IA 2026 deel 1

Consumption timing

Financial markets allow individuals to shift consumption between periods through saving and borrowing

Saving reduces current consumption to increase future consumption

Borrowing increases current consumption at the expense of future consumption

Indifference curve

Represents an investor’s preferences for consumption in two periods

Every combination of period 1 and 2 consumption along the same indifference curve provides the investor with the same utility

Optimal is the combination in periods 1 and 2 that reaches the highest attainable indifference curve

IPO process

Road shows

Bookbuilding

Underwriter bears price risk

IPOs are commonly underpriced

Some IPOs are overpriced

Sometimes an issue cannot be fully sold

SEO

Seasoned Equity Offer

Additional public offering of shares by an already public traded company

SPAC

Special Purpose Acquisition Company

Raises funds through its own IPO and goes public without underlying commercial operations

Then searches for an acquisition target and merges it into the publicly traded SPAC

Also called blank-check company

If no suitable acquisition is found within 2 years, money must be returned to investors

Bid-Ask spread

Difference between the ask and bid price

Spread represents a source of trading costs

HFT

High Frequency Trading

Tech driven trading in which speed of order execution is crucial

Seeks to be first in the market

Acting on opportunities before slower traders’ orders are fully executed

Dark pool

A venue where a broker can match buyers and sellers internally, without sending the orders to a public exchange

IEX trading

Uses an intentional delay to reduce the speed advantage of HFT

Orders arrive at the exchange simultaneously

Short sale

Profit from a decline in a security’s price

1. Borrow the stock

2. Sell the borrowed stock at current price

3. Later buy the stock back

4. Return the stock to the holder

5. Pay the borrowing fee

Profit = P(t=0) - P(t=1) - fee

Originate to hold vs originate to distribute

Originate to hold: Local thrift institution originates mortgages and keeps the mortgages loans in its own portfolio

Originate to distribute: Mortgages are originated and then sold and securitized, allowing claims on the mortgage pool to be sold to investors

Securitization

Process of pooling loans and transforming their cash flows intro tradable securities

RMBS

Residential Mortgage Backed Security

The mortgage pool is divided into tranches with different levels of default risk and expected return

Senior, Mezzanine, Equity

RMBS tranches

Tranching distributes default risk across investors

Senior tranch, last to absorb losses, lowest risk, lower yield

Equity tranch, first to absorb, highest risk, highest yield

The structure therefore concentrates losses in junior tranches before they reach senior tranches

CDO

Collateralized Debt Obligations

A structured product in which a pool is divided into tranches to concentrate default risk

• Senior tranch, lowest risk and highest rating

• Junior tranch, highest risk and lowest rating

Essentially the same as RMBS

Made up of the risky tranches of a RMBS to be pooled into a CDO and divided into new tranches

Mortgages, RMBS, risky RMBS tranches, CDO, new tranches

CDO squared

Risky tranches from CDOs pooled again and divided into new tranches in a CDO squared

Why credit risk was underestimated

Not expected that entire housing market collapsed simultaneously

Geographic diversification reduced risk less than anticipated

Agency problems existed with rating agencies

Credit Default Swaps did not reduce risk as anticipated

CDS

Credit Default Swap

Insurance contract against the default of the borrower

The protection buyer makes regular premium payments to the protection seller

If a credit event occurs

• Seller pays the buyer for the loss

• Buyer stops paying premiums

Synthetic CDO

A CDO structure based on CDS exposure rather than directly pooling the underlying mortgages

Failure of CDS

Some major CDS issuers did not have enough capital to back their CDS contracts when the market collapsed

Rise of systemic risk

Potential breakdown of the financial system in which problems in one market spill over and disrupt others

One default triggering further defaults

Waves of selling, falling asset prices, further selling

Contagion between institutions and markets

Factors increasing systemic risk

Maturity/liquidity mismatch

High bank leverage

Excessive reliance on credit enhancement through structured products as CDS

CDS mostly traded over the counter, less transparency and no posted margin requirements

Opaque linkages between financial instruments and institutions

Financial crisis timeline

2000-2006 Sharp increase in housing prices

2004 Interest rates begin rising

2006 Home prices peak

2007 Mortgage defaults and MBS losses surge, Bear Stearns announces problems

2008 Major institutions experience distress, money market breakdown, frozen credit market and federal intervention

Responses to systemic risk

Add liquidity to reduce insolvency risk and break vicious circle of valuation/counterparty/liquidity risk

Increase transparency of structured products such as CDS

Change incentives to discourage excessive risk-taking

Reduce agency problems at rating agencies

Net Asset Value

NAV = MV of assets - Liabilities / Shares Outstanding

Used as the basis for valuation of investment company shares when selling new shares or redeeming existing shares

Open-End vs Closed-End Mutual Funds

Open-End

• Shares outstanding change when shares are sold or redeemed

• Priced at NAV

Closed-End

• Shares outstanding do not change unless new stock is offered

• Can trade at premium or discount to NAV

Mutual Fund Costs

Front-end load: Fee when purchasing the fund

Back-end load: Fee when exiting, starts around 5-6% and decrease with years invested

Operating expenses: 0,2-2% paid through reduced portfolio value

12b-1 charges: Distribution/advertising costs paid by the fund; limited to 1% of average net assets per year

Mutual fund performance

Empirical evidence shows

• Average mutual fund performance i generally less than broad market performance

• Evidence that performance is consistent from one period to the next is suggestive but inconclusive

Most funds underperform

Some show stronger performance depending on measurement interval and time period

Morningstar style box

Classifies funds/stocks along tow dimensions

Size

• Large

• Medium

• Small

Investment style

• Value

• Blend

• Growth

Value vs Growth stock

Value

• Low ratios of market price per share, low price to fundamental value

Growth

• High ratios of market price per share, investors must expect rapid growth to justify their prices

Holding period return

R_t+1 = (P_t+1 - P_t + D_t+1) / P_t

Measures the realized total return

Log Return

r_t+1 = ln(P_t+1) - ln(P_t)

Empirical distribution

A probability distribution constructed using historical realized returns

By counting how often realized returns fall within particular ranges, we can estimate the underlying probability distribution

Probability distribution of returns

A risky investment can generate different returns in different states of the world

A probability distribution assigns a probability P_R to each possible return R

Expected return is the probability-weighted average

E(R) = SUM P_R * R

Empirical vs probability distribution

Empirical is based on historical realized returns, represents only one possible path

Probability considers the possible future states of the world and their probabilities

Normal distribution of returns

Returns for many asset classes can be approximated by the normal distribution

Advantage normality assumption;

• full distribution can be characterized by mean and variances of returns only

Fat-Tailed Distribution

A return distribution with fatter tails than the normal distribution

Mean extreme returns are more likely than predicted by the normal distribution

Risk Profile

Defined as the cumulative distribution of future wealth

P(W_t+1 < X) for every X

Future Wealth

Current wealth grown by the return during the investment period

W_t+1 = P_t+1W_t / P_t = W_t (1 + r_t+1)

Risk Profile under Normally Distributed Returns

Starting from:

P(W_t+1 < X)

and

W_t+1 = W_t (1 + r_t+1)

We obtain

P(W_t+1 < X) = P(r_t+1 < X/W_t - 1)

If returns are normally distributed, standardize:

= P( z < (X/W_t - 1 - mu) / sigma)

Value at Risk

A measure that summarized the portfolio risk in one number, combining the expected return and risk

For a 95% one-year VaR

P(W_t+1 < W_t - VaR_t) = 5%

P(r_t+1 < - VaR_t/W_t) = 5%

95% VaR under normally distributed returns

P(z < -1.64) = 5%

Combining this with standard returns gives

VaR_t = W_t (1.64sigma - mu)

Interpretation of 95% VaR

If VaR = 19.81

There is a 5% probability that the portfolio lose more than 19.81 over the specified period

VaR gives the threshold but does not tell us how large the loss will be

Log Return

r_t+1 = ln(P_t+1) - ln(P_t)

Using a first-order Taylor approximation:

r_{t+1 =~} P_t+1 - P_t / P_t = R_t+1

Additivity of Log Returns

Log returns are additive across time

For four quarterly returns, the annual log return is:

r_t+4⁽⁴⁾ = ln(P_t+4) - Ln(P_t)

and

r_t+4⁽⁴⁾ = SUM r_t+h⁽¹⁾

So the annual log return is the sum of the quarterly log returns

Expected return across investment horizons

If returns are identical and independently distributed (IID)

mu^LF = H mu^HF

Therefore, expected returns scales linearly with the investment Horizon

example: mu^annual = 12 mu^montly

Covariance and SD across investment horizons

Under IID returns, covariance scales linearly

sigma_ij^LF = H sigma_ij^HF

Therefore, variance also scales with H, while sd scales with sqrt(H)

sigma^LF = sqrt(H) sigma^HF

example: sigma^annual = sqrt(12) sigma^monthly

VaR across investment horizons

Two effects determine how VaR changes with investment horizons

1. Time-diversification: sigma^LF = sqrt(H) sigma^HF

2. Expected return effect: mu^LF = H mu^HF

Therefore: VaR^LF = W_t (1.64 sqrt(H) sigma^HF - H mu^HF)

Risk premium vs Excess return

Excess return: Difference between the actual return on a risky asset and the rf rate

R(p) - R(f)

Risk premium: The expected excess return

E{R(p) - R(f)}

Mean-Variance Utility Function

U = E( r ) - 0.5 A sigma²( r )

A coefficient of risk aversion

Mean-Variance Indifference Curve

Shows combination of E( r ) and sigma that provide an investor with the same level of utility

Greater risk sigma must be compensated by a higher expected return

A higher A produces a steeper indifference curve

Therefore, a more risk-averse investor requires a larger increase in expected return as compensation for taking additional risk

Diversification

Holding multiple risky assets can reduce portfolio risk because firm specific (idiosyncratic) risk can be diversified away

Diversification can not eliminate systematic risk

Total Risk = Systematic Risk + Idiosyncratic Risk

Covariance matrix SUM

Diagonal: variances, where sigma²_i = sigma_ii

Off-diagonal: covariances sigma_ij

SUM is symmetric because sigma_ij = sigma_ji

Variance of N-asset portfolio

sigma²_p = SUM_j=1SUM_i=1 w_iw_jsigma_ij

w = portfolio weights

sigma_ij covariance between returns of assets i and j

portfolio variance depends on both individual asset variances and the covariances between assets

Equal-weighted portfolio variance

If all n assets have equal weights w_i = 1/n

Portfolio variance consists of:

• n variance terms

• n(n-1) covariance terms

sigma²_p = 1/n sigma^-2 + (n-1)/n cov^-

Diversification as n → inf

sigma²_p → cov^-

Individual variance is diversified away, but the common covariance components remain

Certainty Equivalent

The return that a rf investment would need to offer to give the investor the same utility as the risky portfolio

Combining 1 risky asset and the rf asset

Let y be the proportion invested in risky portfolio P, so 1-y is invested in rf asset

E(r_c) = yE(r_C) + (1-y)r_f

= r_f + y[E(r_p) - r_f]

Because the rf asset has no volatility, increasing y increases both the expected return and risk of the complete portfolio proportionally

y = sigma_c / sigma_p

Capital Allocation Line

Describes all risk return combinations obtainable by combining risky portfolio P wiht the rf asset

E(r_C) = r_f + ([E(r_P) - r_f] / sigma_P ) sigma_C

Intercept r_f

Slope [E(r_P) - r_f] / sigma_P

The CAL is a straight line through the rf asset and the risky portfolio P

Sharpe Ratio

Reward-to-Volatility Ratio

S = [E(r_P) - r_f] / sigma_P

Measures the expected risk premium per unit of risk

Higher S, more expected excess return per unit of risk

Optimal allocation to the risky asset

Maximize y

U = r_f + y[E(r_P) - r_f] - 0.5A y²sigma²_p

Optimal proportion invested in risky portfolio P is

y^* = [E(r_P) - r_f] / A sigma²_p

Higher risk premium → higher y^*

Higher risk aversion → lower y^*

Higher variance → lower y^*

Expected Return two risky assets

E(r_p) = w_DE(r_D) + w_EE(r_E)

w_D + w_E = 1

Variance two risky assets

sigma²_p = w²_Dsigma²_D + w²_Esigma²_E + 2w_Dw_Esigma_DE

sigma_DE = rho_DEsigma_Dsigma_E

sigma²_p = w²_Dsigma²_D + w²_Esigma²_E + 2w_Dw_Erho_DEsigma_Dsigma_E

Risk depends on how two assets move together

Perfect positive correlation

When rho = 1

portfolio sd becomes sigma_p = w_Dsigma_D + w_Esigma_E

The assets move perfectly together so combining them provides no diversification benefit

Perfect Hedge Asset

A hedge asset had negative correlation with the other asset

Great at reducing risk

rho = -1 , so sigma_p = (w_Dsigma_D - w_Esigma_E)

w_D = sigma_E / (sigma_D + sigma_E), which gives sigma_p = 0

Global Minimum Variance Portfolio

The portfolio of risky assets with the lowest possible variance

w^min_D = (sigma²_E - sigma_DE) / (sigma²_D + sigma²_E - 2sigma_DE)

GMV portfolio depends on the assets’ variances and covariance, not on risk aversion

Efficient Frontier of risky assets

Part of the portfolio opportunity set containing portfolios that provide the

• Highest expected return for a given level of risk

• Lowest risk for a given expected return

It begins at the Global Minimum Variance Portfolio and consists of the upper part of the mean-variance frontier

Optimal portfolio without a risk free asset

Chooses portfolio on the efficient frontier that maximizes

U = E(r_p) - 0.5Asigma²_p

w^*_D = (E(r_D) - E(r_E) + A(sigma²_E - sigma_DE) / A(sigma²_D + sigma²_E - 2sigma_DE)

Graphically, this is where the investor’s highest attainable indifference curve is tangent to the efficient frontier

Therefore, optimal risky portfolio depends on risk aversion A

Tangency Portfolio, two-risky plus a rf asset

When a rf asset is available, choose the risky portfolio that maximizes the Sharpe ratio

max S_p = (E(r_p) - r_f) / sigma_p

w_D = ([E(r_D) - r_f]sigma²_E - [E(r_E) - r_f]sigma_DE) / ([E(r_D) - r_f]sigma²_E + [E(r_E) - r_f]sigma²_D - [E(r_D) - r_f + E(r_E) - r_f]sigma_DE)

Graphically, here starts CAL at r_f is tangent to the efficient frontier of risky assets

Tangency portfolio = Optimal risky portfolio

It gives the steepest possible CAL