modern finance weeks 1-5 (class test) Flashcards
CAPM formula
ERi = Rf + bi(ERm - Rf)
r= rf+b(rm - rf)
-expected return of asset/portfolio = RFR + beta asset * market risk premium (expected return market - RFR)
NPV formula
-NPV = sum of : Ct/ (1+r)^t - intital investment
-Ct = cash, r = discount rate
Net Present Value (NPV) is the total of all future cash flows discounted to the present, minus the initial investment.
Formula Equivalent Annual Rate (EAR) with continuous compounding
EAR (continous compounding) = e^r - 1
-r is the quoted rate
define Equivalent Annual Rate (EAR)
(EAR) is the annual interest rate reflects the effect of compounding over a given period, showing the true rate of return/ cost of borrowing taking into acccount compoiunding.
reflects the true interest rate when compounding occurs more than once per yea
formula for present value + PV with multiple years of cash flow + defintiion
pv = FV/Cashflow / (1+r)t, ct is cash flow at time t.
-with multiple years sum the discounted cash flows which have been discounted to the correct year.
- future cash flows discounted to the present by an appropriate discount rate.
(PV) is the current value of future cash flows, discounted to the present by an appropriate discount rate, which reflects the time value of money and the risk associated with future cash flows.
formula for Equivalent Annual Rate (EAR) with discrete compounding
EAR (normal compounding) = (1 + quoted rate r/n)^n - 1.
r = Nominal annual interest rate (as a decimal)
n = compounding periods per year (e.g., 12 for monthly, 4 for quarterly)
Risk premium v simple + define risk free rate
RiskPremium=PortfolioReturn−Risk-FreeReturn
-RFR refers to the basic interest rate assuming no uncertainty/inflation and reflects the pure time value of money.
The rate of return is also called…
discount rate, hurdle rate, and opportunity cost of capital
expected holding-period return HPR normal + with different states/probabilities
-Multiple states:
-e(HPR) = sum(pi * HPRi)
-HPRi = expected holding-period return state i, pi = prob state i occuring
-Normal:
HPR = (P1+ D1))/ P0 - 1
-future price + div all over initial price all monus 1
or HPR = p1 - p0 + d1 / p0
-p1:Expected price of the asset at the end of the holding period, p0: initial asset price, d1: expected dividend income during the period
formula for expected standard deviation and expected variance
Variance σ^2 = ∑pi * ( ri - E(R) )^2 )
-sum of prob state pccuring * (return in that state - baseline/expected return)squared
-Standard deviation σ is square root of the variance sqrt(variance)
formula for expected standard deviation and variance of a assest
E[σ^2] = ∑ pj * σi^2
σ^2i = variance of returns in state i, pi = prob state i occuring
-expected sd is just sqrt(E[σ^2])
When comparing investments with different horizons, the ____________ provides the more accurate comparison.
The effective annual rate provides the more accurate comparison of investments with different horizons because it expresses the returns in a common period.
What process is the inverse of compounding.
discounting
-discount invesre of compounding
fisher effect
describes relationshipn between nominal returns/interst rate and real returns/rates and inflation
1+R = (1+r)*(1+h)
-R - nominal rate, r- real rate, h inglation rate
arithmetic and geometric mean + difference
-The arithmetic mean (often just called the “average”) is calculated by adding up all the values and dividing by the number of values. Tells return in ‘average year’ over a time period.
-R = [(1+r1)(1+r2)…*(1+rn)]^1/2 - 1
geometric average returns way to measure the average rate of return over multiple periods, accounting for the compounding effect/volatility.
-Arithmetic assumes no compounding, while the geometric mean accounts for compounding and provides a more accurate measure of the average rate of return over time, especially when returns fluctuate/volatile.
describe apr
Annual Percentage Rate (APR) refers to the yearly cost of borrowing or the return on investment (which is expressed as a percentage) including interest and certain fees (management/payment fees) but excluding compounding effects.
-within uk all loans must state apr and lender must state total amount paid at end of loan.
Calculating expected standard deviation/variane using HPR
-First find the Expected Holding Period Return E(HPR):
∑ pi*HPRi
-Variance σ^2 = ∑ pi*(HPRi - E(HPRi))^2
-SD is sqrt of variance
expected return + portfolio return
∑pi*ri
rp =∑ wi *ri
covariance
-Covariance, cov, refers to an absolute measure of the extent to which two sets of numbers, assets, move together over time (directional relationshiop of two assets over time)
COVab =∑(Rai - e(Ra)) * (Rbi - e(Rb)) / n
e(ra/b) is the mean/average return of assets a and b over the time period.
correlation coeffieint
refers to a relative measure of both strength and direction of a relationship between two variables, assets. Value between -1-1.
ρab = COVab /σa*σb,
given bycovariance between 2 assets divided by product of the standard deviations
define systematic and unsystematic risk + relation to correlation coeffieints
-ssytematic risk refers to overall market risk which is undiversfiable and effects returns of all assets. Eg) policy, econmic business cycles, finaltion etc.
-Unsystematic risk, also known as diversifiable risk, is risk that is industry/asset specific and can be reduced/eliminated through diversification. Unsystematic risk is decreasing in number of portfolio assets.
what does the portfolio possibilites frontier show
The Portfolio Possibilities Frontier shows the set of all possible portfolios that can be created by combining two or more assets. It shows the highest expected return for each level of risk (or the lowest risk for each level of return) that can be achieved with different combinations of assets.
define evelope curve and process in which leads to the effieint frontier
-The envelope curve shows all possible combinations of N stocks in a return/risk space (Rp, σp). Plots all potential portfolios in a risk return space.
-To find efficient frontier, start by rulling out all combinations of stocks (portfolios) that are never the best option to another. aim is to elimate combinations that are always dominated i.e., those that never provide a better return for the same or lower level of risk.
-Result is a set or singular portfolio, combo of stocks, that offer the highest expected return for each level of risk, forming the efficient frontier.”
what does the effieint frontier show/defien
The efficient frontier shows the set of portfolios that offer the highest expected return for a given level of risk or the lowest risk for a given level of return. portfolios considered “optimal” as they represent the best possible trade-offs between risk and return.
-Any portfolio/point on the efficient frontier maximises expected return for a given risk level, or minimises risk for a given return level, any point/portfolio on frontier is better than any portfolio below it, as it provides more return for the same or less risk.
what does the capital market line cml show and describe each point/draw + explain each point
CML shows the optimal combination of a risk-free asset and a portfolio of risky assets, offering higest return for each risk level
-CML given by Rp = Rf + σp* (Rm - Rf)/σm
Rm market port return, σm: sd of market portfolio.
slope of CML is sharpe ratio
shows that investors can
both lend and borrow at a risk-free rate of return.
Portfolios along the capital market line are derived
by borrowing and lending at the risk free rate of interest in the capital
market.
describe tangency between capital market line and efficient frontier + relatiioon to sharpe ratio at this point
Tangency represents the optimal combination of risky assets and the risk-free asset that provides the highest expected return for a given level of risk.
tangency also has the highest Sharpe ratio, which means it gives the best return for the least amount of risk.
define sharp ratio + in terms of market portfolio
-how much excess return an investor is receiving for each unit of risk taken sd.
-tells you how much excess return the market portfolio is generating compared to the risk-free rate for each unit of market risk (standard deviation of the market returns).
-slope of the cml
define the market portfolio + market portfolio beta + interpet
The market portfolio M is A portfolio that includes all risky assets in the economy, not just stocks, weighted by their market values. M lies on the eff frontier providing the highest return for any given level of risk
beta of mkt port is 1.0, serving as a benchmark for assessing the systematic risk of individual assets relative to the market.
Beta > 1.0: Indicates that the security or portfolio is more volatile than the market. For example, a beta of 1.2 suggests the asset is 20% more volatile than the market.
Beta < 1.0: Signifies that the security or portfolio is less volatile than the market.
Beta = 1.0: Implies that the security or portfolio’s volatility matches that of the market.
formula for variance of a 2 asset portfolio
2 asset portfolio variance:
-σp^2 = w1^2σ1^2 + w2^2σ2^2 + 2(w1w2p12σ1σ2)
relation between diversification and correlation coefficient
Diversification most efficient when the correlation coefficient p between assets is low or negative because this means the assets don’t move together, helping to reduce overall risk.
Perfect positive correlation means no risk reduction.
Perfect negative correlation allows for complete risk elimination.
correlation coefficient between the market return and a risk-free asset would be..
0
The most important criteria when adding new investments to a portfolio is..
correlation of the new investment with the rest of the portfolio
The optimal portfolio is identified at the point of tangency between the efficient frontier and the
-highest possible utility curve
what does the steepness of utility curves indicate?
Steep utility curve: Indicates high risk aversion; the investor requires a higher reward for taking on additional risk.
Flat utility curve: Indicates low risk aversion; the investor is more willing to risk for smaller increases in expected return.
Utility curve slope: Represents the trade-off between risk and return—steeper means more risk-averse, flatter means more risk-tolerant.
Markowitz portfolio equilbirum outcome
in equilibrium, regardless of individual preferences, all investors will hold a combination of the market portfolio (M) and the risk-free asset (RFR). Investors may adjust positions by borrowing or lending at RFR in order to maximize their utility, depending on their individual risk level.
Formula for estimated rates of return
- can be used to see if asset is over or undervalued.
estimated return =( pt+1 - pt ) / pt + dt+1/pt
aka =( p1 - p0) / p0 + d1 / p0
-If estimated return > required rate of return –> asset is undervalued thus BUY.
-If estimated return < required rate of return –> asset overvalued thus sell.
- Sharpe ratio
-Refers to the excess return per unit of risk, sd.
-Measure of risk adjusted return on investment.
Rp - Rf / σp
Treynor ratio/measure + why more widely used
-Measure of risk adjusted return on investment.
-Refers to additional/excess return per unit of systematic risk, beta. More widely used as penalises low diversification.
= Rp - Rf / Bp
- Information ratio/measure
-Ratio shows how an investment performs compared to benchmark relative to the additional risk taken.
IR = Rj - Rb / σER ,
σER = sd of excess return, numerator is ability to generate portfolio return > market, numerator shows risk taken to generate this excess return.
Good +ve, 1 exceptional
Securities with returns that lie above the security market line are
true
-If securities have returns (estimated) that sit above the SML they are –> undervalued.
-If securities with returns that lie below the SML are overvalued.
sortino ratio
-measure of risk adjusted performance and measures additional/excess return per unit of downside negative risk.
-Focuses on only negative fluctuations rather than total risk.
-ST = Ri - t / DRi or ST = Ri - t / σd
where t is comparision rate, could be RFR and σd is standard deviation of downside risk.
- Jensens alpha + interpretations of value
-Measures how much a portfolio out/underprerforms compared to predictions made by the CAPM.
-Doesnt take into acconunt managers ability to diversify, calculates risk premiums relative to beta/systematic risk.
Rit - Rf = ai + Bi(Rmt - Rf) + Eit
-rearrange for a (Rit = portfolio i return at time t, RM market return etc.)
-If a = 0, port perfroms exactly as predicted by capm. portfolio in equilibrium.
-if a > 0, portfolio outperforms predctions of capm, investor has ‘superior ability’.
-if a < 0, port underperforms predictions of capm, suggesting poor ablity.
With the sortino ratio/measure explain downside risk and a popular measure of it
Measures the volatility of negative returns (below a target, mean return/thresh-hold), unlike standard deviation, which considers both gains and losses.
-Captures what investors consider truely risky.
-Semi deviation is a pop measure, is the sd of only negative returns,
σd = sqrt[1/n * ∑ (Ri - R^)^2 ]
-ri returns if fall belwo the threshold r^, ignore if greater than this threshhold.
-r^ threshold return
-n number returns below the mean/threshold
formula for beta B + special cases
- B = COV (Ri, Rm) / VAR(Rm) =
-ri return of asset, rm market return.
-B = pim * σi / σm
σi = sd of assets returns, σm sd of market returns
beta = 1 mkt beta, The asset’s return moves in line with the market./ same volility as market.
Beta = 0 The asset’s returns are uncorrelated with the market. It does not move with the market at all.
applications of CAPM
-Risk assessment
-Estimating the required return on an investment based on its risk (beta) and the market risk premium.
-Measuring portfolio performance
-Stock evaluation of individual securities or portfolios using metrics like Jensen’s Alpha
-Cost of equity calculation for companies to determine the expected return required by shareholders.
-Capital budgeting to assess the whether an investment should take place based on their risk-adjusted return.
Comparing the SML and the CML
-CML used for efficient portfolios (made up of risky assets and riskless assets) and relates total risk, sd to return, while SML used for individual stocks/portfolios and relates return to systematic risk, beta.
-Inefficient portfolios do not lie on the CML.
-CML used Sd (total risk) as measure of risk where SML uses systematic risk, beta.
In equilibrium, the CML represents the optimal portfolios that balance total risk and return, while the SML reflects the correct pricing of assets based on their systematic risk (beta) and expected return according to the CAPM.
Security market line SML
-SML is a graphical represenation of the CAPM model.shows the relationship between the expected return of a security and its systematic risk (beta), and the pure time value of money measured by RFR.
-SML shows the reward for bearing systematic risk, beta, is the market premium Rm - Rf.
-Plotted in (Beta, Ri) space
Formula for portfolio beta
Bp = ∑ wi* bi, weight of asset i in port, beta of asset i
assumptions of the CAPM
-No taxes, transaction costs and information publicly available.
-Investors rational, utolity maxers, mean variance optimisers.
-Investors can borrow and lend at Rf, due to existence of the riskless asset.
- investors have a one-period investment horizon (they plan to invest for the same time frame).
The market consists of all risky assets, and there is a market portfolio that contains all risky assets weighted by their market value.
if an individual owns only one security the most appropriate measure of risk is
-Standard deviation, as it captures the total volatility of that security’s returns, reflecting both systematic and unsystematic risks.
define apt + general formula
model which states the expected return of an asset is a function of multiple macroeconomic risk factors, where each factor has a corresponding sensitivity (beta) and risk premium, and any mispricing will be corrected through arbitrage.
-APT does not specify the specific factors—it only states that asset returns are influenced by multiple systematic risk factors.
-APT does not require the market portoflio as a factor.
making it more adaptable but also more complex and requiring empirical estimation
Studies strongly suggest that the CAPM be abandoned and replaced with the APT T/F
-false
. While APT is a more flexible multi-factor model, CAPM remains widely used in finance due to its simplicity and practical applications. Instead of replacing CAPM, APT serves as an alternative approach for asset pricing.
some emprical findings of the capm
-returns tend to have a positive linear relationship with beta, including a beta squared term not much explnatory power.
-inclusion of controls for unsystematic risk doesnt improve model for explaining past returns.
-Security market line less steeply sloped than in theory
-intercept typically greater than theoretical comparison.
Describe roll 1997 critique of the emprical tests of the capm’s validity.
-argue emperical tests of the capm are limited.
-predictions rely on imprerfect market represernations.
-proxy for market port, indicies, fail to incude all tradeable shares and other assets, weak proxy.
-Cannot accurately observe true mkt port so proxy important for validity.
describe Farma and french 1993 critique of validity of capm.
-studied performance of 2k+ stocks (1941-1990) conclude beta alone cannot explain returns over time,
-adding macroeconomic risk factors assosiated with the secuirty such as firm size, value etc may improve predictions between risk and return.
eg) they found small cap stocks earned higher average returns and stocks with higher book to amrket ratio expereinced hiugher average returns.
strengths and limitations of capm
-Simple & Easy to Use – Requires only one factor (market beta) for asset pricing.
-Clear Theoretical Foundation – Based on well-defined risk-return trade-off.
-emphasises likely impact systematic risk has on expected returns.
-Widely Used in Practice
-ves:
-Unrealistic Assumptions – Assumes a risk-free rate, no transaction costs, and rational investors.
Single Risk Factor – Only considers market risk, ignoring other factors like size, value, and momentum. providing additional factors may better help explain returns
describe estimating the apt + results of estimating both apt and capm
- Identify macroeconomic risk factors
- using historical data estimate the risk premiums lamda i assosiated with each risk factor.
- using historical data estimate the factor sensitivities betai, finding out how sensitibe the securities returns are to the respective factor.
- calculate the expected return using the estimated apt model
-both models yield similar results for some industries, oil gas) however large differences in others (machinery)
-Both models express the importance of divresidfication for reducing un systeamtic rislk.
