Investment Analysis Ch 5-7

What determines interest rates?

• Supply of funds (savers)

• Demand for funds (businesses investing)

• Government demand + Fed actions

Nominal vs Real interest rate?

• Nominal (rn) = growth of money

• Real (rr) = growth of purchasing power

Approximate real rate formula?

rr ≈ rn − i

Exact real rate formula?

rr = rn-i/1+i

Fisher Equation

rn=rr+E(i)

What shifts real interest rates in equilibrium?

• Increase in demand → rates ↑

• Increase in supply → rates ↓

After-tax real rate formula?

After-tax real return = rn(1 − t) − i

Expanded:
= rr(1 − t) − it

Return on zero-coupon bond?

rf(T)=100/P(T)-1

What is EAR (Effective Annual Rate)?

True annual return accounting for compounding

EAR formula?

1+EAR=[1+rf(T)]^1/T

APR formula?

(1+EAR)^T-1/T

What is APR?

the rate charged or earned on an annual basis.

Continuous compounding formula?

EAR = e^(APR) − 1

Continuous Compounding

compounding occurs on a continuous basis.

Expected return formula

E(r)=∑s​p(s)r(s)

Variance formula?

σ2=∑s​p(s)[r(s)−E(r)]^2

Standard deviation formula?

σ = √variance

What does standard deviation measure?

Risk (volatility of returns)

Risk-free rate?

Return with no risk (Treasuries)

Risk premium?

Extra return for taking risk

Excess return?

Return − risk-free rate

Risk aversion?

Preference for less risk

Arithmetic average return formula

(Simple average)
= sum of returns ÷ years

Average Arithmetic Returns

The return earned in an average year over a multiyear period.

Geometric Average Returns

The average compound return earned per year over a multiyear period.

Geometric average return formula

GAR=[(1+R1​)(1+R2​)...(1+RT​)]^1/T−1

Difference between arithmetic vs geometric?

• Arithmetic = average

• Geometric = true compounded return

Sharpe Ratio formula?

= Risk premium / SD of excess return

What does Sharpe measure?

This ratio measures the trade-off between reward (the risk premium)

and the risk (the standard deviation or SD). return per unit of risk

Annualized Sharpe from monthly?

SRₐ = SRₘ × √12

Key normal distribution rules?

• 68% within 1σ

• 95% within 2σ

• 99.7% within 3σ

If returns are NOT normal, what happens?

• SD is incomplete risk measure

• Sharpe is incomplete

• Must consider skewness

Positive vs Negative skew?

• Positive = big upside

• Negative = crash risk

What is VaR?

Loss at a given percentile (e.g. 5%)

Example meaning of 5% VaR?

5% chance losses exceed this amount

When can VaR be easily calculated?

When returns are normally distributed

What does it mean that VaR is a “quantile” of a distribution?

A quantile is a cutoff value such that q% of outcomes fall below it

What is Expected Shortfall (ES)?

Average loss in worst-case scenarios

What is another name for Expected Shortfall?

conditional tail expectation (CTE)

ES vs VaR?

• VaR = cutoff

• ES = average beyond cutoff

What is Sortino Ratio?

Uses ONLY downside risk

Why is the Sortino Ratio better than the Sharpe Ratio?

Doesn’t penalize upside volatility

What is meant by Life Cycle Investment Goals?

They are are generally broken down into the

following categories:

• Near-term, high-priority goals

• Long-term, high-priority goals

• Lower-priority goals

What is a risk-averse investor?

Someone who prefers lower risk for a given return.

What is a gamble/speculation?

Taking risk in hopes of higher returns.

What is the utility function for investors?

U = E(r) − ½Aσ²

What do each of these represent in the utility formula?

• E(r) = expected return

• A = risk aversion level

• σ² = variance (risk)

What happens to utility when risk increases?

Utility decreases (risk-averse investors hate risk).

What is a risk-free asset?

An investment with no uncertainty (guaranteed return).

Examples of risk-free assets?

• T-bills

• Bank CDs

• Commercial paper

Key characteristics of risk-free assets?

• Very safe

• Low return

• Highly liquid

• Short-term

What is the risk premium?

E(rp) − rf

What does y represent?

% invested in risky asset

What does (1 − y) represent?

% in risk-free asset

Expected return of complete portfolio?

E(rc) = rf + y[E(rp) − rf]

Portfolio risk (std dev)?

σc = yσp

What happens when y increases?

• Return ↑

• Risk ↑

What is the Capital Allocation Line (CAL)?

A line showing risk vs return combinations of portfolios.

What does the slope of CAL represent?

Reward-to-risk ratio

What happens if borrowing rate > lending rate?

CAL becomes kinked

What determines optimal portfolio?

Risk tolerance (A), expected return, and variance

Optimal allocation to risky asset?

y* = (E(rp) − rf) / (Aσ²p)

If investor is more risk-averse (higher A), what happens to y*?

y* decreases (less risky investment)

What is the Capital Market Line (CML)?

a theoretical, upward-sloping line representing the most efficient portfolios combining risk-free assets and the market portfolio, maximizing return for a given level of risk (standard deviation). It defines the trade-off between expected return and risk, with the slope indicating the market portfolio's Sharpe ratio.

CAL using the market portfolio (like S&P 500)

What is a passive strategy?

Investing without trying to beat the market

Example of passive investing?

Index funds (like S&P 500)

What is a bull market?

Prices rising

What is a bear market?

Prices falling

What % change defines bull/bear market?

~20% move

Why are they called bulls and bears?

• Bull = attacks upward

• Bear = swipes downward

What is diversification

Spreading your money across different investments so one bad investment doesn’t kill your whole portfolio.

Why does diversification reduce risk?

Because different assets don’t move perfectly together — losses in one can be offset by gains in another

When does diversification work BEST?

When assets are not highly correlated (don’t move together).

Can diversification eliminate ALL risk?

No — only firm-specific risk can be eliminated.

What is systematic risk (market risk) (non diversifiable risk)?

Risk that affects the entire market (recession, inflation).

What is nonsystematic risk (firm-specific risk) (unique risk) (diversifiable risk)?

Risk specific to a company (bad CEO, lawsuit).

Which risk matters more for investors?

Systematic risk — because you can’t eliminate it.

How do you calculate expected return of a 2-asset portfolio?

E(rp) = wA·E(rA) + wB·E(rB)

What does the E(rp) formula really mean?

Portfolio return is just a weighted average of the assets.

If you put more weight on a high-return asset, what happens?

Expected return increases.

What does standard deviation (σ) measure?

• A measurement of risk

• A measure of the variation of possible rates of return from the

expected rate of return

What does a higher standard deviation mean?

More volatility → more risk.

What does σ = 0 mean?

No risk (ex: U.S. Treasuries).

Does portfolio risk = weighted average of risks?

No — depends on how assets move together (correlation).

What is the Sharpe Ratio?

S = (E(r) − rf) / σ

Is a higher Sharpe ratio better or worse?

Better — more return for each unit of risk

If two portfolios have same return, which is better?

The one with lower risk → higher Sharpe

How do you find optimal % in risky asset?

y = (E(rp) − rf) / (Aσ²)

What happens if risk aversion (A) increases?

y decreases → invest less in risky assets

What happens if expected return increases?

y increases → invest more in risky assets

What happens if risk (σ²) increases?

y decreases → less investment in risky assets

What did Harry Markowitz prove?

You can build an optimal portfolio using diversification.

What is the goal of the Markowitz model?

Maximize return for a given level of risk

What does “efficient portfolio” mean?

Best possible return for its level of risk

What happens as you add more stocks to a portfolio?

• Firm-specific risk ↓

• Total risk ↓ (but only up to a point)

What risk remains no matter how diversified you are?

Systematic (market) risk

What is the portfolio management process?

Steps to build and manage investments over time

Portfolio Management Step 1: Policy statement?

Define goals, time horizon, and risk tolerance

Portfolio Management Step 2: Analysis?

Study economic and market conditions

Portfolio Management Step 3: Construction?

Build the actual portfolio