Quick Sheet

Question 1

Implementation shortfall

Answer 1

= (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

Question 2

Execution Cost

Answer 2

= actual sale profit - (shares traded x decision price)
OR
= delay costs + trading costs

Question 3

Opportunity Cost

Answer 3

= shares remaining unexecuted x (closing price - decision price)

Question 4

Trading Cost

Answer 4

= 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

Question 5

Delay Cost

Answer 5

= (decision price - arrival price) x actual shares traded

Question 6

Market Adjusted Cost

Answer 6

= arrival cost - (beta x index cost)

–> negative number would indicate a savings

Question 7

Arrival Cost

Answer 7

= side x [(avg px - arrival price)/ arrival price]

—> arrival cost is noted in bps vs trading cost is dollars

Question 8

index cost

Answer 8

= side x [(index VWAP - index arrival price)/ arrival price]

Question 9

Sharpe Ratio

Answer 9

= (asset return - risk free rate) / std. deviation

OR (avg return - min acceptable return) / s.d.

Question 10

standard deviation

Answer 10

= (asset return - rf rate) / sharpe ratio

OR square root of the variance

Question 11

Sortino Ratio

Answer 11

= (asset return - risk free rate) / s.d. of negative returns

Question 12

Target Semideviation

Answer 12

= (asset return - rf rate)/ sortino ratio

Question 13

Beta

Answer 13

= 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.

Question 14

Active Return

Answer 14

= 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

Question 15

Active Risk

Answer 15

= square root of [ (sum of all :port returns - benchmark returns)^2) / (n-1) ]

–> n = number of return periods

Question 16

Active Share

Answer 16

= .5 x (sum of all absolute value of weight of security in portfolio - weight in benchmark)

Question 17

Macaulay Duration

Answer 17

= 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

Question 18

Modified Duration

Answer 18

= 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

Question 19

Effective Convexity

Answer 19

= [ (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.

Question 20

Money Duration vs BPV

Answer 20

money duration = market value x modified duration

BPV = market value x modified duration x .0001

Question 21

Human Capital

Answer 21

= [ 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)

Question 22

Core Capital needs

Answer 22

= [ spending x (1+g) x (probability of survival in mortality table) ] / [ (1+rf rate & any other premiums)^n ]

Question 23

Grinold Kroner Model

Answer 23

= 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

Question 24

Expected income return

Answer 24

= div yield - change in shares outstanding

Question 25

Taylor rule

Answer 25

target fed funds nominal rate = neutral fed funds rate + [.5 (exp GDP growth - trend GDP growth) + [.5 (exp inflation - target inflation) ]

Question 26

Rolling Yield

Answer 26

= div yield + roll down return

–> roll down return = (ending price - beg price) / beg price

Question 27

horizon yield

Answer 27

= 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

Question 28

Expected change in price based on yield view

Answer 28

= (-MD x change in yield) + (.5 x convexity x (change in yield^2))

Question 29

Credit vs deduction tax method

Answer 29

credit = max or source and resident tax rate 
deduction = source  + [resident (1-source tax rate)]

Question 30

FV assuming deferred capital gains

Answer 30

= 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.

Question 31

of futures contracts needed to hedge a portfolio from change in fx rates

Answer 31

= 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

Question 32

of futures contracts needed to remove a duration gap

Answer 32

= (liability BPV - asset BPV ) / BPV of futures

Question 33

of futures required to hedge a portfolio from a change in interest rates

Answer 33

= (- 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

Question 34

Cost of applying leverage

Answer 34

= [ value x (1- 1/leverage ratio) x borrow rate ] / value of portfolio

–> i.e. if 3x levered than 1/3

Question 35

return of a leveraged portfolio

Answer 35

= return + [ (debt/equity) x (return - borrow rate)]

Question 36

excess return

Answer 36

= (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%

Question 37

spread duration

Answer 37

= (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.

Question 38

credit loss

Answer 38

= probability of default x loss severity

Question 39

information ratio

Answer 39

= active return / active risk

can also be calculated:
IR = ( portfolio return - benchmark return ) / tracking error

Question 40

estimated LT return of real estate

Answer 40

= cap rate + NOI Growth rate

cap rate = NOI / prop value

Question 41

emerging market red flags

Answer 41

1. debt to GDP greater than 70-80% (minor)
2. persistent annual real growth less than 4%
3. current account deficit greater than 4%
4. foreign debt greater than 50% of GDP or greater than 200% of current account receipts
5. fx reserves less than 100% of short term debt (should be greater than 200%)

Question 42

Modified dietz

Answer 42

= (ending port value - begin portfolio value - all cash flows
) / portfolio value at beg + (sum of all cf x their weight)

Question 43

Fund with highest probability of meeting a target return

Answer 43

= (exp return - target) / s.d.

Question 44

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)?

Answer 44

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

Question 45

target portfolio beta

Answer 45

= [ (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))

Question 46

How to calculate probability of joint survival

Answer 46

Prob (Joint survival) = probabilty of husband survives + probability wife survives - prob husband survives x prob wife survives

Question 47

FV of a tax free gift

Answer 47

= PV x [ (1 + (pretax return of recipient x (1 - recipients tax rate))] ^n

Question 48

FV of a bequest

Answer 48

= [present value x [ 1+ (pre tax return of estate x (1 - estate tax rate)]^n] x (1- inheritance tax)

Question 49

how to determine if to gift now or bequest

Answer 49

FV of gift/ FV of bequest

Question 50

Treynor Ratio

Answer 50

= (portfolio return - rf rate) / portfolio beta

Question 51

growth adjusted discount rate used to find the PV of a surviving spouses living expenses

Answer 51

= [(1+r)/(1+g)] -1

Question 52

how to compare g-spreads

Answer 52

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)

Question 53

variance notional

Answer 53

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

Question 54

payoff to a variance buyer

Answer 54

= 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

Question 55

slippage cost

Answer 55

the difference between an execution price and the mid point of the bid/ask spread when the order was entered into the market

Question 56

Turnover ratio

Answer 56

The greater of portfolio purchases OR sales / average portfolio value

Question 57

equation to find CD Price

Answer 57

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.

Question 58

How to find the change in portfolio value given an effective duration and change in yield?

Answer 58

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%

Question 59

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?

Answer 59

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

Question 60

Net asset value and how to calculate the expected net asset value at the end of the year?

Answer 60

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)

Question 61

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

Answer 61

1. 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
2. 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)

3. 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

Question 62

Modified duration of shareholders capital

Answer 62

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

Question 63

constituents of portfolio return

Answer 63

portfolio return = market index return + manager style return + active management return

return due to active management = portfolio return - benchmark

Question 64

semideviation

Answer 64

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.

Question 65

How to find the estimated return for an asset using the Singer–Terhaar approach to the international extension of the CAPM

Answer 65

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.

Question 66

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.

Answer 66

(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.

Question 67

Herfindahl–Hirschman Index (HHI)

Answer 67

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.

Question 68

Fama French Model

Answer 68

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

Question 69

the 5 BB&K Client Classifications

Answer 69

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.

Question 70

How to calculate the change in convexity given the sale of one bond in exchange for the purchase of another?

Answer 70

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)

Question 71

How to calculate the equivalent value $ one should allocate from one bond to another in order to maintain a duration neutral position

Answer 71

in order to maintain a neutral duration position

1. find the money duration of the bond being exchanged out = $value / PVBP
2. 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

Question 72

how to determine the inflation adjusted annual cash flow generated by a sub portfolio

Answer 72

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)

Question 73

Risk Parity asset allocation

—> how to calculate the risk attributable from each asset?

Answer 73

weight of asset i x covariance of asset i with the portfolio =
variance of the portfolio / number of assets)

Question 74

1. how to determine the value needed to invest given the desire to buy a 5M house in 5 years earning 4.4% annually
2. 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

Answer 74

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)

Question 75

Long Term Economic Growth Rate

Answer 75

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)

Question 76

Risk Premium Approach to expected bond return

Answer 76

expected bond return =

risk-free rate + term premium + credit premium + liquidity premium

Question 77

Appraisal Ratio

Answer 77

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

Question 78

How to best implement a minimum-variance hedge vs another currency

Answer 78

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

Question 79

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?

Answer 79

1. 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
2. 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)

Question 80

percent capital gain exposure (PCGE)

Answer 80

= 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%

Question 81

How to find the EXCESS return due to active factor weighting

–> aka active effect

Answer 81

1. find the return contribution of the factor in the portfolio (portfolio sector weight x portfolio sector return)
2. find the benchmark sector contribution (benchmark sector weight x benchmark sector return)
3. subtract value one (portfolio rector attribution) - value two (benchmark sector attribution)
4. 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)

Question 82

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)

Answer 82

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%

Question 83

rolldown return of a CD strategy

Answer 83

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

Question 84

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

Answer 84

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

Question 85

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

Answer 85

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

Question 86

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

Answer 86

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

Question 87

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

Answer 87

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

Question 88

Behavioral biases

Answer 88

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)

Question 89

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.

Answer 89

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:

0. 6667 reflects the 3 times leverage factor - 66.67% borrowed and 33.33% cash usage (1-1/leverage ratio)
1. 15% reflects the ETF borrowing rate (3-month Libor of 1.80% + 35 bps), and
2. 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.

Question 90

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%.

Answer 90

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.

Question 91

Actual contribution to total risk

Answer 91

ACTR = weight x marginal contribution to total risk

Question 92

Marginal contribution to total risk

Answer 92

MCTR = beta of sector x portfolio standard deviation

Question 93

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

Answer 93

1. Find BPV if liabilities
   = 16B x 12 x .0001 = 19.2M
2. 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
3. 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

Question 94

how to calculate the traders added value

Answer 94

added value = arrival cost - estimated pre trade cost

arrival cost = side * [ (average - arrival) / arrival] x .0001

Question 95

Carhart model

Answer 95

Carhart model = Fama French + Momentum

Expected return =
risk free rate + (factor coefficient) Market risk premium+ (factor coef) SMB + (factor coef) HML + (factor coef) momentum

Question 96

cognitive biases

Answer 96

• 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)

Question 97

emotional biases

Answer 97

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

Question 98

PIC multiple

Answer 98

= since inception paid in capital / cumulative committed capital

Question 99

alpha

Answer 99

alpha = fund return - [(beta A x factor coefficient A) + (beta B x factor coefficient B) + (beta C x factor coefficient C) …]]