power, the Federal Reserve
System was created with many checks and balances to diffuse power.






The structure of the Federal Reserve System consists of twelve regional Federal Reserve banks,
around 2,500 member commercial banks, the Board of Governors of the Federal Reserve
System, the Federal Open Market Committee (FOMC), and the Federal Advisory Council.
Although on paper the Federal Reserve System appears to be decentralized, in practice it has
come to function as a unified central bank controlled by the Board of Governors, especially the
board’s chairman.
The Federal Reserve is more independent than most agencies of the U.S. government, but it is
still subject to political pressure because the legislation that structures the Fed is written by
Congress and can be changed at any time.
The case for an independent Federal Reserve rests on the view that curtailing the Fed’s
independence and subjecting it to more political pressures would impart an inflationary bias to
monetary policy. An independent Fed can afford to take the long view and not respond to shortrun problems that will result in expansionary monetary policy and political business cycle. The
case against an independent Fed holds that it is undemocratic to have monetary policy (so
important to the public) controlled by an elite that is not accountable to the public. An
independent Fed also makes the coordination of monetary and fiscal policy difficult.
The theory of bureaucratic behavior suggests that one factor driving central banks’ behavior
might be an attempt to increase their power and prestige. This view explains many central bank
actions, although central banks may also act in the public interest.
The European System of Central Banks has a similar structure to the Federal Reserve System,
with each member country having a National Central Bank, and an Executive Board of the
European Central Bank being located in Frankfurt, Germany. The Governing Council, which is
made up of the six members of the Executive Board (which includes the president of the
European Central Bank) and the presidents of the National Central Banks, makes the decisions
on monetary policy. The Euro system, which was established under the terms of the Maastricht
Treaty, is even more independent than the Federal Reserve System because its charter cannot
be changed by legislation. Indeed, it is the most independent central bank in the world.
There has been a remarkable trend toward increasing independence of central banks
throughout the world. Greater independence has been granted to central banks such as the
Bank of England and the Bank of Japan in recent years, as well as to other central banks in such
diverse countries as New Zealand and Sweden. Both theory and experience suggest that more
independent central banks produce better monetary policy.
Summary – Chapter 15 M10



The three players in the money supply process are the central bank, banks (depositary
institutions), and depositors.
Four items in the Fed’s balance sheet are essential to our understanding of the money supply
process: the two liability terms, currency in circulation and reserves, which together make up
the monetary base, and the two asset items, securities and loans to financial institutions.
The Federal Reserve controls the monetary base through open market operations and
extensions of loans to financial institutions and has better control over the monetary base than
over reserves. Although float and Treasury deposits with the Fed undergo substantial short-run
fluctuations, which complicate control of the monetary base, they do not prevent the Fed from
accurately controlling it.




A single bank can make loans up to the amount of its excess reserves, thereby creating an equal
amount of deposits. The banking system can create a multiple expansion of deposits, because as
each bank makes a loan and creates deposits, the reserves find their way to another bank, which
uses them to make loans and create additional deposits. In the simple model of multiple deposit
creation in which banks do not hold on to excess reserves and the public holds no currency, the
multiple increase in checkable deposits (simple deposit multiplier) equals the reciprocal of the
required reserve ratio.
The simple model of multiple deposit creation has serious deficiencies. Decisions by depositors
to increase their holdings of currency or of banks to hold excess reserves will result in a smaller
expansion of deposits than the simple model predicts. All three players – the Fed, banks, and
depositors – are important in the determination of the money supply.
The money supply is positively related to the non-borrowed monetary base MBn, which is
determined by open market operations, and the level of borrowed reserves (lending) from the
Fed, BR. The money supply is negatively related to the required reserve ratio, rr; holdings of
currency; and excess reserves. The model of the money supply process takes into account the
behavior of all three players in the money supply process: the Fed through open market
operations and setting of the required reserve ratio; banks through their decisions to borrow
from the Federal Reserve and hold excess reserves; and depositors through their decisions
about holding currency.
The monetary base is linked to the money supply using the concept of the money multiplier,
which tells us how much the money supply changes when the monetary base changes.
Summary – Chapter 16 M10



A supply and demand analysis of the market for reserves yields the following results: When the
Fed makes an open market purchase or lowers reserve requirements, the federal funds rate
declines. When the Fed makes an open market sales or raises reserve requirements, the federal
funds rate rises. Changes in the discount rate and the interest rate paid on reserves may also
affect the federal funds rate.
Conventional monetary policy tools include open market operations, discount policy, reserve
requirements, and interest on reserves. Open market operations are the primary tool used by
the Fed to implement monetary policy in normal times because they occur at the initiative of
the Fed, are flexible, are easily reversed, and can be implemented quickly. Discount policy has
the advantage of enabling the Fed to perform its role of lender of last resort, while raising
interest rates on reserves to increase the federal funds rate avoids the need to conduct massive
open market operations to reduce reserves when banks have accumulated large amounts of
excess reserves.
Conventional monetary policy tools no longer are effective when the zero-lower-bound problem
occurs, in which the central bank is unable to lower short-term interest rates because they have
hit a floor of zero. In this situation, central banks use nonconventional monetary policy tools,
which involve liquidity provision, asset purchases, and commitment to future policy actions.
Liquidity provision and asset purchases lead to an expansion of the central bank balance sheet,
which is referred to as quantitative easing. Expansion of the central bank balance sheet by itself
is unlikely to have a large impact on the economy, but changing the composition of the balance

sheet, which is what liquidity provision and asset purchases accomplished and is referred to as
credit easing, can have large impact by improving the functioning of credit markets.
The monetary policy tools used by the European Central Bank are similar to those used by the
Federal Reserve System and involve open market operations, lending to banks, and reserve
requirements. Main financing operations – open market operations in repos that are typically
reversed within two weeks – are the primary tool to set the overnight cash rate at the target
financing rate. The European Central Bank also operates standing lending facilities, which ensure
that the overnight cash rate remains within 100 basis points of the target financing rate.
Lecture 11 – Transmission of monetary policy. Repos. (Macroprudential regulation)
Transmission of monetary policy




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Monetary policy plays a key role in attempts to avoid recessions – and to get out of recessions if
the economy has slipped. There are several possible ways – or transmission channels through
which monetary policy affects the real economy. Still there is a great source of debate which
channels work and which do not – or when they work.
The traditional interest rate channel
o Real rate = nominal rate – expected inflation
o Lowering the real rate should lead to a higher investment. Highest investment should
lead to a higher aggregate demand. Both business investment and households’ decisions
to invest in housing and durables (cars, refrigerators,…) should be affected
o Long vs. short rates
 Traditional monetary tools (such as changing the target Fed funds rate) have
focused on nominal short rates
 If inflation does not immediately react to changes in the nominal rate (perhaps
due to sticky prices), then the real rate also changes
 Since long-term rates depend on the path of future short rates, a change in
short real rates may affect long term rates
 Long rates affect investment decisions
o At the zero bound
 When the zero lower bound for nominal rates is hit, commitments to keep the
nominal rate low in the future can affect long term real rates. This can be done
by:
 Perhaps affecting inflation upward (if expansionary monetary policy
stimulates the economy)
 Signaling a low future level of rates
Exchange rate channel
o If domestic real rates are low, investment in domestic assets becomes less attractive (or
attracts less capital from abroad).
o If exchange rate floats, this means that the currency depreciates
o This should stimulate exports and lower imports, thus stimulating domestic demand
Effects through stock prices
o Tobin’s q: The ratio between the market value of a firm and the replacement value of its
capital. If q is high, it is very profitable to invest in new equipment by issuing equity. If q
is low, then you get little new investment in capital.
o Possible mechanism: Lower rates increases stock values (required return falls which
increases prices)
Wealth effects
o If wealth increases, it increases consumption. Consumers’ smooth consumption over
their lifetime, and the level depends on wealth.
o Note that in the current crisis, home equity (i.e. house owners’ equity in their house(s))
had a huge effect. Hence housing “stock” is also a type of stock here.



The credit view
o Bank lending channel
 Higher bank reserves and deposits, increase in bank loans, more investment,
more output.
 Less obvious in times where banks can finance themselves through capital
markets (role in crisis because of capital constraints and thus couldn’t access
these capital markets)
 Losses force cuts in lending (as banks are getting afraid) and liquidity hoarding
also cut lending (because banks want to keep their liquidity).
o Balance sheet channel
 Lower net worth of firms implies that there is less collateral value and thus a
lower ability to borrow.
 Doubts about the value of firms makes banks reluctant to lend (aggravates
adverse selection problem)
 Both of the above cases will effect raise in interest rates
 If monetary policy helps increase the net worth of firms, this can mitigate the
effects above.
Summary:
o
Recession 2007-09
o Although rates were kept low through the crisis, other monetary tools have been critical
o Bank lending has been hit because
 Losses on asset side that forces bank to decrease leverage
 Asset sales cause new price declines in asset markets and a negative spiral
occurs


Fear of liquidity shocks that cause banks to hoard liquidity (afraid to lend to
firms and to other financial institutions)
Uncertainty of other banks’ credit risk also an issue

Repo haircuts
o A channel not mentioned in the book – important for leverage that happens through
trading operations and the ability of arbitrageurs to implement “arbitrage” trades
o Repos are also important in providing liquidity in trading
o Financing a bond purchase using a repo: understand the mechanics.
 A repo (“repurchase agreement”) is a collateralized loan, really.
 Date 0: Sell bond and receive cash
 Date 1: Buy back bond and deliver cash
 Difference in the selling price and buyback price defines an interest rate.
 Low risk for lender: If borrower defaults, he has the bond instead.
 But may require the value of the bond to be larger than the loan to be robust to
price fluctuations.
 Imagine you can borrow 90 for a bond worth 100 (haircut = 10%). You can then
finance the purchase of a bond worth 100 by supplying 10 of your capital And
borrowing 90 on repo. Leverage potential is a function of the haircut (i.e. the
lower the haircut is -> the more you can lever up)!
Macroprudential tools
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Prudential = exercising prudence
Prudence = wisdom in practical matters, cautiousness
Macroprudential policies are tools that affect the financial system as a whole.
In contrast with “macroprudential” which concern the safety of individual institutions
Why use these tools?
o Crisis has emphasized that the real economy can suffer if the financial system is
unstable. Hence the stability of financial systems are important. The monetary tools
(interest rates, liquidity) cannot achieve financial stability on their own.
o For example, asset bubbles could build up because of fiscal policies (including tax
policies).
o Incentives (compensation schemes) in the system may play a role
o Political economy may play a role (the emphasis on house ownership in the US and
improving access to borrowing for low income families)
Toolkit available from Claessens’ survey paper (2014)
o
Restrictions related to
borrower, instrument,
or activity
Expansio
nary
phase
 DTI = Debt to Income
 LTI = Loan to Income
 LTV = Loan to Value
 Indicators of ability to
pay leverage.
 Margin (haircuts: How
much can you level
financial instruments?
(repo)
 Credit growth (in the
aggregate)
Contracti
 Should the rules for
Restricti
ons on
financial
sector
balance
sheet
 We
have
seen
reserve
require
ments
 Pegging
exchang
e rates
 Most
drastic
FX tool
involves
using
capital
controls
 We
Capital
requirements,
provisioning,
surcharges
Taxation,
levels
 Capital buffers Transaction
that are
taxes?
allowed to
Property
decrease when taxes?
economy goes Limiting tax
down
advantage
 Exists in Basel
of debt?
III
 Leverage
restrictions are
part of capital
requirements
 Leverage ratio
being
contemplated
 Already
 Higher
Other (including institutional
infrastructure)
 Should assets and liabilities
always be marked to market?
 If you plan to hold an asset to
maturity (if possible), should
you recognize losses along the
way?
 Pension liabilities can change
a lot in turbulent markets –
should the pension fund be
allowed to “ride the storm”?
 Restriction on bonuses
 Standardized products are
onary
phase:
firesales,
credit
crunch
when you are allowed
to recognize (or forced
to recognize) a loss on
loan vary?
 Big controversy in
Danish crisis
have
discuss
ed
those!
Contagio
n, or
shock
propagat
ion from
SIFIs or
networks
 Volcker rule: In
abstract term a rule
that restricts a
number of activities
related to securities
trading associated
with investment banks
 Dodd-Frank regulates
such activities but
does not prohibit
them
 Can continue market
making, underwriting,
hedging, trading of
government securities,
acting as agents,
brokers or custodians
and other things
 Leverag
e ratio
 Standar
dized
risk
weights
discussed
 Already
implemented
in many ways
 SIFIs have
higher capital
requirements
 Regulators
have
discretion to
levy additional
capital charges
under Pillar II
of the Basel
accord.
property
taxes in
upturn?
 Limiting
interest
deductibilit
y on debt?
being promoted by having
higher capital charges on
exotic products (that are not
cleared)
 This is already affecting
trading operations in financial
firms
 Tobin taxes
(taxes on
financial
transaction
s)
 Other
transaction
taxes
 CCP = Central Counter Party
 Increasing usage (repo, swaps,
CDS)
 Example: Bank resolution and
restructuring directive. Making
sure a regulator can quickly
assess assets and liabilities of
distressed financial institution.
 Bail-in: Methods for saving
tax-payer money by making
sure junior claimholders
(equity, hybrid capital,
unsecured debt, etc. are hit
first)
 Restrictions on high-frequency
trading? Or changing the
auction mechanism
Summary Chapter 26 – M10
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
The transmission mechanism of monetary policy include traditional interest-rate channels that
operate through the real cost of borrowing and affect investment; other asset price channels
such as exchange rate effects, Tobin’s q theory, and wealth effects; and the credit view channels
– the bank lending channel, the balance sheet channel, the cash flow channel, the unanticipated
price level channel, and household liquidity effects.
Four lessons for monetary policy can be drawn from this chapter: (a) It is dangerous always to
associate monetary policy easing or tightening with a fall or a rise in short-term nominal interest
rates; (b) other assets prices besides those on short-term debt instruments contain important
information about the stance of monetary policy because they are important elements in the
monetary policy transmission mechanisms; (c) monetary policy can be effective in reviving a
weak economy even if short-term interest rates are already near zero; and (d) avoiding
unanticipated fluctuations in the price level is an important objective of monetary policy, thus
providing a rationale for price stability as the primary long-run goal for monetary policy.
Lectures on monetary policy. March 29, CBS, Departmeny of Finance
Lecture 1: Monetary policy objectives and practice.
Read: ECB Economic Bulletin, p. 3-15.
https://www.ecb.europa.eu/pub/pdf/ecbu/eb201601.en.pdf
Prepare: 6 power point slides on USB stick on what the ECB president should do in terms of
monetary policy (E.g. raise, lower or keep interest rates unchanged) based on the above. Two
of you will be asked to present their slides in front of class.
FX swaps and CCY (cross-currency swaps)
Several Swedish banks use FX swaps and CCY swaps to fund (in international currency) their lending in
domestic currency. There are some risks associated with this:
Key ratios
Liquidity ratio
Net stable funding ratio (NSFR)
Key formulas:
General formulas:
The structural approach of pricing equity and debt (pricing as options)
Equity is a call option with payoff:
Debt corresponds to risk free debt minus a put-option. The value of the put option corresponds to the
value of a deposit insurance (that covers all losses). The payoff to debt is:
The value of equity and debt can also be calculated when we have several risk classes of debt:
The price of a European call option is:
And put call parity gives the value of a put option:
The yield of a zero coupon bond is calculated from:
When asset value follows a Brownian motion, the probability of default can be calculated using:
Note that in the above formula, mu affects the probability of default. Mu is also connected with the
volatility. To make the model more realistic we can calculate mu with CAPM. The asset volatility is then:
Modeling distributions of loan portfolios when we introduce a common risk variable (M):
Firm i default if:
The probability of default for a given outcome or the market factor is: (large portfolio approximation)
We can then calculate the probability that the fraction of defaulting firms (p(M)) is below a certain
threshold (theta)
The above formula can be rearranged to calculate the fraction of defaults that we can be 99.9% sure of
ending below:
)
If we have
Formulas for the multi-pool correlation model. (might be a bit too excessive)
When several pools are considered simultaneously we need to define a “between pool factor” (Mbp)
and a “whitin pool” (Mwp) factor. Mortage I in pool j defaults if:
Where rho is the total within pool correlation and alpha is the proportion of defaults correlation that
comes from a factor common to all pools.
The realized default rate for pool j, conditional on Mbp and Mwp is:
Where phi is the cumulative distribution function of Zij.
We can then calculate the probability that the fraction of defaulting firms (p(M)) is below a certain
threshold (theta).Type equation here.
𝑃(𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑙𝑜𝑠𝑠 ≤ 𝜃) = 𝑃((1 − 𝑅)𝛷 [
𝛷−1 (𝑝) − √𝛼𝜌𝑚 − √(1 − 𝛼)𝜌𝑚𝑗
√1 − 𝜌
]≤𝜃
Note that this formula includes the recovery rate.
Securitization formulas:
One way to calculate the value of equity/debt in securitization is to discount the sum of expected payoff.
This expected payoff is:
Where Ti(D) is the payoff to the tranche i and P(D=k) is the probability that k defaults occur.
To calculate P(D=k) we can use binomial probabilities:
Where:
the pool.
and p is the common default probability (independent) of the loans in
The mean and variance of this distribution is:
When we want to account for a common market variable in the binomial model (mixed binomial
modeling) we get the distribution:
Where f(p) can be seen as the probability of a certain state occurring. When we only have 2 possible
states of the market we get:
Where f(p1) is the probability of state 1 occurring and p1 is the individual default probability given state
1 occurred.
The mean and variance (in terms of number of defaults) of this is:
Where:
The variance of the loss fraction is:
This means that when N is very large, the variance of loss frequency is the variance of
.
Formulas for rating based pricing of CDOs and bonds:
The sale price (S) of a tranche I is:
Where y* is the yield to maturity of the reference bond with rating k and Bi,ki is the face value of the
tranche. If we denote Wi,ik as the market value of the trance (can be calculated with merton option
formulas) the marketing gain of tranche i is:
And the total marketing gain is:
Rating based pricing in S&P:
Note that in the S&P system ratings corresponds to an exogenously given default probability Π. The face
value of a bond with rating k issued from the reference firm is implicitly defined in:
The market value of the reference bond is:
The yield of the reference bond can be calculated as usual from:
The probability that the reference firm defaults is:
By inverting the formula above, the face value of the reference bond over total asset may be expressed
as:
Rating based pricing in Moody’s Method (expected default loss):
Define the expected default loss as:
A given rating k corresponds to an exogenously given expected default loss rate Λ. We can find the face
vaule B* as the solution to:
Where the expected default loss (Lk*) on the reference bond is given by:
The market value of the reference bond is given by the usual model:
Where d(Q) is defined above substituting rf for mu. Finally, we get the bond yield from:
Formulas for pricing subsidies, taxes and bankruptcy cost:
The payoff and “value” (at time 0) of a bankruptcy cost that is proportional to firm value (alpha) in
default states is calculated from:
Payoff:
Value:
Where d1 as usual is calculated from:
The payoff and “value” of taxes that is proportional to firm value (alpha) in non-default states is
calculated from:
Payoff:
Value:
The payoff and value of a tax subsidy which depends on debt principal (constant pay-off) conditional on
no default is given from:
Payoff:
Value:
The payoff and “value” of a subsidy in default (deposit insurance for example) which depends on the
size of debt only is calculated from:
Payoff:
Value:
The payoff of a subsidy (deposit insurance) and bankruptcy cost is given by:
Where the bankruptcy cost (payoff) is Alpha*V and a compensation is v*max(D-(1-alpha)*V,0).
The payoff of debt and equity can then be written as:
Formula for risk capital as put option (se appendix in Merton and Perold)
Under the assumption that the gross assets and customer liabilities both follow geometric Brownian
motions the amount of risk capital, is given by:
Risk capital = 𝑉0 𝐶(𝑆 = 1, 𝑋 = 1, 𝑟𝑓 = 0, 𝑇, 𝑣𝑜𝑙)  Call option on stock
Where “vol” refers to the volatility of profits.
This formula can ba approximated with:
Risk capital = 0.4 ∗ 𝑉0 ∗ 𝑣𝑜𝑙√𝑇
Covered Interest rate Parity states that
Or equivalently
Other useful formulas
The value of the firm is often assumed to follow a geometric Brownian motion
Calculation of mean and variance
In the mixed binomial model we have:
Formulas for ratios:
Risk adjusted return on capital (RAROC)
Capital ratio = Capital (tier 1 and additional tier 1 and tier 2)/RWA
Leverage ratio: Capital(tier 1 and additional tier 1 and tier 2)//total amount of liabilities
Liquidity coverage ratio:
Net Stable funding ratio (NSFR):