This course as a portfolio

1. Introductory Material (19%)
2. Financial Markets (23%)
3. Optimal Portfolios (23%)
4. Equity Topics (15%)
5. Fixed Income Topics (12%)
6. Performance Evaluation (4%)
7. Taxes (4%)

Savings problems

• Annuity calculations
• Basic bond prices
• Saving for retirement
• Real and nominal cash flows and rates

Returns

• How to calculate
• Compounding returns
• Arithmetic vs. geometric averages
• Dispersion: variance and standard deviation
• Comovement: covariance and correlation
• Calculating portfolio return characteristics
  • Expected returns; variance; SD

Equity Market

• Over long horizons, average returns in the US stock market have exceeded those of bonds.
• Stock returns are risky; that is, volatile.
• Stock return distributions are fat-tailed and negatively skewed.
• Past aggregate returns do not predict future aggregate returns.
• Volatility is time-varying and persistent.

Treasury Markets

• Term structure of interest rates
• Spot rates
  • zero-coupon bonds
  • bootstrapping from coupon bonds

Arbitrage

• Free risk-free return
• Bootstrapping spot rates based on no-arbitrage pricing
• Law of One Price says that price of bond = price of replicating portfolios
  • If not, there exists an arbitrage

Markets and Trading

• Adverse selection: taking advantage of information asymmetry
• Winner’s Curse: you might regret winning an auction!
• Bid-ask spreads in limit order books are partially due to adverse selection concerns

Leverage and Margin

• Leverage is investing with borrowed money
  • amplifies good and bad returns
• Margin: borrowing money from your broker to buy assets
• Brokers and regulators require initial and maintenance margins to protect against default risk
• Price movements against your position may generate margin calls.

Short-selling and Limits to Arbitrage

• Borrow the asset, sell it short, then buy back later
• Margin accounts on short positions require extra collateral to protect against price increases (liability increases)
• In practice, arbitrage trades are limited by frictions like equity borrowing fees and margin requirements.
• Prices might move the wrong way before they correct!

Diversification

• Diversification: portfolios of assets may reduce overall risk
• Efficient Frontier: the set of portfolios that minimize portfolio risk for a given target expected return
• Global Minimum Variance: portfolio of risky assets with the smallest variance

Theory

• Capital allocation with a risk-free asset
• Tangency portfolio: portfolio of risky assets with the highest Sharpe ratio
• Capital Allocation Line: set of portfolios combining risk-free asset and tangency portfolio
• Location on CAL depends on investor’s risk aversion

Borrowing Frictions

• Borrowing rates usually exceed savings rates
• For a single risky asset, this leads to a kinked CAL
• Efficient frontier consists of
  • a CAL consisting of saving and maximum Sharpe ratio portfolio w.r.t savings rate
  • a portion of the all-risky-asset frontier
  • a CAL consisting of borrowing and maximum Sharpe ratio portfolio w.r.t borrowing rate

Shorting Constraints

• Some investors may not be able to short assets
• This reduces the investment opportunity set
• Recall how to implement efficient and tangency portfolios with position limits

Rebalancing

• Assuming our inputs stay constant over time, price movements will cause portfolios to drift from optimal weights over time.
• Rebalancing portfolios back to optimal weights improves expected performance.

Input Sensitivity

• Mean-variance optimization is sensitive to inputs
• Position limits can mitigate error-maximization problem
• In practice, some try to estimate a subset of the inputs
  • GMV (assume equal means)
  • Risky parity (assume equal means and zero corr)
  • Equal-weighted (equal means & SDs; zero corr)

Market Model

• \(\beta\) measures sensitivity to market returns
• \(\alpha\) measures historical average abnormal return
• \(\beta\)’s can be used in estimating the covariance matrix with fewer parameters

CAPM

• Widely used model for expected equity returns
• Requires 3 inputs
  • Risk-free rate
  • Beta
  • Market risk premium
• Performs poorly in explaining differences in stock return empirically
  • Security market line is too flat

Cross-sectional Predictability

• Sorting stocks on characteristics has been more successful in explaining cross-sectional differences in returns than beta
  • Market cap, book-to-market, momentum, liquidity, idiosyncratic volatility
• Cross-sectional regressions of returns on lagged characteristics are another method of explaining returns

Multifactor Models

• Beyond the market excess return as a factor
• Size: Small Minus Big
• Value: High B/M Minus Low
• Momentum: Winners Minus Losers
• Op. profitability: Robust Minus Weak
• Investment: Conservative Minus Aggressive

Duration

• Duration is a weighted average time to maturity
• Duration allows us to quickly compare interest rate risk for bonds with different coupons, maturity, yields, etc.
• Duration is also the horizon at which reinvestment risk and interest risk cancel out

Convexity

• Convexity measures the curvature of the pricing function
• Duration + Convexity allow for better approximation of pricing function
• Investors like positive convexity (standard coupon bond)
• Issues prefer negative convexity (callable bonds / MBS)

Credit Risk

• Credit risk: risk issuer will not pay promised CFs
• Credit ratings are a standard way to measure
• Yields higher for lower rated debt
• Yield \(\neq\) expected return!

Evaluating Asset Managers

• \(\alpha\)’s from a factor model are average benchmark-adjusted average returns
• Atrribution analysis: performance can be decomposed into the benchmark component(s) (factor loadings times factor realizations) and the active component.

Tax-efficient Investing

• Calculating Taxes
• Effects of deductibility of tax-advantaged savings
  • Traditional IRA/401(k)
• Effects of deferral of taxation
  • Non-dividend stock; non-deductible IRA