Quick Sheet Flashcards
Implementation shortfall
= (execution cost + opportunity cost + fees)/ (total shares in order x decision price)
= paper return - actual return
execution cost = actual sale profit - (shares traded x decision price)
OR = delay costs + trading costs
Execution Cost
= actual sale profit - (shares traded x decision price)
OR
= delay costs + trading costs
Opportunity Cost
= shares remaining unexecuted x (closing price - decision price)
Trading Cost
= actual cost of trade (found sum of all qty x price) - (shares traded x arrival price)
—> arrival and benchmark can be interchanged if specified - assume benchmark is the arrival price
—> trading cost is typically noted in dollars vs arrival cost in bps. If they ask for bps, you divide by the arrival cost
Delay Cost
= (decision price - arrival price) x actual shares traded
Market Adjusted Cost
= arrival cost - (beta x index cost)
–> negative number would indicate a savings
Arrival Cost
= side x [(avg px - arrival price)/ arrival price]
—> arrival cost is noted in bps vs trading cost is dollars
index cost
= side x [(index VWAP - index arrival price)/ arrival price]
Sharpe Ratio
= (asset return - risk free rate) / std. deviation
OR (avg return - min acceptable return) / s.d.
standard deviation
= (asset return - rf rate) / sharpe ratio
OR square root of the variance
Sortino Ratio
= (asset return - risk free rate) / s.d. of negative returns
Target Semideviation
= (asset return - rf rate)/ sortino ratio
Beta
= Covariance/variance
** note that Beta references variance of the BROAD MARKET - if given the s.d. of a sector, note that this is not the s.d. to be plugged into the formula. The s.d. of the market portfolio should be used to calculate variance.
Active Return
= info coef x square root of breadth x s.d. of active return x transfer coef
- info coef: The information coefficient shows how closely the analyst’s financial forecasts match actual financial results. The IC can range from 1.0 to -1.0, with -1 indicating the analyst’s forecasts bear no relation to the actual results, and 1 indicating that the analyst’s forecasts perfectly matched actual results.
- Breadth: The number of truly independent decisions made each year.
- -> if a manager selects ten stocks every month, his breadth is 10 x 12 = 120. If a manager makes a selection every quarter, his breadth is 4
- s.d. of active return: the difference between the benchmark and the actual return aka the active risk
- transfer coef: defined as the correlation between the risk-adjusted alphas and active weights. The TC is an objective measure of how much of the alphas’ information is transferred into a portfolio and is a measure of portfolio construction efficiency
Active Risk
= square root of [ (sum of all :port returns - benchmark returns)^2) / (n-1) ]
–> n = number of return periods
Active Share
= .5 x (sum of all absolute value of weight of security in portfolio - weight in benchmark)
Macaulay Duration
= modified duration x [ 1 + (yield/annual frequency)
Macaulay duration calculates the weighted average time before a bondholder would receive the bond’s cash flows vs modified duration is the price sensitivity to a change in rates
Modified Duration
= macaulay duration / [ 1 + (yield/frequency)
-> change of 20bp increase = -ModDur x .0020
modified duration measures the price sensitivity of a bond when there is a change in the yield to maturity.
–> change in a bonds value given x% interest rate change
Effective Convexity
= [ (P-) + (P+) - (2xPo) ] / ΔCurve^2 x Po)
A second-order effect that describes how a bond’s interest rate sensitivity changes with changes in yield. Effective convexity is used when the bond has cash flows that change when yields change (as in the case of callable bonds or mortgage-backed securities). Similarly, we use the effective convexity to measure the change in price resulting from a change in the benchmark yield curve for securities with uncertain cash flows.
Money Duration vs BPV
money duration = market value x modified duration
BPV = market value x modified duration x .0001
Human Capital
= [ wages x (1+g) x (probability of survival) ] / [ (1+rf rate & any other premiums)^n ]
–> you have to discount each year individually if you are asked to calculate over a span of x years (i.e. 3 years: year one would be discounted at 1.01^1 year 2 1.01^2 year 3 1.01^3 etc)
Core Capital needs
= [ spending x (1+g) x (probability of survival in mortality table) ] / [ (1+rf rate & any other premiums)^n ]
Grinold Kroner Model
= div yield - change in shares outstanding + nominal earnings growth + % change in P/E multiplier
- -> expected income return = dividend yield - change in shares outstanding
- *make sure to consider whether shares are being reduced - i.e. if reducing by 1% you would add dividend yield + 1%
- -> Earnings growth rate = expected inflation + expected real total corporate earnings growth rate
- -> %ΔP/E Multiplier = expected repricing return
- -> dividend yield = dividend / price
Expected income return
= div yield - change in shares outstanding
Taylor rule
target fed funds nominal rate = neutral fed funds rate + [.5 (exp GDP growth - trend GDP growth) + [.5 (exp inflation - target inflation) ]
Rolling Yield
= div yield + roll down return
–> roll down return = (ending price - beg price) / beg price
horizon yield
= div yield + roll down return + expected change on price based on yield view + fx return - credit losses
expected change in price based on yield view =
[–MD** × ΔYield] + [0.5 × Convexity × (ΔYield)^2]
** if MD is expected to change, use the expected effective duration for portfolio at the horizon. Duration can be swapped out for MD
Expected change in price based on yield view
= (-MD x change in yield) + (.5 x convexity x (change in yield^2))
Credit vs deduction tax method
credit = max or source and resident tax rate deduction = source + [resident (1-source tax rate)]
FV assuming deferred capital gains
= portfolio value x [ (1+return)^n x (1- tax rate) + tax rate - (1-B) x tax rate ]
-> B = cost basis / current value
OR you can do the algebra -
You have $1 million growing for 20 years at 5%. The tax basis is $200,000, and the capital gains tax rate is 25%. What’s the final value?
Future value before taxes = $1,000,000(1.05)²º = $2,653,298.
Capital gain = $2,653,298 − $200,000 = $2,453,298.
Capital gains tax = $2,453,298 × 25% = $613,324.
Future value after taxes = $2,653,298 − $613,324 = $2,039,974.
of futures contracts needed to hedge a portfolio from change in fx rates
= currency to be exchange / futures contract size
- > futures contract size = futures price x multiplier
- -> this is assuming that the portfolio is seen as an asset with no liabilities held against it
of futures contracts needed to remove a duration gap
= (liability BPV - asset BPV ) / BPV of futures
of futures required to hedge a portfolio from a change in interest rates
= (- BPV of portfolio / BPV of treasury) x conversion factor
conversion factor = ($mkt value of portfolio / $ futures contract)
BPV of Portfolio = Mod Dur of Portfolio x .0001 x Market Value of Portfolio
BPV of treasuries = Mod Dur of Tres. futures x .0001 x Market Value of Futures
Market Value of Futures = (Contract price / 100) x $100,000
Cost of applying leverage
= [ value x (1- 1/leverage ratio) x borrow rate ] / value of portfolio
–> i.e. if 3x levered than 1/3
return of a leveraged portfolio
= return + [ (debt/equity) x (return - borrow rate)]
excess return
= (OAS or Z spread x time held/1 year) - (ΔSpread x spread duration)
- (probability of default x credit loss x time held/1 year)
ex: find excess spread of a bond with an A2 rating, 5.25 effective spread duration, 3.5% YTM, 100 OAS spread, 0.25 probability of default and 205 estimated loss severity if there is a 30% tightening in yield spreads
ER = .01 - [5.25 x (0.01 x -0.3)] - (0.0025 x .2) = 2.53%
- -> *spread duration = duration x spread / current OAS spread
- -> i.e. 30% recovery rate implies a 70% loss severity
- -> probability of default = typically default yield on bond rating
- -> change in spread when a 30% tightening change is expected and the current OAS or Z spread is 1% = -30bps
To find excess return of a bond portfolio:
apply individual allocation weights to each respective excess return and sum them all up
i.e.
Portfolio EXR ≈ (70% × 0.05%) + (15% × 0.04%) + (10% × 0.10%) + (5% × 0.23%) = 0.06%
spread duration
= (duration x spread) / current spread
—> duration calculated when treasures = 0
For example, you have a portfolio with:
$1,000,000 market value of 9-year Treasury Notes, with a modified duration of 7 years
$2,000,000 market value of a 7-year corporate bond with a modified duration of 5 years
$3,000,000 market value of a 2-year corporate with a modified duration of 1.8 years
The spread duration of the portfolio is:
($1,000,000/$6,000,000) × 0 years + ($2,000,000/$6,000,000) × 5 years + ($3,000,000/$6,000,000) × 1.8 years
= 2.57 years.
For comparison, the modified duration of the portfolio is:
($1,000,000/$6,000,000) × 7 years + ($2,000,000/$6,000,000) × 5 years + ($3,000,000/$6,000,000) × 1.8 years
= 3.73 years.
credit loss
= probability of default x loss severity
information ratio
= active return / active risk
can also be calculated:
IR = ( portfolio return - benchmark return ) / tracking error
estimated LT return of real estate
= cap rate + NOI Growth rate
cap rate = NOI / prop value
emerging market red flags
- debt to GDP greater than 70-80% (minor)
- persistent annual real growth less than 4%
- current account deficit greater than 4%
- foreign debt greater than 50% of GDP or greater than 200% of current account receipts
- fx reserves less than 100% of short term debt (should be greater than 200%)
Modified dietz
= (ending port value - begin portfolio value - all cash flows
) / portfolio value at beg + (sum of all cf x their weight)
Fund with highest probability of meeting a target return
= (exp return - target) / s.d.
of contracts needed to achieve a target duration
how to calculate the notional principal of an interest rate swap to increase (or reduce) portfolio duration (target duration)?
of contracts = [(tgt BPV - current BPV of portfolio) / BPV of futures contract] x conversion factor
conversion factor = ($mkt value of portfolio / $ futures contract)
Notional value = [(target mod dur - current portfolio mod dur)/ (mod dur of swap)] x market value of portfolio
- BPV target = Mod Dur of target x .0001 x market value of target
- BPV of Portfolio = Mod Dur of Portfolio x .0001 x Market Value of Portfolio
- BPV of treasuries = Mod Dur of Tres. futures x .0001 x Market Value of Futures
- Market Value of Futures = (Contract price / 100) x $100,000
target portfolio beta
= [ (tgt beta - current beta) / beta of stock index] x ($mkt value of portfolio / $ futures contract)
conversion factor = ($mkt value of portfolio / $ futures contract)
–> note if trying to hedge a 10M USD Portfolio that you’d like to invest in a brazilian equity index and the USD/BRL fx rate is 4.2 - the portfolio value is 42M BRL
Futures contract value = futures price x multiplier
example:
you want to hedge 10M FTSE exposure to 8M SPY exposure. UK FTSE beta to spy is 1.1, S&P 500 E-Mini Futures One-Month Contract Price is US$3,000 w/ $50 multiplier and FTSE IDX One-Month Futures Contract Price is £7,300 with $10 multiplier
number of futures to short FTSE = ( (0-1.1)/1 ) x (8M / (7.3k x 10)) = -121
Number of SPY futures to buy = ((1-0)/1) x (10M / (3k x 50))
How to calculate probability of joint survival
Prob (Joint survival) = probabilty of husband survives + probability wife survives - prob husband survives x prob wife survives
FV of a tax free gift
= PV x [ (1 + (pretax return of recipient x (1 - recipients tax rate))] ^n
FV of a bequest
= [present value x [ 1+ (pre tax return of estate x (1 - estate tax rate)]^n] x (1- inheritance tax)
how to determine if to gift now or bequest
FV of gift/ FV of bequest
Treynor Ratio
= (portfolio return - rf rate) / portfolio beta
growth adjusted discount rate used to find the PV of a surviving spouses living expenses
= [(1+r)/(1+g)] -1
how to compare g-spreads
if given two bonds that don’t have matching maturities - interpolate them to average a maturity of the bond in question
find the yield of the new bond using weight x yield + weight x yield and compare this to the fair g spread (which will be given)
variance notional
variance notional = vega notional / ( 2x volatility strike price)
–> vega notional is the approximate gain or loss for a 1% change in volatility for a variance swap
—> in this instance, the strike price is not squared. Ie if they say annual vol is 17%, then you take Vega notional/ 34
payoff to a variance buyer
= variance notional x PV of interest factor x (realized variance - variance strike)
–> if volatility strike is quoted in s.d. it needs to be squared ( vol = s.d. )
i.e. what is the value of a variance swap 6 mos after initiation?
Vega notional = 4M
strike of variance swap = 17%
6 month realized vol = 21%
fair strike of 6 month var swap after 6 mos: 18%
annual interest rate = 2.02%
step 1: find variance notional = vega notional / 2 x strike vol
= 4M / (17 x 2) = $117,647
= $117,647M x [1/(1+ (0.0202 x (6mos/12))] x (6/12mos at 21^2 + 6/12mos x 18^2) - (17^2)
= 4M x [0.99 x (0.5 x 441) + (0.5 x 324) - 289] = $10,889,995
slippage cost
the difference between an execution price and the mid point of the bid/ask spread when the order was entered into the market
Turnover ratio
The greater of portfolio purchases OR sales / average portfolio value
equation to find CD Price
CD price = 1 + [(Fixed Coupon − CDS Spread) × EffSpreadDurCDS]
standard fixed CDS spreads are 1% for investment-grade issuers and 5% for high-yield issuers.
ex: An active portfolio manager seeking to purchase single-name CDS protection observes a 1.75% 10-year market credit spread for a private investment-grade issuer. The effective spread duration is 8.75 and CDS basis is close to zero.
What should the protection buyer expect to pay or receive to enter a new 10-year CDS contract?
CD price = 1+ ((Fixed Coupon − CDS Spread) × EffSpreadDurCDS)
= 1+ [ (.01-0.0175) x 8.75] = .934375
–> The fixed notional amount upon contract initiation; the initial CDS price is therefore 93.4375 per 100 of notional with a CDS spread of 175 bps.
How to find the change in portfolio value given an effective duration and change in yield?
change in portfolio value = (−EffDur × ΔYield)
- -> this also gives you the instantaneous holding period return
- -> effective duration is the sum of all of the key rate durations along a curve
i. e. if the sum of all of portfolio’s holding key rates add up up 6.115 and you expect a yield change of 50bps - the portfolio value would decrease by -6.115 x 0.005 = -0.030575 or 3.06%
How to find the notional value (in US dollar millions) of the interest rate swap necessary to match the duration profile of a given liability?
notional value of swap = (mod dur of liability - mod duration of assets / mod dur of the swap) x plan assets
–> keep in mind when calculating the mod dur of the assets that this should only consider the FI portion - i.e. if a plans assets are worth 100M with a mod dur of 7 but only holds 60M in FI and the rest in equities, the mod duration of assets is found 7 x 60% = 4.2
Net asset value and how to calculate the expected net asset value at the end of the year?
Value established at the end of each trading day based on the fund’s valuation of all existing assets minus liabilities, divided by the total number of shares outstanding.
Expected NAV = [Prior-year NAV × (1 + Growth rate) + Capital contributions – Distributions)] × (1 + Growth rate)
Expected distribution = [Prior-year NAV × (1 + Growth rate)] × (Distribution rate)
Capital contributions in period t = percentage to be called in period x (committed capital - capital previously called)
How to find the attribution of a certain sector’s security selection to an overall portfolio
–> i.e. a certain regions attribution to the overall portfolio
- calculate the allocation effect:
Allocation = (factor weight in port - factor weight in Benchmark)(benchmark factor return – total return of benchmark)
–> evaluates a PMs ability to effectively allocate capital among various segments - Calculate the Selection + interaction effect
Selection effect = benchmark weight x (Factor return in Port - factor return in benchmark)
Interaction effect = (factor weight in port - factor weight in Benchmark) x (Factor return in Port - factor return in benchmark)
- Sum the two up to find the total. The largest negative number, that sector’s security selection detracted from the portfolio the most, and positive numbers - security selection in those sectors added to performance
Modified duration of shareholders capital
MD shareholders capital = MD of equity
equity MD = asset MD - [ liability MD x (change in interest / change in yield)
asset MD = (A/E) x asset MD
i.e. if equity capital ratio = 10% then 1/.1 is asset weight and (1/.1 - 1) is portion of asset that is financed (liability weight)
if MD of A is 3 years and MD of liability is 2 years - a 70bp movement in liabilities per 1% move in assets
= (1/.1) x 3 - [ (1/.1 - 1) x 2 x .7] = 17.4
constituents of portfolio return
portfolio return = market index return + manager style return + active management return
return due to active management = portfolio return - benchmark
semideviation
semideviation = square root of [ (sum of all average - value)^2] / n]
n= the number of observations below the mean average = the mean or target value of a set of data
–> Semi-deviation will reveal the worst-case performance to be expected from a risky investment. Semi-deviation is a method of measuring the below-mean fluctuations in the returns on investment. Semi-deviation is an alternative to the standard deviation for measuring an asset’s degree of risk. Semi-deviation measures only the below-mean, or negative, fluctuations in an asset’s price. This measurement tool is most often used to evaluate risky investments.
How to find the estimated return for an asset using the Singer–Terhaar approach to the international extension of the CAPM
How to find the estimated return for an asset using the Singer–Terhaar approach to the international extension of the CAPM
Step 1:
Find the fully segmented risk premium = s.d. x sharpe ratio
Step 2:
find the fully integrated risk premium (the industry risk premium) = s.d. x sharpe ratio x correlation
Step 3:
Combine the two, considering the degree of integration = fully integrated risk premium x integration portion + fully segmented risk premium x non integrated portion
–> non integrated portion = 1 - fully integrated portion
Step 4: add the risk free rate and any other given premiums (i.e. if given an illiquidity premium, add that)
note: sharpe ratio = return vs risk =
Expected risk premium for overall portfolio /
Expected standard deviation for the portfolio
–> the portfolio or industry with the highest estimated return would be the most attractive (expected return reflects compensation for systematic risk)
–> In integrated financial markets, domestic investors can buy foreign assets and foreign investors can buy domestic assets
–> All else being equal, the Singer–Terhaar model implies that when a market becomes more globally integrated (segmented), its required return should decline (rise) as a reflection of its risk. As prices adjust to a lower (higher) required return, the market should deliver an even higher (lower) return than was previously expected or required by the market. Therefore, the allocation to markets that are moving toward integration should be increased. If a market is moving toward integration, its increased allocation will come at the expense of markets that are already highly integrated. This will typically entail a shift from developed markets to emerging markets.
Sarah Ko, a private wealth adviser in Singapore, is developing a short-term interest rate forecast for her private wealth clients who have holdings in the US fixed-income markets. Ko needs to understand current market expectations for possible upcoming central bank (i.e., US Federal Reserve Board) rate actions. The current price for the fed funds futures contract expiring after the next FOMC meeting is 97.175. The current federal funds rate target range is set between 2.50% and 2.75%.
Q: Explain how Ko can use this information to understand potential movements in the current federal funds rate.
(implied rate - current mid point) / (new mid point - current mid point
First, Ko knows that the Federal Fund Effective (FFE) rate implied by the futures contract price of 97.175 is 2.825% (= 100 – 97.175). This is the rate that market participants expect to be the average federal funds rate for that month.
Second, Ko should determine the probability of a rate change. She knows the 2.825% FFE rate implied by the futures signals a fairly high chance that the FOMC will increase rates by 25 bps from its current target range of 2.50%–2.75% to the new target range of 2.75%–3.00%. She calculates the probability of a rate hike as follows:
(2.825%−2.625%) / (2.875%−2.625%) =0.80, or 80%
(implied rate - current mid point) / (new mid point - current mid point)
–> 2.625 is mid point, 2.825 is implied rate, 2.875 new mid point
Ko can now incorporate this probability of a Fed rate hike into her forecast of short-term US interest rates.
Herfindahl–Hirschman Index (HHI)
The HHI measures stock concentration risk in a portfolio. It is calculated by squaring the market share of each firm competing in a market and then summing the resulting numbers. It can range from close to zero to 10,000.
The number of effective stocks in a portfolio is 1/HHI
i.e. if there was only one firm in the market, its market cap would be 100% of the market share so 100^2 = HHI of 10,000 vs if there were 3 firms, with firm 1 being 5% of the market share, firm 2 being 60% of the market share and firm 3 being 35% of the market share, the HHI would be .05^2 + .60^2 + .35^2 = .4850 and the effective number of stocks is 2.06
–> keep in mind that if a stock is 30% of the equity exposure but only 70% of the portfolio is in equities, the contribution to HHI would be (.3*.7)^2 = .21^2 = .0441
- -> regulators reference this when considering mergers etc (high score = more concentration in the market and mergers and acquisitions might be considered bad for the consumer due to concentration of firm power and influence)
- -> Using the HHI, one can estimate the effective number of stocks, held in equal weights, that would mimic the concentration level of the respective index. The effective number of stocks for a portfolio is calculated as the reciprocal of the HHI. The HHI is 0.0286; the reciprocal (1/0.0286) is 34.97. Therefore, the effective number of stocks to mimic the US large-cap benchmark is approximately 35.
Fama French Model
Expected ROR =
risk free rate + (factor coefficient) Market risk premium+ (factor coef) SMB + (factor coef) HML
where:
- Market risk premium: return market - risk free
- SMB (Small Minus Big) = Historic excess returns of small-cap companies over large-cap companies
- –> positive value indicates small cap tilt, negative would indicate large cap tilt
- HML (High Minus Low) = Historic excess returns of value stocks (high book-to-price ratio) over growth stocks (low book-to-price ratio)
- -> positive value would indicate value tilt, negative value would indicate a growth tilt
the 5 BB&K Client Classifications
Individualists – They are confident and careful. They generally do not go to a consultant to manage their investments but do it by themselves. Individualists are unlikely to easily take advice without doing their own analysis but are pleasant to work with because they process information rationaly
Adventurers – Adventurers generally go for only big bets and end up with concentrated portfolios. They have the resources to do so and are willing to take risks. The investment made by this type of investors are generally focused and not diversified. They are difficult to work with.
Celebrities – Celebrities are those that are swayed too much by the trend and do not have any expertise or opinion about investments. However, not having the expertise and the confidence required to manage the portfolio on their own, they approach investment managers frequently.
–> celebrities hold opinions about some things but may be willing to take advice about investing. They only recognize their investment limitations to a certain extent.
Guardians – Guardians are both anxious and careful. Lacking confidence in themselves, they approach investment counsels. They generally emphasize on safety of the capital while making the investments and a significant proportion of their investments is generally devoted to government securities and guaranteed return investments. People tend to become guardians as they age.
Straight arrows – The average investor. These are halfway between complete confidence and anxiety, and extreme carefulness and impetuousness. Straight arrows are sensible and secure. They fall near the center of the graph. They are willing to take on some risk in the expectation of earning a commensurate return.
How to calculate the change in convexity given the sale of one bond in exchange for the purchase of another?
1) find the mixture of new bonds which matches the duration of the bond being replaced. I.e. say you are replacing 10 year bonds with a mixture of 3 year and LT bonds
duration 10 = (duration 3 X weight) + duration LT X (1-weight)
weight = allocation to 3 year bonds
2) after you have the duration matching weights, the G/L in convexity is calculated:
Gain in convexity = (Weight of the 3-year) × (Convexity of the 3-year) + (Weight of the long-term bond) × (Convexity of the long-term bond) – (Weight of the 10-year) × (Convexity of the 10-year)
How to calculate the equivalent value $ one should allocate from one bond to another in order to maintain a duration neutral position
in order to maintain a neutral duration position
- find the money duration of the bond being exchanged out = $value / PVBP
- exchange that money duration value / PBVP of the bond to be switched in
- -> assuming PVBP is given in millions, divide the value of the bond position by 1M to make sure you are comparing apples to apples
i.e. if you are exchanging 150M LT bonds with a PVBP ($million) of 1,960 for an unknown value of 2 year bonds with a PVBP of 197 math = 150M/1M = 150 150 x 1960 = 294,000 294,000/197 = 1,492.39 1492.39 x 1M = $1.492M
how to determine the inflation adjusted annual cash flow generated by a sub portfolio
inflation adjusted annual cash flow generated by a sub portfolio =
[ amount invested x (minimum expected return - inflation) ] /
[ 1 - [ (1+inflation)/(1+expected return) ]^n ] x (1+expected return)
–> minimum adjusted return should be the rate expected at the appropriate horizon and probability of success (you will be given multiple)
Risk Parity asset allocation
—> how to calculate the risk attributable from each asset?
weight of asset i x covariance of asset i with the portfolio =
variance of the portfolio / number of assets)
- how to determine the value needed to invest given the desire to buy a 5M house in 5 years earning 4.4% annually
- investment needed to keep up with current annual expenditures of $100,000 for the next 10 years, assuming annual inflation of 3% from Year 2 onward and 2.2% earned annually
how to determine the value needed to invest given the desire to buy a 5M house in 5 years earning 4.4% annually
N=5 i=4.4 PMT = 0 FV=-5,000,000 cpt PV
PV= $100,000/(1.022) + [ ($100,000(1.03)^1) / (1.022^2) ] + [($100,000(1.03)^2) / (1.02^3)] + etc PV = $1,013,670 (or $1.01 million)
Long Term Economic Growth Rate
long term economic growth rate = population growth + labor force participation + new cap spending + total factor productivity
–> each input is stated in terms of growth (percentage change)
Risk Premium Approach to expected bond return
expected bond return =
risk-free rate + term premium + credit premium + liquidity premium
Appraisal Ratio
An appraisal ratio is a ratio used to measure the quality of a fund manager’s investment-picking ability.
AR = alpha / s.d. of the residual/unsystematic risk
s.d. of the residual/unsystematic risk = the standard error of regression
How to best implement a minimum-variance hedge vs another currency
The minimum variance hedge ratio, also known as the optimal hedge ratio, is a formula to evaluate the correlation between the variance in the value of an asset or liability and that of the hedging instrument that is meant to protect it.
minimum variance hedge =
portfolio value x [correlation x (s.d. of domestic currency/ s.d. of domestic to foreign currency)]
–> hedge position is ratio above x asset value
An investor is considering the portfolio impact of a new 12-year corporate bond position with a $75M face value, a 3.25% coupon, current YTM of 2.85, modified duration of 9.887 and a price of 104.0175 per 100 of face value
What is the approximate VaR for the bond position at a 99% confidence interval (equal to 2.33 standard deviations) for one month (with 21 trading days) if daily yield volatility is 1.50% and returns are normally distributed?
- The expected change in yield based on a 99% confidence interval for the bond and a 1.50% yield volatility over 21 trading days equals
16 bps = (1.50% × 2.33 standard deviations × √.0021)
= daily yield vol x sd of confidence interval x square root of day
= .015 x .0233 x 4.583 - We can quantify the bond’s market value change by multiplying the familiar (–ModDur × ∆Yield) expression by bond price to get
$1,234,105 = $75 million × 1.040175 ⨯ (–9.887 × .0016).
bond price = 75M x (104.0175/100)
change in yield = (–9.887 × .0016)
percent capital gain exposure (PCGE)
= pnl / total assets including pnl
i.e. if the fund starts with $2000 and gains $500 before losing $100 than the PCGE is 400/2400 = 17%
How to find the EXCESS return due to active factor weighting
–> aka active effect
- find the return contribution of the factor in the portfolio (portfolio sector weight x portfolio sector return)
- find the benchmark sector contribution (benchmark sector weight x benchmark sector return)
- subtract value one (portfolio rector attribution) - value two (benchmark sector attribution)
- repeat this for all of the factors where there is a difference in sector weighting between the portfolio and then benchmark and sum them up to find total excess return
(note: BHB and BF find total return)
How to evaluate a utility function
U= E(R)m - .005 x Y x sdm^2
i.e. if elected return is 8% and st of portfolio is 12% with utility function of 4 (all given)
U= E(R)m - .005 x Y x sdm^2
i.e. if elected return is 8% and st of portfolio is 12% with utility function of 4 (all given)
8- 0.005 x 4 x 12^2 = 5.12 or 5.12%
rolldown return of a CD strategy
i.e. a person wants to see what their overall pnl would be if they bought a 10 year bond and held it for 1 year
gain would be coupon income + price appreciation
price appreciation is (px at 10year - px at 9 years) / px at 10 years
CD price = 1 + [(fixed coupon - CDS spread) x eff spread of CDS]
if you are not given a spread, you can interpolate two bonds - i.e. a 5 year bond with a 1.5% spread and a 10 year bond with a 2% spread - to find the equivalent 9 year spread
9 = (10 x S) + (5 x (1-S))
S = .8 so an equivalent spread is 80% 10 year at 2% and 20% 5 year at 1.5 = 1.9%
to find notional value of return you would =
roll down return + coupon x portfolio value
Of the 4 different types of behavioral investor types, list the behavioral biases prevalent (emotional and cognitive) and level of risk aversion of a friendly follower
friendly follower: primarily cognitive (responsive to data and new information and will tend to follow the advice of their adviser - aka passive moderate)
- low to moderate risk tolerance. More Passive. They often want to be in the latest, most popular investments without regard to suitability for long-term goals.
- emotional biases: regret aversion, status quo
- cognitive biases: availability, hindsight, framing bias. A friendly follower will typically respond to data backed investment advice
Of the 4 different types of behavioral investor types, list the behavioral biases prevalent (emotional and cognitive) and level of risk aversion of a independent individualist
independent individualist - primarily cognitive
- wiling to take on slightly more risk with a growth investment style. More active investor (aka active growth)
- emotional biases: overconfidence and self attribution
- cognitive biases: conservatism, availability, confirmation, representative bias
The independent individualist is most difficult to understand, they are independent risk taker, they should not have conservatism bias
Of the 4 different types of behavioral investor types, list the behavioral biases prevalent (emotional and cognitive) and level of risk aversion of a active accumulator
active accumulator (aka active aggressive): primarily emotional
- high risk tolerance and aggressive investment style. More active investor leads to higher turnover. Typically high net worth.
- emotional biases: overconfidence, self control
- cognitive biases: illusion of control
Of the 4 different types of behavioral investor types, list the behavioral biases prevalent (emotional and cognitive) and level of risk aversion of a passive preserver
passive preserver (aka guardian):
primarily emotional –
- low risk tolerance and conservative investment style. More passive. Does better with big picture information
- emotional biases: endowment, loss aversion, status quo, regret aversion
- cognitive biases: mental accounting and anchoring and adjustment
Behavioral biases
A tendency to behave in a way that is not strictly rational.
- -> Behavioral biases can be
1. cognitive - result from incomplete information or inability to analyze - i.e. belief perseverance biases (representativeness, illusion of control, conservatism, confirmation and hindsight bias) and Information processing biases (Framing bias, anchoring and adjustment, mental accounting, availability) (R.I.C.C.H.F.A.M.A)
2. emotional - spontaneous reactions that affect how individuals see information - i.e. loss aversion bias, overconfidence bias, self control bias, status quo bias, endowment bias, regret-aversion bias) (L.O.S.S.E.R)
In its quarterly policy and performance review, the investment team for the Peralandra University endowment identified a tactical allocation opportunity in international developed equities. The team also decided to implement a passive 1% overweight ($5 million notional value) position in the asset class. Implementation will occur by either using an MISC EAFE Index ETF in the cash market or the equivalent futures contract in the derivatives market.
The team determined that the unlevered cost of implementation is 27 basis points in the cash market (ETF) and 32 bps in the derivatives market (futures). This modest cost differential prompted a comparison of costs on a levered basis to preserve liquidity for upcoming capital commitments in the fund’s alternative investment asset classes. For the related analysis, the team’s assumptions are as follows:
Investment policy compliant at 3 times leverage
Investment horizon of one year
3-month Libor of 1.8%
ETF borrowing cost of 3-month Libor plus 35 bps
Q. Recommend the most cost-effective strategy. Justify your response with calculations of the total levered cost of each implementation option.
As the lower cost alternative, the endowment’s investment team should implement the 1% overweight position using futures.
The additional cost of obtaining leverage for each option is as follows:
= [ notional value of investment x (1 - (1/leverage ratio) x ETF Borrowing rate ] / notional value of investment
–> for both, if you are given an unlevered cost of implementation, add that to the cost of obtaining the leverage and compare
ETF: ($5 million × 0.6667 × 2.15%) / $5 million = 1.43% (or 143 bps) and
Futures: ($5 million × 0.6667 × 1.80%) / $5 million = 1.20% (or 120 bps),
where the inputs are derived as follows:
- 6667 reflects the 3 times leverage factor - 66.67% borrowed and 33.33% cash usage (1-1/leverage ratio)
- 15% reflects the ETF borrowing rate (3-month Libor of 1.80% + 35 bps), and
- 80% reflects the absence of investment income offset (at 3-month Libor) versus the unlevered cost of futures implementation.
The total levered cost of each option is the sum of the unlevered cost plus the additional cost of obtaining leverage:
ETF: 27 bps + 143 bps = 170 bps and
Futures: 32 bps + 120 bps = 152 bps.
This 18 bps cost advantage would make futures the appropriate choice for the endowment’s investment team.
How to calculate the gain on a variance swap:
Regan next suggests that Monatize could alternatively hedge Portfolio B using variance swaps. Monatize’s CFO asks Regan to calculate what the gain would be in five months on a purchase of $1,000,000 vega notional of a one-year variance swap on the S&P 500 at a strike of 15% (quoted as annual volatility), assuming the following:
Over the next five months, the S&P 500 experiences a realized volatility of 20%;
At the end of the five-month period, the fair strike of a new seven-month variance swap on the S&P 500 will be 18%; and
The annual interest rate is 1.50%.
Values for the inputs are as follows:
Volatility strike on existing swap = 15
Variance strike on existing swap = 15^2 = 225
Variance Notional = (Vega Notional/2×Strike) = $1,000,000/ 2×15=$33,333.33
RealizedVol(0,t)2 = 20^2 = 400 x (5/12)
ImpliedVol(t,T)2= 18^2 = 324 x (7/12)
discount rate for interest = 1/ [1+[1.50%(7/12)] = 0.991326, which is the present value interest factor after five months (i.e., discounting for seven remaining months, from t to T), where the annual interest rate is 1.50%.
Thus, the value of the swap in five months is calculated as follows:
VarSwapt=$33,333.33×0.991326 × {(5/12)× 400 + ((12−5)/12) × 324 − 225}=$4,317,774.59
Given that Monatize would be long the swap, the mark-to-market value would be positive (i.e., a gain) for Monatize, equal to $4,317,775.
Actual contribution to total risk
ACTR = weight x marginal contribution to total risk
Marginal contribution to total risk
MCTR = beta of sector x portfolio standard deviation
A funds assets are split into two categories and 56% of the fund is allocated to return seeking assets with the remainder in treasury bonds. The asset portfolio has a duration of 13.
If a defined benefit pension is underfunded with a deficit of 3.5B, a current PBO of 16B and a duration of 12, how many futures are needed to close the duration gap?
CTD Bond: Price: 96.32 Conversion factor:.8017 Duration: 14.95 Contract size:100,000 Futures price: 118.5
- Find BPV if liabilities
= 16B x 12 x .0001 = 19.2M - Find BPV of assets
If the fund is underfunded by 3.5B assets must equal 12.5B
Value of the bond portion = 12.5B x .34 = 5.5B
BPV a = 5.5B x 13 x .0001 = 7.15M - BPV of CTD Bond
= 14.95 x .9632 x 100k x .0001
[(BPV L - BPV A) / BPV CTD ] x conversion factor
= (12.05M/144) x 0.8017 = 67087 futures
how to calculate the traders added value
added value = arrival cost - estimated pre trade cost
arrival cost = side * [ (average - arrival) / arrival] x .0001
Carhart model
Carhart model = Fama French + Momentum
Expected return =
risk free rate + (factor coefficient) Market risk premium+ (factor coef) SMB + (factor coef) HML + (factor coef) momentum
cognitive biases
- REPRESENTATIVENESS BIAS (base rate neglect or sample size neglect- cognitive bias in which people tend to classify new information based on past experiences and classifications. If-then stereotype heuristic used to classify new information)
- ILLUSION OF CONTROL BIAS (the tendency to overestimate one’s control over events)
- CONSERVATISM BIAS (where people emphasize original, pre-existing information over new data. This can make decision-makers slow to react to new, critical information and place too much weight on base rates.)
- CONFIRMATION BIAS (looking for what confirms one’s beliefs)
- HINDSIGHT BIAS - selective memory of past events, remember correct views and forget errors
- FRAMING BIAS - viewing info differently depending on how it is received
- ANCHORING AND ADJUSTMENT (the tendency to reach a decision by making adjustments from an initial position, or “anchor”)
- MENTAL ACCOUNTING BIAS - each goal is considered separately
- AVAILABILITY (the probability of events is influenced by the ease with which examples of the event can be recalled)
emotional biases
L oss-Aversion
O verconfidence and familiarity (illusion of knowledge)
S tatus Quo (preference for no change)
S elf-control
E ndowment (a tendency to ask for much more money to sell something than one would be willing to pay to buy it)
R egret aversion
PIC multiple
= since inception paid in capital / cumulative committed capital
alpha
alpha = fund return - [(beta A x factor coefficient A) + (beta B x factor coefficient B) + (beta C x factor coefficient C) …]]
