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教育背景:英国埃塞克斯大学金融学硕士、通过CFA三级、PMP( Project Management Pr
ofessional 项目管理专业认证)
Topic Weightings in CFA Level II
Session NO.
Study Session 1-2
Quantitative Methods
Study Session 3
Study Session 4-5
Financial Statement Analysis
Study Session 6-7
Corporate Issuers
Study Session 8-10
Study Session 11-12
Fixed Income
Study Session 13
Study Session 14
Alternative Investments
Study Session 15-16
Portfolio Management
Study Session 17
Ethical and Professional Standards
 SS15: Portfolio Management (1)
• R38 Exchange-Traded Funds:
Mechanics and Applications
• R39 Using Multifactor Models
• R40 Measuring and Managing
Market Risk
• R41 Backtesting and Simulation
 SS16: Portfolio Management (2)
• R42 Economics and Investment
• R43 Analysis of Active Portfolio
• R44 Trading Costs and Electronic
Exchange-Traded Funds: Mechanics and Applications
1. ETF Mechanics
• Creation/redemption process
• Arbitrage
• Advantages of the ETF mechanics
• Trading and settlement
2. Understanding ETF
• Expense ratio
• Tracking Error
• Trading costs
• Tax issues
• Total Costs of ETF Ownership
• Types of ETF risks
3. ETFs in Portfolio Management
1. ETF Mechanics
 ETF transactions take place in two interrelated markets.
Primary Market
Investor 1
ETF sponsor
(creation basket)
Securities 1
Securities 2
Securities n
Investor 2
Securities 1
Securities 2
Securities n
Investor 3
Buy & Sell
Secondary Market
1.1 The Creation/Redemption Process
 The primary market for ETF trading is that which exists on an over-thecounter basis between authorized participants (APs), and the ETF issuer,
or sponsor.
 The only investors who can create or redeem new shares of an ETF are a
special group of institutional investors called APs.
 APs are large broker/dealers, often market makers, who are
authorized by the ETF issuer to participate in the process.
 The AP creates new ETF shares by transacting in-kind with the ETF issuer.
This in-kind swap happens off the exchange, in the primary market for
the ETF, where APs transfer securities to (for creations) or receive
securities from (for redemptions) the ETF issuer, in exchange for ETF
1.1.1 Creation Process
 Each business day, the ETF manager publishes a list of required in-kind
securities (creation basket) for each ETF.
 To create new shares, an AP acquires the securities in the creation basket
in the specified share amounts (generally by transacting in the public
markets or using securities the AP happens to have in inventory).
 The AP then delivers this basket of securities to the ETF manager in
exchange for an equal value in ETF shares.
 The price the AP might have paid to acquire that stock or what its
price happens to be at the end of the day is not relevant to the
exchange taking place.
 These transactions between the AP and the ETF manager are done in large
blocks called creation units, usually but not always equal to 50,000 shares
of the ETF.
1.1.2 Redemption Process
 The process also works in reverse: the AP presents these shares for
redemption to the ETF manager and receives in return the basket of
underlying securities.
 The basket of securities the AP receives when it redeems the ETF shares is
called the redemption basket.
 This basket often has the same security composition as the creation
basket, but it may be different if the ETF portfolio manager is trying
to sell particular securities for tax, compliance, or investment reasons.
 Although actual process of exchanging baskets and blocks of ETF shares
happens after the markets are closed, the AP is able to execute ETF trades
throughout the trading day because the AP knows the security composition
of the basket needed for ETF share creation or redemption.
1.2 Arbitrage
 Because prices of the ETF and the basket securities are continuously
changing on the basis of market conditions, APs monitor both for
discrepancies, looking for opportunities to make arbitrage profits.
 The arbitrage gap—the price(s) at which it makes sense for ETF market
makers to step in and create or redeem shares—vary with the liquidity
of the underlying securities and a variety of related costs.
 Time difference, illiquid underlying securities would have higher
arbitrage gaps, while liquid securities will have a lower arbitrage gap.
 Service fee that AP pay to ETF manager for creation/redemption
1.2 Arbitrage
ETF shares creation
ETF shares redemption
1.2 Arbitrage
 Arbitrage keeps the ETF trading at or near its fair value.
 NAV > ETF price (ETF share is undervalued, trading at a discount)
 current per-share market value of the basket of underlying securities
is greater than the quoted price of the ETF shares
 AP can simultaneously sell (or short) the basket of securities and buy
ETF shares, to make a profit.
 NAV < ETF price (ETF is trading at a premium)
 shares of the ETF are quoted at a higher price than the per-share
market value of the basket of securities
 AP can make a profit by simultaneously selling the ETF shares in the
market and buying the basket of securities.
1.3 Advantage of the ETF Mechanics
 1. Lower cost
 The creation/redemption process does not force the ETF manager to
sell/purchase portfolio investments. The manager do not incur any
resulting transaction cost.
 AP absorbs all costs of transacting the securities for the fund’s portfolio.
 APs pass these costs to investors in the ETF’s bid–ask spread.
 Only transacting shareholders pay the cost.
 ETF structure is fair: Frequent ETF traders bear the tax of their
activity, whereas buy-and-hold ETF shareholders are shielded from
those capital gain tax.
 In contrast, the mutual fund manager incurs costs to buy or sell
investments arising from this activity, which affect all fund
1.3 Advantage of the ETF Mechanics
 2. Tax efficiency
 Because creation and redemption happen in kind, they allow the ETF’s
portfolio managers to manage the cost basis of their holdings by
selecting low-basis holdings for redemptions, leading to greater tax
 Issuers may choose to publish customized redemption baskets,
which allows them to target specific low-basis securities for removal
from the portfolio.
 By delivering out shares that were originally acquired at low costs,
the issuer can continuously raise the average acquired cost (or cost
basis) of each position, thereby minimizing the position’s
unrealized gains.
 3. Arbitrage keeps the ETF trading at or near its fair value.
1.4 Secondary: Trading and Settlement
 Process of investment in the secondary market.
 Step 1: You place a buy order in your brokerage account the same way
you would place an order to buy any publicly listed equity security,
 Step 2: Your broker submits that order to the public market to find a
willing seller: another investor or a market maker (i.e., a broker/dealer
who stands ready to take the opposite side of the transaction).
 Step 3: The order is executed, and you receive shares of the ETF in your
brokerage account just as if you transacted in a stock.
 The selling activities of individual investors in the secondary market do not
require the fund to trade out of its underlying positions.
1.4 Secondary: Trading and Settlement
 US settlement: centralized
 National Security Clearing Corporation and Depository Trust
Company. ( T+2 ) : guarantor of that transaction—the entity that
ensures all parties are immunized against the financial impact of any
operational problems—on the evening of the trade, and the trade is
considered “cleared”.
 The Depository Trust Company (DTC), of which the NSCC is a
subsidiary, holds the book of accounts—the actual list of security
holders and ownership.
 EU settlement: fragmented
 The majority of trading happens in negotiated over-the-counter trades
between large institutions.
 In Europe, they are cleared to one of 29 central securities depositories
(or CSDs).
 A complex system results in wider spreads and higher local market
trading costs.
 Apolo ETF is currently trading at $25 per share. Its NAV is $23.
Dex bank, an AP in the ETF, would most likely:
A. Do nothing
B. Redeem shares if the arbitrage gap is more than $2
C. Create shares if the arbitrage gap is less than $2
 Correct Answer: C
The AO would earn a profit by selling the shares in the market at
$25 while creating shares at $23 plus cost. The cost would have
to be less than $2 per share for the AP to make a profit.
2. Understanding ETFs
 The best-managed ETFs
 charge low and predictable investment costs,
 closely track the indexes on which they are based,
 provide investors with the lowest possible tax exposure for the
investment objective.
 To best understand an ETF’s ability to meet expectations, one should
consider its:
 expense ratio;
 index tracking;
 tax treatment;
 potential costs and risks.
2.1 Expense Ratios
 The actual costs to manage an ETF (management fee) vary, depending
on portfolio complexity , issuer size (economies of scale apply), and the
competitive landscape.
 ETFs generally charge lower fees than mutual funds
 ETF providers do not have to keep track of individual investor accounts,
since ETF shares are held by and transacted through brokerage firms.
 Nor do ETF issuers bear the costs of communicating directly with
individual investors.
 Index-based portfolio management, used by most ETFs, does not
require the security and macroeconomic research carried out by active
managers, which increases fund operating costs.
 Expense ratio does not reflect the cost of portfolio rebalancing or other fees.
2.2 Index Tracking
 Daily differences=Rp (measured by NAV)-RB.
 Measure how close an ETF can track relative index using one-day
difference in return
 Periodic tracking
 1. Tracking error = annualized standard deviation of daily
 Typically for a 12-month period.
 Tracking error does not reveal
 the extent to which the fund is under- or over performing its
 the distribution of errors.
2.2 Index Tracking: Periodic tracking
 Periodic tracking (cont.)
 2. Rolling tracking difference
 Tracking differences calculated over a longer holding period.
 This approach allows investors to
see the cumulative effect of portfolio management and
expenses over an extended period.
Represent both central tendencies and variability.
 It allows for comparison with other annual metrics, such as expense
2.2 Index Tracking: Source of Tracking Error
 1. Fees and expenses
 A fund’s operating fees and expenses reduce the fund’s return relative
to the index.
 2. Representative sampling/optimization
 For funds tracking index exposure to small or illiquid markets, owning
every index constituent can be difficult and costly.
 Therefore, fund managers may choose to optimize their portfolios by
holding only a portion, or representative sample, of index securities.
 Compared with a full replication approach, representative
sampling/optimization introduces greater potential for tracking error.
2.2 Index Tracking: Source of Tracking Error
 3. Depositary receipts and other ETFs
 Funds may hold securities that are different from those in the index
 such as American depositary receipts (ADRs), global depositary
receipts (GDRs), and other ETFs.
 Differences in trading hours and security prices create discrepancies
between portfolio and index values.
 An ETF may invest in other ETFs, thus inherit the tracking error of those
2.2 Index Tracking: Source of Tracking Error
 4. Index changes
 Funds may trade index changes at times and prices that are different
from those of the benchmark tracked.
 Since rebalance is infrequent, this part is often the smallest contributor.
 5. Fund accounting practices
 Differences in valuation practices between the fund and its index can
create discrepancies that magnify daily tracking differences.
 6. Regulatory and tax requirements
 Funds may be subject to regulatory and tax requirements that are
different from those assumed in index methodology, such as with
foreign dividend withholding.
2.2 Index Tracking: Source of Tracking Error
 7. Asset manager operations
 ETF issuers may attempt to offset costs through security lending and
foreign dividend recapture. These act as “negative” costs, which enhance
fund performance relative to the index.
 Security lending: Many ETFs (and mutual funds) lend a portion of
their portfolio holdings to short sellers. In exchange, the ETF
receives a fee and earns interest on the collateral posted by the
borrower. Since securities-lending income is not accounted for in
the index calculation, it is a source of tracking error.
 Foreign dividend recapture: Asset managers may work with
foreign governments to minimize tax paid on distributions received.
2.3 Tax
 1. Capital Gains Distributions: In general, funds must distribute any capital
gains realized during the year. On average, ETFs distribute less in capital
gains than competing mutual funds for two primary reasons.
 Tax fairness:
 For traditional mutual fund, shareholders may have to pay tax
liabilities triggered by other shareholders redeeming out of the fund.
 For ETF:
 The selling activities of individual investors in the secondary
market do not require the fund to trade out of its underlying
 If an AP redeems ETF shares, this redemption occurs in kind
and is not a taxable event. Thus, redemptions do not trigger
capital gain realizations.
 “Tax fair”: The actions of investors selling shares of the fund do not
influence the tax liabilities for remaining fund shareholders.
2.3 Tax
 1. Capital Gains Distributions (cont.)
 Tax efficiency:
 Tax lot management allows portfolio managers to limit the
unrealized gains in a portfolio.
 Tax lot management: By choosing shares with the largest
unrealized capital gains—that is, those acquired at the lowest cost
basis—ETF managers can use the in-kind redemption process to
reduce potential capital gains in the fund.
2.3 Tax
 2. Other Distributions
 Security dividend distributions can trigger tax liabilities for investors but
the treatment varies by region.
 Return-of-capital (ROC) distributions are amounts paid out in excess of
an ETF’s earnings and serve to reduce an investor’s cost basis by the amount
of the distribution.
 These distributions are generally not taxable.
 3. Taxes on Sale
 In most jurisdictions, ETFs are taxed according to their underlying
holdings. However, there can be nuances in individual tax jurisdictions that
require investor analysis.
2.4 ETF Trading Costs
 ETF trading costs
 Commission
 Bid–ask spread
 Premium and discount
2.4.1 ETF Bid–Ask Spreads
 Factors that determine the width of the Bid–Ask Spreads:
 the amount of ongoing order flow in the ETF, as measured by daily share
volume (more flow means lower spreads);
 the amount of competition among market makers (more competition
means lower spreads);
 the actual costs and risks for the liquidity provider.
 Maximum quoted spread on ETF:
 ± Creation/redemption fees and other direct trading costs, such as
brokerage and exchange fees
 + Bid–ask spreads of the underlying securities held in the ETF
 + Compensation (to market maker or liquidity provider) for the risk of
hedging or carrying positions for the remainder of the trading day
 + Market maker’s desired profit spread, subject to competitive forces
 – Discount related to the likelihood of receiving an offsetting ETF order
in a short time frame
2.4.1 ETF Bid–Ask Spreads
 ETF Bid–Ask Spreads: One of the most important drivers of ETF bid–ask spreads
and liquidity is the market structure and liquidity of the underlying securities
 Fixed-income securities, which trade in a dealer market, tend to have much
wider bid-ask spreads than large-capitalization stocks.
 Large, actively traded ETFs have narrow bid–offer spreads and the capacity
(or liquidity) for large transaction sizes. The bid–ask spread for liquid ETFs can
be significantly tighter than the spreads on the underlying securities.
 For larger trades, posted spreads may not reflect trading costs, and these
trades may best be handled by negotiation.
 International equity and international fixed-income spreads are wider
2.4.2 ETF Premiums and Discounts
 Each ETF has an end-of-day NAV at which shares can be created or
redeemed and with which the ETF’s closing price can be compared.
 End−of−day ETF premium or discount (%) =
ETF price – NAV per share
NAV per share
 NAV is intended to be an accurate assessment of the ETF’s fair value.
 During the trading day, “indicated” NAVs (iNAVs)
 I𝑛𝑡𝑟𝑎day ETF premium or discount (%) =
ETF price – iNAV per share
iNAV per share
 iNAVs are intraday “fair value” estimates of an ETF share based on its
creation basket composition for that day.
 An ETF is said to be trading at a premium when its share price is higher than
iNAV and at a discount if its price is lower than iNAV.
2.4.2 ETF Premiums and Discounts
 ETF premiums and discounts are driven by a number of factors, including
timing differences and stale pricing.
 Timing differences
 NAV is often a poor fair value indicator for ETFs that hold foreign
securities because of differences in exchange closing times between
the underlying (e.g., foreign stocks, bonds, or commodities) and the
exchange where the ETF trades.
 Stale pricing
 ETFs that trade infrequently may also have large premiums or
discounts to NAV. If the ETF has not traded in the hours leading up
to the market close, NAV may have significantly risen or fallen
during that time owing to market movement.
 The maximum quoted spread on an ETF is most likely to be
negatively related to:
A. the AP's profit margin.
B. the quoted spreads of securities underlying the tracked index.
C. the probability of completing an offsetting trade in the
secondary market.
 Correct Answer: C
 A higher probability of completing an offsetting trade results
in a reduction (i.e., discount) in the quoted spreads. The other
two components are positively related to the quoted spread.
2.5 Total Costs of ETF Ownership
Fund Cost Factor
Management fee
Function of
√ (often less)
Tracking error
√ (often less than
comparable index mutual
√ (some free)
Bid–ask spread
Premium/discount to NAV
Portfolio turnover (from
investor flows and fund
√ (often less)
Taxable gains/losses to
√ (often less)
Security lending
√ (often more)
2.5 Total Costs of ETF Ownership
 Holding period cost
𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑 𝑐𝑜𝑠𝑡 %
= 𝑅𝑜𝑢𝑛𝑑 𝑡𝑟𝑖𝑝 𝑡𝑟𝑎𝑑𝑒 𝑐𝑜𝑠𝑡 % + 𝑀𝑎𝑛𝑎𝑔𝑒𝑚𝑒𝑛𝑡 𝑓𝑒𝑒 𝑓𝑜𝑟 𝑝𝑒𝑟𝑖𝑜𝑑 %
 Consider the 3-month versus 12-month versus 3-year holding period
costs for an ETF with a 0.15% annual fee, one-way commissions of
0.05%, and a bid–ask spread of 0.15%. Holding period costs can be
calculated as follows:
 Round-trip trading cost (%) = (One-way commission % × 2) + (½
Bid–ask spread % × 2) = (0.05% × 2) + (½ × 0.15% × 2) = 0.25%
 3-month holding period cost (%) = 0.25%+3/12×0.15% = 0.29%
 12-month holding period cost (%) = 0.25%+12/12×0.15% = 0.40%
 3-year holding period cost (%) = 0.25%+36/12×0.15% = 0.70%
2.5 Total Costs of ETF Ownership
 Trading costs vs. management fees
 The longer an ETF is held, the greater the proportion of total costs
represented by the management fee component.
 The size of the management fee is typically a more significant
consideration for longer-term buy-and-hold investors.
 Shorter-term tactical traders may use an ETF with a higher
management fee but a tighter bid–ask spread and lower commission.
 Of the various components of ETF cost, a long-term buy-and-hold
investor is most likely to focus on:
A. management fees.
B. trading costs.
C. creation/redemption service fees.
 Correct Answer: A
 While all costs are important, long-term investors should be
more concerned with recurring annual management fees as
opposed to one-time trading costs.
 Creation/redemption fees are paid by the AP to the ETF
manager and are reflected in the quoted spread (which is part
of trading costs).
2.6 Types of ETF risks
 1. Counterparty Risk: A counterparty failure can put the investor’s principal
at risk of default or affect a portion of the assets via settlement.
Counterparty activity can affect a fund’s economic exposure.
 Settlement risk: A fund that uses OTC derivatives, such as swaps, to
gain market exposure has settlement risk. To minimize settlement risk,
OTC contracts are typically settled frequently—usually on a daily or
weekly basis. This frequent settlement reduces the exposure the swap
partners face if a company goes bankrupt.
 Security lending: ETF issuers lend their underlying securities to short
sellers, earning additional income for the fund’s investors. Securities lent
are generally overcollateralized, so that the risk from counterparty
default is low. Cash collateral is usually reinvested into extremely shortterm fixed-income securities with minimal risk.
 Exchange-traded notes (ETNs) is a type of unsecured debt obligations of
the institution that track on index and are structured as a promise to pay a
pattern of returns based on the return of the stated index minus fund
expenses. (similar to bond but do not pay interest)
 ETNs have a creation/redemption mechanism, they are not truly funds
because they do not hold underlying securities.
 ETNs have the largest potential counterparty risk of all exchange-traded
products because they are unsecured, unsubordinated debt notes and,
therefore, are subject to default by the ETN issuer.
 Theoretically, an ETN’s counterparty risk is 100% in the event of an
instantaneous default by the underwriting bank, and should an
issuing bank declare bankruptcy, any ETNs issued by the bank would
be worthless. (e.g. Lehman Brothers in 2008)
2.6 Types of ETF risks
 2. Fund Closures: Primary reasons for a fund to close include regulation,
competition, and corporate activity. "Soft" closures—which do not involve
an actual fund closing—include creation halts and changes in investment
 (1) Regulations. Security regulators can change the regulations
governing certain types of funds, resulting in forced closure of those
 For example, in 2018, the Israeli security regulator banned the ETN
structure, forcing over 700 products to close and reopen as
traditional ETFs.
 (2) Competition. Investors have benefited from a growing number of
ETFs and increased competition. As ETFs proliferate, some funds fail to
attract sufficient assets and are shut down by the ETF issuer.
2.6 Types of ETF risks
 2. Fund Closures: (Cont.)
 (3) Corporate actions. Mergers and acquisitions between ETF providers
can prompt fund closures.
 (4) Creation and redemption halts. ETN issuers may halt creations and
redemptions when the issuer no longer wants to add debt to its balance
sheet related to the index on which the ETN is based.
 (5) Change in investment strategy. Some ETF issuers find it easier to
repurpose a low-asset ETF from their existing lineup than to close one
fund and open another. Issuers simply announce a change in the fund’s
underlying index—a common occurrence in the ETF industry.
2.6 Types of ETF risks
 3. Investor-Related Risk (expectation-related risks)
 ETFs may introduce risks to investors who do not fully understand
complex asset classes and strategies.
 Eg. leveraged and inverse ETFs.
 Leveraged and inverse funds generally offer levered (or geared),
inverse, or levered and inverse exposure to a given index and have a
daily performance objective that is a multiple of index returns.
 These products must reset or adjust their exposure daily to deliver
the target return multiple each day.
 Inverse leveraged ETFs are most likely to be described as having a
A. A expectation-related risk.
B. counterparty risk.
C. fund closure risk.
 Correct Answer: A
 A Inverse and leveraged ETFs may not be well understood by
their investors, leading to a gap between expectation and
actual outcome; this is expectation-related risk.
3. ETFs in Portfolio Management
 ETFs are used for both top-down and bottom-up investment approaches.
In addition, ETFs are used for tactical tilts, portfolio rebalancing, and risk
 Such factors as tax efficiency, low fees, and available product make ETFs
competitive alternatives to traditional mutual funds and active managers.
 The primary applications in which ETFs are used include the following:
 Portfolio efficiency: The use of ETFs to better manage a portfolio for
efficiency or operational purposes.
 Asset class exposure management: The use of ETFs to achieve or
maintain core exposure to key asset classes, market segments, or
investment themes on a strategic, tactical, or dynamic basis.
 Active and factor investing: The use of ETFs to target specific active or
factor exposures on the basis of an investment view or risk management
3.1 Efficient Portfolio Management
 (1) Portfolio liquidity management (cash flow management)
 ETFs can be used to invest excess cash balances quickly (cash equitization),
enabling investors to remain fully invested in target benchmark exposure,
thereby minimizing potential cash drag.
 Cash drag refers to a fund’s mis-tracking relative to its index that results
from holding uninvested cash.
 (2) Portfolio rebalancing
 Many investors rebalance portfolios on the basis of a specified time interval,
and some may adjust whenever the allocation deviates from its target weight
by a threshold.
 For tighter rebalancing thresholds and more frequent rebalancing time
intervals, using liquid ETFs with tight bid–ask spreads allows the portfolio
manager to execute the rebalance in a single ETF trade and ensures the
portfolio remains fully invested according to its target weights.
Efficient Portfolio Management
 (3) Portfolio completion strategies
 ETFs can also be used for completion strategies to fill a temporary gap
in exposure to an asset class, sector, or investment theme or factor.
 E.g. Manager moves out of small-caps, and investor still seek smallcap exposure
 (4) Transition management
 the process of hiring and firing managers—or making changes to
allocations with existing managers—while trying to keep target
allocations in place.
 A newly appointed transition manager can invest in an ETF to
maintain market exposure as she undergoes the process of selling the
unwanted positions of the manager she is replacing (the terminated
 The new transition manager can then take her time to invest in positions
for her strategy and gradually reduce the ETF holding.
3.2 Asset Class Exposure Management
 (1) Core exposure to an asset class or sub-asset class
 The primary strategic use of ETFs is to gain core index exposure to
various asset classes and sub-asset classes, and investors regularly use
ETFs for broad portfolio diversification.
 Investors also use ETFs for more targeted strategic exposure to such
segments as high-yield debt, bank loans, and commodities.
 (2) Tactical strategies
 ETFs can also be used to implement market views and adjust portfolio
risk on a more short-term, tactical basis (opportunistic trading).
 ETFs that have the highest trading volumes in their asset class
category are preferred for tactical trading applications. Trading costs
and liquidity are the important criteria in selecting an ETF for tactical
 Thematic ETFs hold stock passively but allow investors to take an active
view on a market segment they believe will deliver strong returns.
3.3 Active and Factor Investing
 (1) Factor (smart beta) ETFs
 Factor strategy ETFs are usually benchmarked to an index created with
predefined rules for screening and/or weighting stock holdings and are
considered longer-term, buy-and-hold investment options rather than
tactical trading instruments.
 The strategy index rules are structured around return drivers or factors,
such as value, dividend yield, earnings or dividend growth, quality, stock
volatility, or momentum.
 Capture risk premium for one or more factors driving returns or risk.
 Investors using factor-based investing seek outperformance versus a
benchmark or portfolio risk modification.
Active and Factor Investing
 (2) Risk management
 Some smart beta ETFs are constructed to deliver lower or higher risk
than that of their asset class benchmark.
 With respect to interest rate risk management, several smart beta
fixed-income ETFs hold long positions in corporate or high-yield bonds
and hedge out the duration risk of these bonds with futures or short
positions in government bonds. These ETFs enable investors to add a
position to their portfolio that seeks returns from taking credit risk with
minimal sensitivity to movements in interest rates.
 (3) Alternatively weighted ETFs
 ETFs that weight their constituents by means other than market
capitalization can also be used to implement investment views—for
example, ETFs that weight constituent stocks on the basis of their
dividend yields.
Active and Factor Investing
 (4) Discretionary active ETFs
 Access discretionary active management in an ETF structure.
 The largest active ETFs are in fixed income, where passive
management is much less prominent than in equities.
 Due to low liquidity of most fixed income securities.
 (5) Dynamic asset allocation and multi-asset strategies
 Seek returns from active allocation across asset classes or factors
based on return or risk outlook.
 Invest in a multi-asset-class strategy in single product.
Using Multifactor Models
1. Arbitrage Pricing Theory (APT)
2. Multifactor models
3. Application
• Return attribution
• Risk attribution
• Portfolio construction
1. Arbitrage Pricing Theory (APT)
 Exactly formula
𝐸 𝑅𝑃 = 𝑅𝐹 + 𝜆1 𝛽𝑃,1 + 𝜆2 𝛽𝑃,2 + ⋯ + 𝜆𝑘 𝛽𝑃,𝑘
 λj = the expected reward for bearing the risk of factor j
 βj = the sensitivity of the portfolio P to factor j
 APT introduced a framework that explains the expected return of portfolio P
in equilibrium as a linear function with multiple systematic risk factor.
 CAPM can be considered as a restrictive case of APT with only one risk factor.
 Assumptions
 A factor model describes asset returns
 With many assets to choose from, investors can form well-diversified
portfolios that eliminate asset-specific risk
 No arbitrage opportunities exist among well-diversified portfolios
1. Arbitrage Pricing Theory (APT)
 The factor risk premium (or factor price, λj) represents the expected reward
for bearing the risk of a portfolio with a sensitivity of 1 to factor j and a
sensitivity of 0 to all other factors.
 Such a portfolio is called a pure factor portfolio for factor j.
 The parameters of the APT equation are the risk-free rate and the factor
risk-premiums .
Example- Arbitrage Pricing Theory (APT)
 Suppose that two factors, surprise in inflation (factor 1) and surprise in
GDP growth (factor 2), explain returns. According to the APT, an
arbitrage opportunity exists unless
𝐸 𝑅𝑝 = 𝑅𝐹 + 𝛽𝑝,1 𝜆1 + 𝛽𝑝,2 𝜆2
 Well-diversified portfolios, J, K, and L, given in table.
Expected return
Sensitivity to
inflation factor
Sensitivity to GDP
𝐸 𝑅𝐽 = 0.14 = 𝑅𝐹 + 1.0𝜆1 + 1.5𝜆2
𝐸 𝑅𝐾 = 0.12 = 𝑅𝐹 + 0.5𝜆1 + 1.0𝜆2
𝐸 𝑅𝐿 = 0.11 = 𝑅𝐹 + 1.3𝜆1 + 1.1𝜆2
𝐸 𝑅𝑝 = 0.07 − 0.02𝛽𝑝,1 + 0.06𝛽𝑝,2
Example 2
 One-factor APT model: 𝐸 𝑅𝑃 = 0.05 + 0.05𝛽𝑃,1
Expected Return
Factor Sensitivity
0.5A + 0.5C
 According to the assumed APT model, the expected return on Portfolio
D should be 𝐸 𝑅𝐷 = 0.05 + 0.05𝛽𝐷,1 = 0.05 + 0.05 × 0.45 = 0.0725,
or 7.25%. Portfolio D is undervalued relative to its factor risk.
Example 2
 Arbitrage: We purchase D using the proceeds from selling short an
equally weighted portfolio of A and C with exactly the same 0.45 factor
sensitivity as D.
 Strategy: invest $10,000 in Portfolio D and fund that investment by
selling short an equally weighted portfolio of Portfolios A and C.
Initial Cash
Portfolio D
Portfolios A and C
Final Cash
Arbitrage Pricing Theory (APT)
 Arbitrage is a risk-free operation that requires
 no net investment of money,
 earns an expected positive net profit.
 An arbitrage opportunity is an opportunity to conduct an arbitrage — an
opportunity to earn an expected positive net profit without risk and with no
net investment of money.
Carhart Four-Factor Model
 𝑅𝑝 − 𝑅𝐹 = 𝛼𝑝 + 𝑏𝑝1 𝑅𝑀𝑅𝐹 + 𝑏𝑝2 𝑆𝑀𝐵 + 𝑏𝑝3 𝐻𝑀𝐿 + 𝑏𝑝4 𝑊𝑀𝐿 + 𝜀𝑝
 𝛼𝑝 =“alpha” or return in excess of that expected given the portfolio’s level of
systematic risk
 𝑏𝑝 =the sensitivity of the portfolio to the given factor
 RMRF=the return on a value-weighted equity index in excess of the one-month
T-bill rate
 SMB = small minus big, a size (market capitalization) factor
 HML = high minus low, the average return on two high book-to-market
portfolios minus the average return on two low book-to-market portfolios
 WML = winners minus losers, a momentum factor
 𝜀𝑝 = an error term that represents the portion of the return to the portfolio,p,
not explained by the model
2. Multifactor Model
 Multifactor models have gained importance for the practical business
of portfolio management for two main reasons.
 Multifactor models explain asset returns better than the market model
 Multifactor models provide a more detailed analysis of risk than does a
single factor model.
Types of Multifactor Models
 1. Macroeconomic factor model: the factors are surprise in macroeconomic
variables that significantly explain returns.
 For example: interest rates, inflation risk, business cycle risk, and credit
 2. Fundamental factor model: the factors are attributes of stocks or
companies that are important in explaining cross-sectional differences in
stock prices.
 For example: book-value-to-price ratio, market capitalization, the priceto-earnings ratio, and financial leverage.
 3. Statistical factor model: use statistical methods to explain asset returns.
2.1 Macroeconomic Factor Model
 Macroeconomic Factor
Surprise = actual value – predicted (expected) value
 Example formula for return of asset i
𝑅𝑖 = 𝐸 𝑅𝑖 + 𝑏𝑖1 𝐹𝐺𝐷𝑃 + 𝑏𝑖2 𝐹𝑄𝑆 + 𝜀𝑖
bi1, bi2
Regression (time
E(Ri )= expected return for asset i
FGDP = surprise in the GDP growth
FQS = surprise in the credit quality spread
= return for asset i
bi1 = GDP surprise sensitivity of asset i
= credit quality spread surprise sensitivity of asset i
=an error term with a zero mean that represents the portion of
the return not explained by the factor model
Macroeconomic Factor Model
 Macroeconomic Factor Model – Interpret Parameter
 1. Factor surprise, F: the difference between the predicted value and the
realized value
 Actual value = Predicted value + Surprise value
 The key to Macro factor model is that the variables that explain returns
reflect not the value of macroeconomic variable itself, but rather the
unexpected part (the surprise), because we assume that the predicted value
has already been reflected in stock prices and expected returns.
 E.g. If the government announces that GDP grew at an annual rate of
1.5% and the consensus prediction was 3%, the surprise in GDP growth
is 1.5 – 3 = -1.5%
 The 3% consensus forecast was already reflected in market prices, the
negative surprise, which was bad news to market, should cause stock
price to fall
Macroeconomic Factor Model
 Macroeconomic Factor Model – Interpret Parameter
 2. Slope coefficients, b: sensitivities of the asset to that surprise
 The higher the sensitivity, the larger the change in return for a given
factor surprise
 3. The term εi is the part of return that is unexplained by expected
return or the factor surprises.
 If we have adequately represented the sources of common risk (the
factors), then εi must represent an asset-specific risk.
 E.g. For a stock, it might represent the return from an unanticipated
company-specific event.
Macroeconomic Factor Model
 Macroeconomic Factor Model – How to Regress Slope coefficients
 In macroeconomic factor models, the time series of factor surprises are
constructed first.
 Regression analysis is then used to estimate assets’ sensitivities to the
 Suppose that stock returns are affected by two common factors:
surprises in inflation and surprises in GDP growth. A portfolio manager
is analyzing the returns on a portfolio of two stocks, Manumatic
(MANM) and Nextech (NXT), The following equations describe the
returns for those stocks, where the factors FINFL. and FGDP, represent
the surprise in inflation and GDP growth, respectively:
𝑅𝑀𝐴𝑁𝑀 = 0.09 − 1𝐹𝐼𝑁𝐹𝐿 + 1𝐹𝐺𝐷𝑃 + 𝜀𝑀𝐴𝑁𝑀
𝑅𝑁𝑋𝑇 = 0.12 + 2𝐹𝐼𝑁𝐹𝐿 + 4𝐹𝐺𝐷𝑃 + 𝜀𝑁𝑋𝑇
 One-third of the portfolio is invested in Manumatic stock, and twothirds is invested in Nextech stock.
1. Formulate an expression for the return on the portfolio.
2. State the expected return on the portfolio.
3. Calculate the return on the portfolio given that the surprises in
inflation and GDP growth are 1% and 0%, respectively, assuming
that the error terms for MANM and NXT both equal 0.5 percent.
 Correct Answer 1 :
 The portfolio's return is the following weighted average of the
returns to the two stocks: Rp = (1/3)(0.09) + (2/3)(0 .12) + [(1/3)(- I)
+ (2/3)(2)] FINFL+ [(1/3)(1) + (2/3)(4)]FGDP + (1/3) εMANM + (2/3) εNXT
= 0.11 + 1 FINFL+ 3FGDP + (1/3) εMANM + (2/3) εNXT
 Correct Answer 2 :
 The expected return on the portfolio is 11 percent, the value of
the intercept in the expression obtained in Part 1.
 Correct Answer 3 :
 Rp = 0.11 + 1 FINFL+ 3FGDP + (1/3) εMANM + (2/3) εNXT = 0.11 +
1(0.01) + 3(0) + (1/3)(0.005) + (2/3)(0.005) = 0.125 or 12.5%
2.2 Fundamental Factor
𝑅𝑖 = 𝑎𝑖 + 𝑏𝑖1 𝐹𝑃/𝐸 + 𝑏𝑖2 𝐹𝑆𝐼𝑍𝐸 + 𝜀𝑖
 不同公司的R和对应的bi1,bi2
 回归出FP/E, Fsize
No economic Standardized beta
𝑏𝑖𝑗 =
Regression (cross
sectional data)
𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎𝑡𝑡𝑟𝑖𝑏𝑢𝑡𝑒 𝑗 − 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎𝑡𝑡𝑟𝑖𝑏𝑢𝑡𝑒 𝑗
𝜎 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎𝑡𝑡𝑟𝑖𝑏𝑢𝑡𝑒 𝑗
𝑃/𝐸 1 − 𝑃/𝐸
𝑒. 𝑔. 𝑏𝑖1 =
 Factor return
 𝐹𝑃
= return associated with P/E factor
 𝐹𝑆𝑖𝑧𝑒 = return associated with size factor
 Specify the factor sensitivities first and then estimate the factor returns
through regressions
Standardized Beta
 𝒃𝒊𝒋 : standardized beta
 The scaling permits all factor sensitivities to be interpreted similarly,
despite differences in units of measure and scale in the variables.
 𝑏1 =-2, stock has a P/E that is 2 standard deviation below the mean
 Example: An investment has a dividend yield of 3.5 percent and that
the average dividend yield across all stocks being considered is 2.5
percent. Further, suppose that the standard deviation of dividend yields
across all stocks is 2 percent.
 Standardized beta of dividend yield is (3.5% - 2.5%)/2% = 0.50, or
one-half standard deviation above average.
Standardized Beta: binary variables
 The exception to this interpretation is factors for binary variables such as
industry membership. A company either participates in an industry or it
does not.
 The industry factor sensitivities would be 0 - 1 dummy variables;
 The value of the variable is 1 if the stock belongs to the industry and 0 if
it does not.
2.3 Statistical Factor Models
 Statistical factor models
 Statistical methods are applied to historical returns of a group of securities
to extract factors that can explain the observed returns of securities in the
 The factors are actually portfolios of the securities in the group under
study and are therefore defined by portfolio weights.
 Two major types of factor models are factor analysis models and principal
components models.
 Factor analysis models best explain historical return covariances.
 Principal components models best explain the historical return
 Advantage and Disadvantage
 Major advantage: it make minimal assumptions, can be most easily
applied to various asset classes, including fixed income.
 Major weakness: the interpretation of statistical factors is generally
difficult in contrast to macroeconomic and fundamental factors
Model Comparison
 The relation between APT and Macroeconomic models
Multifactor models
cross-sectional equilibrium
time-series regression that
pricing model that explains the
explains the variation over time in
variation across assets’
returns for one asset
expected returns
equilibrium-pricing model that
assumes no arbitrage
Regression model that the factors
are identified empirically by
looking for macroeconomic
variables that best fit the data.
expected return derived from the
APT equation in macroeconomic
factor model
risk-free rate
2.4 Fixed-Income Multifactor Models
 Macroeconomic Multifactor Models
 Consider a bond factor model with two factors.
 𝑅𝑖 = 𝛼𝑖 + 𝑏𝑖1 𝐹𝐼𝑁𝐹𝐿 + 𝑏𝑖2 𝐹𝐺𝐷𝑃 + 𝜀𝑖
 𝑅𝑖 =the return to bond i
 𝛼𝑖 =the expected return to bond i
 𝑏𝑖1 =the sensitivity of the return on bond i to inflation rate surprises
 𝐹𝐼𝑁𝐹𝐿 =the surprise in inflation rates
 𝑏𝑖2 =the sensitivity of the return on bond i to GDP growth surprises
 𝐹𝐺𝐷𝑃 =the surprises in GDP growth
 𝜀𝑖 =an error term with a zero mean that represents the portion of the
return to bond i not explained by the factor model.
Fixed-Income Multifactor Models
 Fundamental Multifactor Models
 The US Barclays Bloomberg Aggregate index can be divided into sectors,
where each has such unique factor exposures as spread or duration.
 These components can be thought of both macroeconomic and
Fixed-Income Multifactor Models
 Fundamental Multifactor Models
 The simplistic approach:
𝑅𝑖 = 𝑎𝑖 + 𝑏𝑖1 𝐹𝐺𝑣𝑡_𝑆ℎ + 𝑏𝑖2 𝐹𝐺𝑣𝑡_𝐼𝑛𝑡 + 𝑏𝑖3 𝐹𝐺𝑣𝑡_𝐿𝑔 + 𝑏𝑖4 𝐹𝐼𝑛𝑣𝑒𝑠𝑡 + 𝑏𝑖5 𝐹𝐻𝑖𝑌𝑙𝑑 + 𝑏𝑖6 𝐹𝑀𝐵𝑆 + 𝜀𝑖
 𝑅𝑖 =the return to bond i
 𝑎𝑖 =the expected return to bond i
 𝑏𝑖𝑘 =the sensitivity of the return on bond i to factor k
 𝐹𝑘 =factor k, where k represents “Gov’t (Short),” “Gov’t (Long),” and so on
 𝜀𝑖 =an error term with a zero mean that represents the portion of the return
to bond i not explained by the factor model
 The historic style factor weights, 𝑏𝑖𝑘 , are determined by a constrained regression
(the total “weights” add up to 100%) of the portfolio returns against the listed
style factors.
 This framework lends itself readily to performance and risk attribution, along with
portfolio construction, it can also be extended to ESG considerations.
Fixed-Income Multifactor Models
 Risk and Style Multifactor Models
 Another category of multifactor approach incorporates risk, or style,
factors , factors, several of which can thematically apply across asset
 Examples: momentum, value, carry, and volatility
 Statistical models can be most easily applied to various asset classes,
including fixed income, as no asset-class- specific tuning is required
given the minimal required assumption set.
 Macroeconomic and fundamental models both require adjustments and
repurposing to ensure the frameworks are fit for the specifics of bond
Fixed-Income Multifactor Models
 Talia Ayalon is evaluating intermediate duration (between 5 and 7 years)
investment-grade fixed-income strategies using the framework
presented in Exhibit 1. One of the strategies has the following sector
attribution (totaling to 100%):
Gov’t (Short) 2%
Gov’t (Intermediate) 4%
Gov’t (Long) 14%
Investment-Grade Credit 56%
MBS/Securitized 6%
High Yield 18%
 Are these sector exposures consistent with an intermediate duration
investment-grade approach? Why or why not?
Fixed-Income Multifactor Models
 Solution:
 No, the sector exposures are inconsistent with the stated approach
for two reasons:
 1) The 18% exposure to high yield constitutes a significant amount
of below investment-grade exposure. A true investment-grade
portfolio would, for example, not have exposure to high yield.
 2) The loading to longer duration sectors implies a longer-thanintermediate duration for the portfolio.
3.1 Application: Return Attribution
 Multifactor models can help us understand in detail the sources of a
manager’s returns relative to a benchmark.
 Active return = Rp − RB
 With the help of a factor model, we can analyze a portfolio manager’s
active return as the sum of two components.
 Return from factor tilts: reflects the manager’s skill in asset class
 Return from security selection: reflects the manager’s skill in
individual asset selection
Application: Return Attribution
 Active return =Return from factor tilts + Return from security selection
 Return from factor tilts:
 overweight or underweight relative to the benchmark factor
 the factor returns
𝑅𝑒𝑡𝑢𝑟𝑛 𝑓𝑟𝑜𝑚 𝑓𝑎𝑐𝑡𝑜𝑟 𝑡𝑖𝑙𝑡𝑠
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑠𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦
− 𝐵𝑒𝑛𝑐ℎ𝑚𝑎𝑟𝑘 𝑠𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦
× 𝐹𝑎𝑐𝑡𝑜𝑟 𝑟𝑒𝑡𝑢𝑟𝑛
 Return from security selection:
 ability to overweight securities that outperform the benchmark or
underweight securities that underperform the benchmark.
 𝑅𝑝 = 𝐸 𝑅𝑝 + 𝑏𝑃,1 𝐹1 + 𝑏𝑃,2 𝐹2 + 𝑏𝑃,3 𝐹3 + 𝑏𝑃,4 𝐹4 + ε𝑃
 That manager's benchmark is an index representing the performance of
the 1,000 largest US stocks by market value. The manager describes
himself as a “stock selection winner”
Contribution to Active
Factor Sensitivity
(3)= (1)-(2)
Return (4)
(3) X (4)
Proportion of
Total Active
A. Return from Factor Tilts =
B. Security Selection =
C. Active Return (A + B) =
1. Evaluate the sources of the managers active return for the year.
 Correct Answer:
 The dominant source of the manager s positive active return was
his positive active exposure to the 𝐹3 factor. The bet contributed
approximately 98% of the realized active return of about 2.07%.
2. What concerns might Boss discuss with the manager as a result of the
return decomposition?
 Correct Answer:
 Although the manager is a self-described “stock selection winner,”
his active return from security selection in this period was actually
3.2 Application: Risk Attribution
 Active risk
 the standard deviation of active returns.(tracking error ,or tracking risk)
𝐴𝑐𝑡𝑖𝑣𝑒 𝑟𝑖𝑠𝑘 = 𝑆
 Active risk squared is the variance of active return:
𝐴𝑐𝑡𝑖𝑣𝑒 𝑟𝑖𝑠𝑘 𝑠𝑞𝑢𝑎𝑟𝑒𝑑 = 𝑆 2 𝑅𝑃 − 𝑅𝐵
 Active risk squared can be separated into two components:
 Active factor risk is the contribution to active risk squared resulting
from the portfolio's different-than-benchmark exposures. It represents
the part of active risk squared explained by the portfolio’s active factor
 Active specific risk or security selection risk measures the active nonfactor or residual risk assumed by the manager.
 Active risk squared = Active factor risk + Active specific risk
3.2 Application: Risk Attribution
 Information Ratio
 Definition: the ratio of mean active return to active risk
 Purpose: a tool for evaluating mean active returns per unit of active risk
 Exact formula:
𝐼𝑅 =
 Example: To illustrate the calculation, if a portfolio achieved a mean
return of 9 percent during the same period that its benchmark earned a
mean return of 7.5 percent, and the portfolio's tracking risk was 6
percent, we would calculate an information ratio of (9% - 7.5%)/6% =
 The higher the IR, the more active return the manager earned per
unit of active risk.
 Richard is comparing the risk of three US equity managers who share
the same benchmark. Exhibit below presents Richard’s analysis of the
active risk squared of the three managers.
Active Risk Squared Decomposition
Active Factor
Industry Style Factor Total Factor
Active Risk
 Questions:
 Compare the active risk decomposition of Portfolios A and B.
 Characterize the investment approach of Portfolio C.
 Correct Answer:
1. Restate the exhibit to show the proportional contributions of the
various sources of active risk.
Active Factor (% of total active)
Active Specific Active
Style Factor Total Factor (% of total active) Risk
Portfolio A: a higher level of active risk than B (7% versus 5%) and
substantial active industry risk
Portfolio B: approximately industry neutral relative to the
benchmark and higher active bets on the style factors representing
company and share characteristics.
2. Portfolio C appears to be a passively managed portfolio, judging
by its negligible level of active risk.
Application: Portfolio Construction
 Passive management. Analysts can use multifactor models to replicate an
index fund's factor exposures, mirroring those of the index tracked.
 Active management. Many quantitative investment managers rely on
multifactor models in predicting alpha (excess risk-adjusted returns) or
relative return (the return on one asset or asset class relative to that of
another) as part of a variety of active investment strategies.
Application: Portfolio Construction
 Rules-based active management (alternative indexes). These strategies
routinely tilt toward factors such as size, value, quality, or momentum when
constructing portfolios.
 The Carhart four-factor model (four factor model)
 ERP=RF+β1RMRF+ β2SML+ β3HML+ β4WML
 According to the model, there are three groups of stocks that tend
to have higher returns than those predicted solely by their sensitivity
to the market return:
 SMB=Return of Small – Return of Big
 HML: return of high BV/MV – return of low BV/MV
 Stocks whose prices have been rising, commonly referred to as
“momentum” stocks: WML=Return of Winner – return of Loser
Application: Strategic Portfolio Decisions
 Benefits for investors considering multiple risk dimensions
 A multifactor approach can help investors achieve better-diversified and
possibly more-efficient portfolios.
 The characteristics of a portfolio can be better explained by a
combination of SMB, HML, and WML factors in addition to the
market factor than by using the market factor alone.
 Compared with single-factor models, multifactor models offer a richer
context for investors to search for ways to improve portfolio selection.
Measuring and managing market risk
1. Understanding VaR
2. Estimate VaR
3. Extensions of VaR
4. Other key risk measures
5. Applications of risk measures
6. Using constraints in market risk
Understanding VaR
 Value at risk (VaR) is the minimum loss that would be expected a certain
percentage of the time over a certain period of time given the assumed
market conditions.
 Important concept:
 VaR can be measured in either currency units (in this example, the euro)
or in percentage terms.
 VaR is a minimum loss.
 A statement references a time horizon: losses that would be expected to
occur over a given period of time.
Understanding VaR
 Analysis should consider some additional issues with VaR:
 The VaR time period should relate to the nature of the situation. A
traditional stock and bond portfolio would likely focus on a longer
monthly or quarterly VaR while a highly leveraged derivatives portfolio
might focus on a shorter daily VaR.
 The percentage selected will affect the VaR. A 1% VaR would be
expected to show greater risk than a 5% VaR.
 The left-tail should be examined. Left-tail refers to a traditional
probability distribution graph of returns. The left side displays the low or
negative returns, which is what VaR measures at some probability. But
suppose the 5% VaR is losing $ 1.37 million, what happens at 4%, 1%,
and so on? In other words, how much worse can it get?
Understanding VaR
 Given a VaR of $12.5 million at 5% for one month, which of the
following statements is correct?
A. There is a 5% chance of losing $12.5 million over one month.
B. There is a 95% chance that the expected loss over the next month
is less than $12.5 million.
C. The minimum loss that would be expected to occur over one
month 5% of the time is $12.5 million.
 Correct Answer : C
Estimating VaR
 3 methods to estimate VaR:
 Parametric method (variance-covariance)
 Historical simulation method
 Monte Carlo method
Understanding VaR
 The parametric method (or variance-covariance/analytical method) is
based on the normal distribution and the concept of one-tailed confidence
intervals. It uses the expected return and standard deviation of return to
estimate the VaR.
 Example: Parametric VaR
 The expected annual return for a $1 00,000,000 portfolio is 6.0% and the
historical standard deviation is 12%. Calculate VaR at 5% significance.
 5% in a single tail is associated with 1.645, or approximately 1.65,
standard deviations from the mean expected return. Therefore, the 5%
annual VaR is:
𝑉𝑎𝑅 = 𝑅𝑃 − 𝑧 × 𝜎 × 𝑉𝑃
= 6% − 1.65 × 12% × $100,000,000
= $13,800,000
The Confidence Intervals
 68% confidence interval is
𝜇 − 𝜎, 𝜇 + 𝜎
90% confidence interval is
𝜇 − 1.65𝜎, 𝜇 + 1.65𝜎
95% confidence interval is
𝜇 − 1.96𝜎, 𝜇 + 1.96𝜎
98% confidence interval is
𝜇 − 2.33𝜎, 𝜇 + 2.33𝜎
99% confidence interval is
𝜇 − 2.58𝜎, 𝜇 + 2.58𝜎
μ-2.58σ μ-1.96σ
μ+1.96σ μ-2.58σ
For the Exam
 5% VaR is 1.65 standard deviations below the mean.
 1% VaR is 2.33 standard deviations below the mean.
 VaR for periods less than a year are computed with return and standard
deviations expressed for the desired period of time.
 For monthly VaR, divide the annual return by 12 and the standard
deviation by the square root of 12. Then, compute monthly VaR.
 For weekly VaR, divide the annual return by 52 and the standard
deviation by the square root of 52. Then, compute weekly VaR.
 For a very short period (1-day) VaR can be approximated by ignoring the
return component (i.e., enter the return as zero). This will make the VaR
estimate worse as no return is considered, but over one day the expected
return should be small.
 The expected annual return for a $1 00,000,000 portfolio is 6.0% and
the historical standard deviation is 12%. Calculate weekly VaR at 1%.
 The number of standard deviations for a 1% VaR will be 2.33 below
the mean return.
 The weekly return will be 6%/52 = 0.1154%. The weekly standard
deviation will be 12%/521/2 = 1.6641%
 VaR = 0.1154% -2.33(1.6641%) = -3.7620%
Which of the following statements is not correct?
A. A 1% VaR implies a downward move of 1%.
B. A one standard deviation downward move is equivalent to a 16%
C. A 5% VaR implies a move of 1.65 standard deviations less than the
expected value.
 Correct Answer: A
Parametric (Variance-Covariance) Method
 Advantages of the Parametric method include:
 Simplicity and straightforwardness.
 Disadvantages of the Parametric method include:
 It can be difficult to use when the investment portfolio contains options.
 This leads to a non-normal distribution that does not lend itself well
to the parametric method.
Historical Simulation Method
 Use the historical data to find out Value at risk.(See example)
 Advantages of the historical simulation method include:
 It estimates VaR based on what actually happened, so it cannot be
dismissed as introducing impossible outcomes.
 Does not assume a returns distribution.
 The primary disadvantage
 There can be no certainty that a historical event will re-occur, or that it
would occur in the same manner or with the same likelihood as
represented by the historical data.
 You have accumulated 100 daily returns for your $100,000,000
portfolio. After ranking the returns from highest to lowest, you identify
the lower five returns:
-0.0019, -0.0025, -0.0034, -0.0096, -0.0101
Calculate daily VaR at 5% significant using the historical method.
 Answer:
Since these are the lowest five returns, they represent the 5% lower tail
of the “distribution” of 100 historical returns. The fifth lowest return (0.0019) is the 5% daily VaR. We should ay there is a 5% chance of a
daily loss exceeding 0.19%, or $190,000.
Monte Carlo Simulation Method
 The user develops his own assumptions about the statistical characteristics
of the distribution and uses those characteristics to generate random
outcomes that represent hypothetical returns
 A Monte Carlo output specifies the expected 1-week portfolio return and
standard deviation as 0.00188 and 0.0125, respectively.
 Calculate the 1-week value at risk at 5% significance.
𝑉𝑎𝑅 = 𝑅𝑃 − 𝑧 𝜎 × 𝑉𝑃
= 0.00188 − 1.65 0.0125
= 0.018745 $100,000,000
= $1,874,500
Monte Carlo Simulation Method
 Advantage of the Monte Carlo method
 It can incorporate virtually any assumptions regarding return patterns,
correlations, and other factors the analyst believes are relevant (Flexible).
 It avoids the complexity inherent in the parametric method when the
portfolio has a large number of assets.
 Disadvantages:
 The more random value we use, the more reliable our answers are but the
more time-consuming then procedure becomes.
 Must take correlation into account.
Advantages and Limitations of VaR
 Advantages
 Simple concept.
 Easily communicated concept.
 Provides a basis for risk comparison.
 Facilitates capital allocation decisions.
 Can be used for performance evaluation.
 Reliability can be verified.
 Widely accepted by regulators.
Advantages and Limitations of VaR
 Limitation
 Subjectivity.
 Underestimating the frequency of extreme events.
 Failure to take into account liquidity.
 Sensitivity to correlation risk.
 Vulnerability to trending or volatility regimes.
 Misunderstanding the meaning of VaR.
 Oversimplification.
 Disregard of right-tail events.
Extensions of VaR
 Conditional VaR (CVaR): the average loss that would be incurred if the VaR
cutoff is exceeded. CVaR is also sometimes referred to as the expected tail
loss or expected shortfall.
 Incremental VaR (IVaR): how the portfolio VaR will change if a position size
is changed relative to the remaining positions.
 Marginal VaR (MVaR): it is conceptually similar to incremental VaR in that it
reflects the effect of an anticipated change in the portfolio, but it uses
formulas derived from calculus to reflect the effect of a very small change in
the position.
 Ex ante tracking error, also known as relative VaR: a measure of the
degree to which the performance of a given investment portfolio might
deviate from its benchmark.
Other Key Risk Measures
 Sensitivity: how performance responds to a single change in an underlying
risk factor.
 Equity Exposure Measures: Beta from CAPM
 Fixed-income Exposure Measure: duration, convexity
 Options Risk Measures: Delta, Gamma, Vega
 Scenario Risk Measures
 Historical scenarios are scenarios that measure the hypothetical
portfolio return that would result from a repeat of a particular period of
financial market history.
 Hypothetical scenarios—extreme movements and co-movements in
different markets that have not necessarily previously occurred. The
scenarios used are somewhat difficult to believe, and it is difficult to
assess their probability, but they represent the only real method to
assess portfolio outcomes under market movements that might be
imagined but that have not yet been experienced.
Other Key Risk Measures
 The two elements that set scenario risk measures apart from sensitivity risk
measures are
 (1) the use of multiple factor movements used in the scenario
measures versus the single factors movements typically used in risk
sensitivity measures.
 (2) the typically larger size of the factor movement used in the
scenario measures.
Other Key Risk Measures
 Scenario analysis and stress tests
 Stress tests, which apply extreme negative stress to a particular
portfolio exposure, are closely related to scenario risk measures.
 Scenario analysis is an open-ended exercise that could look at positive
or negative events.
 To design an effective hypothetical scenario, it is necessary to identify
the portfolio’s most significant exposures. Targeting these material
exposures and assessing their behavior in various environments is a
process called reverse stress testing.
Other Key Risk Measures
 Compare sensitivity and scenario risk measures to VaR
 VaR is a measure of losses and the probability of large losses
 Sensitivity risk measures capture changes in the value of an asset in
response to a change in something else, they do not tell us anything
about the probability of a given change in value occurring.
 Both VaR and scenatio risk measures estimate portential loss
 VaR is vulnerable if correlation relationship and market volatility
during the period in question are not representative of the
conditions of the portfolio may face in future
 Scenario analysis allows either the risk assessment to be fully
hypothetical or to be linked to a different and more extreme period
of history.
Advantages and Limitations of Other Measures
 Sensitivity
 Advantage
 address some of the shortcomings of position size measures.
For example, duration addresses the difference between a 1year note and a 30-year note, it measures the level of interest
rate risk.
 It does not need to rely on history.
 Limitations
 Do not often distinguish assets by volatility, which makes it less
Advantages and Limitations of Other Measures
 Scenario Risk Measures
 Advantage
 Do not need to rely on history;
 Overcome any assumption of normal distributions;
 Can be tailored to expose a portfolio’s most concentrated positions
to even worse movement than its other exposures;
 Allowing liquidity to be taken into account.
 Limitations
 Historical scenarios are not going to happen in exactly the same way
 Hypothetical scenarios may incorrectly specify how assets will comove, they may get the magnitude of movements wrong;
 Hypothetical scenarios can be very difficult to create;
 It is very difficult to know how to establish the appropriate limits on
a scenario analysis or stress test.
Applications of Risk Measures
 Banks: Liquidity gap, VaR, sensitivities, economic capital, scenario analysis.
 Asset Managers:
 Traditional Asset Managers: position limits, sensitivities, beta sensitivity ,
liquidity, scenario analysis, active share, redemption risk, ex post versus
ex ante tracking error, VaR .
 Hedge Funds: sensitivities, gross exposure ,leverage, VaR, scenario
analysis, maximum drawdown.
 Pension Fund:
 interest rate and curve risk, surplus at risk, glide path, liability hedging
exposures versus return generating exposures.
 Insurers:
 Property and casualty insurers: sensitivities and exposures, economic
capital, VaR, scenario analysis
 Life Insurers: sensitivities, asset and liability matching, scenario analysis.
Using Constraints in Market Risk Management
 Risk budgeting: the total risk appetite of the firm or portfolio is agreed on at
the highest level of the entity and then allocated to sub-activities.
 Position limits place a nominal dollar cap on positions.
 Scenario limits is a limit on the estimated loss for a given scenario, which if
exceeded, would require corrective action in the portfolio
 Stop-loss Limits sets an absolute dollar limits for losses over a certain period.
 Risk Measures and Capital Allocation
 Capital allocation is the practice of placing limits on each of a company’s
activities in order to ensure that the areas in which it expects the greatest
reward and has the greatest expertise are given the resources needed to
accomplish their goals.
 E.g. Economic Capital
Backtesting and Simulation
1. Objectives of backtesting
2. Backtesting process
3. Problems in a backtest
4. Historical scenario analysis
5. Simulation
6. Sensitivity analysis
1. Objectives of Backtesting
 Backtesting approximates the real-life investment process by using
historical data to assess whether a strategy would have produced desirable
 Backtesting can offer investors insight and rigor to the investment process.
 Backtesting can be employed as a rejection or acceptance criterion for an
investment strategy.
 Backtesting fits quantitiative and systematic investment styles, it is also
widely used by fundamental managers.
 Before using a criterion to screen for stocks, a backtest can uncover the
historical efficacy of that criterion by determining if its use would have
added incremental excess return.
2. Backtesting Process
 Steps and procedures
 Step 1: Strategy design
 Specify investment hypothesis and goals
 Determine investment rules and process
 Decide key parameters
 Step 2: Historical investment simulation
 Form investment portfolios for each period according to the rules
specified in the previous step
 Rebalance the portfolio periodically based on pre-determined rules
 Step 3: Analysis of backtesting output
 Calculate portfolio performance statistics
 Compute other key metrics
2. Backtesting Process
 Key parameters
 Investment universe
 It refers to all of the securities in which we can potentially invest
 Return definition
 in what currency the return should be computed
translate all investment returns into one single currency
denominate returns in local currencies
 The benchmark used is often the benchmark for the client mandate or
fund for which the investment strategy under study is applicable.
2. Backtesting Process
 Key parameters
 Rebalancing frequency and transaction cost
 Practitioners often use amonthly frequency for portfolio rebalancing
 daily or higher frequency rebalancing typically incurs higher transaction
 Start and end date
 All else equal, investment managers prefer to backtest investment
strategies using as long a history as possible
 performance over a long data history should be supplemented with
examinations of discrete regimes within the long history using historical
scenario analysis
2. Backtesting Process
 Historical investment simulation
 To simulate rebalancing, analysts typically use rolling windows, in which a
portfolio or strategy is constituted at the beginning of a period using data
from a historical in-sample period, followed by testing on a subsequent,
out-of-sample period (OOS). The process is repeated as time moves
forward and replicates the live investing process, because investment
managers adjust their positions as new information arrives
2. Backtesting Process
 Analysis of backtesting output
 Analyst often use metrics such as the Sharpe ratio, the Sortino ratio,
volatility, and maximum drawdown (the maximum loss from a peak to a
trough for an asset or portfolio), other key performance ouputs are visual
 It is also useful to examine the backtested cumulative performance of an
investment strategy over an extended history, plotting performance using
a logarithmic scale is recommended.
 Backtesting implicitly assumes that the past is likely to repeat itself, but
this assumption does not fully account for the dynamic nature of financial
markets, which may include extreme upside and downside risks that have
never occurred before.
 Stuctural breaks(regime changes) are one reason
 Regarding rolling-window backtesting, which one of the following
statements is inaccurate?
A. The data are divided into just two samples.
B. Out-of-sample data become part of the next period’s in-sample data.
C. Repeated in-sample training and out-of-sample testing allow
managers to adjust security positions on the basis of the arrival
over time of new information.
 Answer: A
3. Problems in Backtesting
 Survivorship bias
 Stocks that have remained in the index over time are referred to as
 Survivorship bias refers to deriving conclusions from data that reflects
only those entities that have survived to that date.
 Point-in-time data allow analysts to use the most complete data for any
given prior time period, thereby enabling the construction (and
backtesting) of the most realistic investment strategies
3. Problems in Backtesting
 Survivorship bias
 Point-in time data VS Survivor
3. Problems in Backtesting
 Look-ahead bias
 Using information that was unknown or unavailable during the historical
periods over which the backtest is conducted.
 Survivorship bias is a type of look-ahead bias, it can be overcome by using
point-in-time data.
 It is likely the most common mistake that practitioners make when
performing backtesting.
 Look-ahead bias has several common forms: reporting lags, revisions,
and index additions
3. Problems in Backtesting
 Look-ahead bias
 Reporting lags
 in conducting a backtest for year-end 2018, EPS results for the quarter
ending 31 December 2018 are unavailable until 2019.
 to avoid look-ahead bias, analysts typically compensate by adding
several months of reporting lag for every company
Usually 30-50 days of quarter end, longer lag will introduce stale
3. Problems in Backtesting
 Look-ahead bias
 Data revisions
 Macroeconomic data are often revised multiple times, and companies
often re-state their financial statements.
 Index Additions
 Data vendors add new companies to their databases. An analyst
backtesting with the current database would be using information on
companies that were not actually in the database during the
backtesting period.
3. Problems in Backtesting
 Data Snooping (p-hacking)
 making an inference after looking at statistical results rather than testing a
prior inference
 It occurs when an analyst selects data or performs analyses until a
significant result is found
 Data snooping may be mitigated by setting a much higher hurdle than
 t-statistic>3, to assess whether a new factor is indeed adding value.
 Another technique to detect and mitigate data snooping is cross
validation. Rolling window backtesting is a form of cross-validation.
 the analyst partitions the dataset into training data and validation data
and tests a model built from the training data on the validation data
 An analyst develops an investment strategy by picking the strategy with the
highest t-statistic and lowest p-value after backtesting dozens of different
strategies. This approach is an example of which common problem in
A. Reporting lag
B. Survivorship bias
C. Data snooping
 Answer: C
 Data snooping refers to making an inference—such as formulating an
investment strategy—after looking at statistical results rather than
testing a prior inference
 Which of the following is an example of cross-validation?
A. Maximum drawdown
B. Backtesting with out-of-sample data
C. Incorporating point-in-time data
 Answer: B
 Cross-validation is a technique that involves testing a hypothesis
on a different set of data than that which was used to form the
inference or initially test the hypothesis. Choice B is the definition
of cross-validation.
4. Historical Scenario Analysis
 Historical scenario analysis is a type of backtesting that explores the
performance and risk of an investment strategy in different structural regimes
and at structural breaks.
 Regime change
 Expansions and recessions
 High-and low-volatility regimes
 Which of the following situations is least likely to involve scenario analysis?
A. Simulating the performance and risk of investment strategies by
first using stocks in the Nikkei 225 Index and then using stocks in
the TOPIX 1000 Index.
B. Simulating the performance and risk of investment strategies in both
“trade agreement” and “no-trade-agreement” environments.
C. Simulating the performance and risk of investment strategies in both
high-volatility and low-volatility environments.
 Answer: A
 A is correct, because there is no structural break or different structural
5. Simulation
 Historical simulation: construct results by selecting returns at random from
many different historical periods (windows) without regard to time-ordering.
 The problem with historical time-series data is that there is only one set of
realized data to draw from, but most financial variables are not stationary.
 In Monte Carlo simulation, each key variable is assigned a statistical
distribution, and observations are drawn at random from the assigned
 Advantage: highly flexible
 Disadvantage: complex and computationally intensive
 Simulation is especially useful in measuring the downside risk of investment
5. Simulation
 Eight steps of simulation
 Determine what we want to understand: the target variable.
 Specify key decision variables.
 Specify the number of trials (N) to run.
 Define the distributional properties of the key decision variables.
 Use a random number generator (inverse transformation) to draw N
random numbers for each key decision variable.
 For each set of simulated key decision variables, compute the value of
the target variable.
 Repeat the same processes from Steps 5 and 6 until completing the
desired number of trials (N).
 Calculate the typical metrics, such as mean return, volatility, Sharpe ratio,
and the various downside risk metrics(CVaR and maximum drawdown)
5. Simulation
 Historical simulation
 Backtesting and historical simulation are different in that rolling-window
backtesting is deterministic, whereas historical simulation incorporates
randomness rather than following each period chronologically.
 First, a decision must be made about whether to sample from the
historical returns with replacement (bootstrapping) or without
 Bootstrapping is used because the number of simulations needed is
often larger than the size of the historical dataset.
 Then perform a historical simulation (eight steps)
5. Simulation
 Monte Carlo simulation
 First, we need to specify a functional form for each key decision variable
 Regression and distribution-fitting techniques are used to estimate the
parameters underlying the statistical distributions of the key decision
variables. This step is called model calibration.
 Considerations for the functional form of the statistical distribution:
 The distribution should reasonably describe the key empirical patterns
of the underlying data.
 It is equally critical to account for the correlations between multiple
key decision variables, use multivariate distribution rather than
modeling each factor or asset on a standalone basis.
 The complexity of the functional form and number of parameters that
determine the functional form are important.
 Which one of the following statements concerning historical simulation and Monte
Carlo simulation is false?
A. Historical simulation randomly samples (with replacement) from the past record
of asset returns, where each set of past monthly returns is equally likely to be
B. Neither historical simulation nor Monte Carlo simulation makes use of a
random number generator.
C. Monte Carlo simulation randomly samples from an assumed multivariate joint
probability distribution in which the past record of asset returns is used to
calibrate the parameters of the multivariate distribution.
 Answer: B
6. Sensitivity Analysis
 Sensitivity analysis is a technique for exploring how a target variable is
affected by changes in input variables (e.g., the distribution of asset or factor
 Monte Carlo simulation fits a multivariate normal distribution, which fails to
account for negative skewness and fat tails. We should conduct a sensitivity
analysis by fitting our factor return data to a different distribution and
repeating the Monte Carlo simulation accordingly
 One alternative to test is a multivariate skewed Student’s t-distribution, it
has the ability to account for the skewness and the excess kurtosis.
 Which of the following situations concerning simulation of a multifactor
asset allocation strategy is most likely to involve sensitivity analysis?
A. Changing the specified multivariate distribution assumption from a
normal to a skewed t-distribution to better account for skewness and fat
B. Splitting the rolling window between periods of recession and
C. Splitting the rolling window between periods of high volatility and low
 Answer: A
 B and C are incorrect because these choices represent scenario analysis.
Economics and investment markets
1. Framework for the analysis of financial
2. The discount rate on real default-free
3. The discount rate on nominal default
free bonds
4. Credit premiums and the business cycle
5. Equities and the equity risk premium
6. Commercial real estate
1. Framework
 The present value model
𝐸𝑡 𝐶𝐹𝑡+𝑠
1 + 𝑙𝑡,𝑠 + 𝜃𝑡,𝑠 + 𝜌𝑡,𝑠
 where:
Pit = the value of the asset i at time t (today)
N = number of cash flows in the life of the asset
= the uncertain, nominal cash flow paid s periods in the future
𝐸𝑡 𝐶𝐹 = the expectation of the random variable CF conditional on the
information available to investors today (t)
lt,s = yield to maturity on a real default-free investment today (t), which
pays one unit of currency s periods in the future
θt,s = expected inflation rate between t and t + s
ρit,s= the risk premium required today (t) to pay the investor for taking on risk in
the cash flow of asset i, s periods in the future ( e.g., credit risk, liquidity risk)
2. The Discount Rate on Real Default-free Bonds
 What sort of return would investors require on a bond that is both
default –free and unaffected by future inflation?
 The choice to invest today involves the opportunity cost of not
consuming today.
 In this case, the investor can:
 Pay price Pt,s today, t, of a default –free bond paying 1 monetary unit
of income s periods in the future, or
 Buy goods worth Pt,s dollars today.
 The tradeoff is measured by the marginal utility of consumption s
periods in the future relative to the marginal utility of consumption
today (t).
 The marginal utility of consumption of investors diminishes as their
wealth increases because they have already satisfied fundamental
The Discount Rate on Real Default-free Bonds
 Inter-temporal rate of substitution
 The ratio of these two marginal utilities - the ratio of the marginal utility
of consumption periods s in the future (the numerator) to the marginal
utility of consumption today (the denominator).
 For a given quantity of consumption, investor always prefer current
consumption over future consumption and m<1.
 The rate of substitution is a random variable because an investor
will not know how much she has available in the future from other
sources of income.
 The Inter-temporal rate of substitution was lower at good state of
the economy, because individuals may have relatively high levels of
current income so that current consumption is high.
The Discount Rate on Real Default-free Bonds
 The investor must make the decision today based on her expectations of
future circumstances.
𝑷𝒕,𝒔 = 𝑬𝒕 1𝒎𝒕,𝒔 = 𝑬𝒕 𝒎𝒕,𝒔
 If this price of the bond was less than the investor’s expectation of the
inter-temporal rate of substitution, then she would prefer to buy more of
the bond today.
 As more bonds are purchased, today’s consumption falls and marginal
utility of consumption today rises, so that expectations conditional on
current information of the inter-temporal rate of substitution, 𝑬𝒕 𝒎𝒕,𝒔 ,
fall. This process continues until the rate of substitution is equal to the
The Discount Rate on Real Default-free Bonds
 If the investment horizon for this bond is one year, and the payoff then is $1,
the return on this bond can be written as the future payoff minus the current
payment relative to the current payment.
𝑇ℎ𝑒 𝑟𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝑡ℎ𝑖𝑠 𝑏𝑜𝑛𝑑 = 𝑙𝑡,1 =
𝑡 𝑚𝑡,1
 The one-period real risk-free rate is inversely related to the inter-temporal
rate of substitution.
 The higher the return the investor can earn, the more important current
consumption becomes relative to future consumption.
The Discount Rate on Real Default-free Bonds
 Pricing a s-period Default-Free Bond
𝑷𝒕,𝒔 =
𝑬𝒕 𝑷𝒕+𝟏,𝒔−𝟏
𝑬𝒕 𝑷𝒕+𝟏,𝒔−𝟏
+ 𝒄𝒐𝒗𝒕 𝑷𝒕+𝟏,𝒔−𝟏 , 𝒎𝒕,𝟏
𝟏 + 𝒍𝒕,𝟏
=Risk neutral present value: the asset’s expected future price
discounted at the risk-free rate.
 It represents a risky asset’s value if investors did not require
compensation for bearing risk.
The Discount Rate on Real Default-free Bonds
 Pricing a s-period Default-Free Bond (cont.)
 𝒄𝒐𝒗𝒕 𝑷𝒕+𝟏,𝒔−𝟏 , 𝒎𝒕,𝟏 =Covariance term is the discount for risk.
 Covariance term is zero for a one-period default-free bond, because
the future price is a known constant ($1).
 Covariance term<0, for most risky assets with risk-averse investors.
 Because the price of bond at time t+1 is uncertain, so the price of
bond at time t should be lower than a risk-free bond, which
indicates that the covariance term is negative.
 Covariance term>0, as economy goes extremely bad.
 When economy goes bad, treasury bond is of high demand because
it can be treated as a safe-haven assets to gain a risk-free return. In
that case, the price of the bond will increase.
 In the meantime, future income is decreased as bad economy
indicates and in turn the marginal utility of future consumption is
increased. In that case, the inter-temporal rate of substitution will
also increase.
The Discount Rate on Real Default-free Bonds
 Default-Free Interest Rates and Economic Growth
 An economy with higher trend real economic growth, other things being
equal, should have higher real default-free interest rates than an
economy with lower trend growth.
 Again, other things being equal, the real interest rates are higher in an
economy in which GDP growth is more volatile compared with real
interest rates in an economy in which growth is more stable.
1 + 𝑙𝑡,𝑠
1. What financial instrument is best suited to the study of the relationship
of real interest rates with the business cycle?
Default-free nominal bonds
Investment-grade corporate bonds
Default-free inflation-indexed bonds
 Correct Answer : C
2. The covariance between a risk-averse investor’s inter-temporal rates of
substitution and the expected future price of a risky asset is typically:
A. Negative
B. Zero
C. Positive
 Correct Answer : A
3. The relationship between the real risk-free interest rate and real GDP
growth is:
A. negative.
B. neutral.
C. Positive.
 Correct Answer: C
4. The relationship between the real risk-free interest rate and the
volatility of real GDP growth is:
A. negative.
B. neutral.
C. positive.
 Correct Answer: C
3. Short-term Nominal Interest Rate
 The pricing formula for a default-free nominal coupon-paying bond
𝑃𝑡𝑖 =
1 + 𝑙𝑡,𝑠 + 𝜃𝑡,𝑠 + 𝜋𝑡,𝑠
 Investors will want to be compensated by this bond for the inflation that
they expect between t and t + s, which we define as θt,s.
 Risk averse investors and thus need to be compensated for taking on
risk as well as seeking compensation for expected inflation, they will
also seek compensation for taking on the uncertainty related to future
inflation. We denote this risk premium by πt.
1. Suppose that an analyst estimates that the real risk-free rate is 1.25%
and that average inflation over the next year will be 2.5%. If the analyst
observes the price of a default-free bond with a face value of £100 and
one full year to maturity as being equal to £95.92, what would be the
implied premium embedded in the bond’s price for inflation
 Correct Answer :
 𝜋𝑡,𝑠 =0.504%=100/95.92-(1+0.0125+0.025)
Short-term Nominal Interest Rate
 Treasury bills (T-bills) are very short-dated nominal zero-coupon government
bonds: Because of their short-dated nature, the uncertainty that investors would
have about the inflation over an investment horizon is low, so we can ignore π.
1 + 𝑙𝑡,𝑠 + 𝜃𝑡,𝑠
 Short-term default-free interest rates tend to be very heavily influenced by:
 the inflation environment and inflation expectations over time
 real economic activity, which, in turn, is influenced by the saving and
investment decisions of households.
 It will also vary with the level of real economic growth and with the
expected volatility of that growth.
 the central bank’s policy rate, which, in turn, should fluctuate around the
neutral policy
 Taylor rule: Policy ratet =lt +ιt +0.5(ιt −ι∗t )+0.5(Yt −Y∗t )
Taylor Rule
 Taylor rule
 Policy ratet = 𝑙t + ιt + 0.5(ιt −ι∗t ) + 0.5(Yt −Y∗t )
 𝑙t is the level of real short-term interest rates that balance long-term
savings and borrowing in the economy
 ιt is the rate of inflation
 ι∗t is the target rate of inflation
 Yt and Y∗t are the logarithmic levels of actual and potential real GDP
 The difference between Yt and Y*t is known as the “output gap”.
 When the output gap is positive, it implies that the economy is
producing beyond its sustainable capacity.
 When the output gap is negative, it implies that the economy is
producing below its sustainable capacity.
 Neutral policy rate: the policy rate that neither spurs on nor impedes real
economic activity.
 ιt = ι∗t
 Output gap = 0
The Yield Curve and Business Cycle
 Break-even inflation rates
 The difference between the yield on, for example, a zero-coupon
default-free nominal bond and on a zero-coupon default-free real bond
of the same maturity is known as the break-even inflation (BEI) rate.
 It should be clear from the discussion earlier that this break-even
inflation rate will incorporate:
 the inflation expectations of investors over the investment horizon
of the two bonds, θt,s, plus
 a risk premium that will be required by investors to compensate
them predominantly for uncertainty about future inflation, πt,s.
The Yield Curve and Business Cycle
 Referring to government yield curves, expectations of increasing or
decreasing short-term interest rates might be connected to expectations
related to future inflation rates and/or the maturity structure of inflation risk
 Yield Curve
 Level, Slope, and Curvature of the Yield Curve
 The shape of yield curve: Upward sloping, Downward sloping, Hump, Flat
 An inverted yield curve is often read as being a predictor of recession.
 During a recession, short rates are often lower because central banks
tend to lower their policy rate in these times with negative output gap.
However, the impact of such monetary policy on longer-term rates will
not be as strong, so long rates may not fall by as much as short rats. Thus,
the slope of the yield curve will typically steepen during a recession.
 The yield spread between non-inflation-adjusted and inflation-indexed
bonds of the same maturity is affected by:
a risk premium for future inflation uncertainty only.
investors’ inflation expectations over the remaining maturity of
the bonds.
both a risk premium for future inflation uncertainty and investors’
inflation expectations over the remaining maturity of the bonds.
 Correct Answer : C
4. Credit Premiums and the Business Cycle
 Credit-risky bonds (corporate bond)
𝑃𝑡𝑖 =
1 + 𝑙𝑡,𝑠 + 𝜃𝑡,𝑠 + π𝑡,𝑠 + 𝛾𝑡,𝑠
 Credit spread: the difference between the yield on a corporate bond and that
on a government bond with the same currency denomination and maturity, and
𝜸𝒊𝒕,𝒔 is the credit premium.
 Bonds spreads do tend to rise in the lead up to and during a recession, and
to decline once the economy comes out of recession.
 If we assume that investors are risk neutral:
 Expected loss = Probability of default × (1 – Recovery rate)
 Recovery rates tend to be higher
 for secured as opposed to unsecured debt holders
 when the economy is expanding and lower when it is contracting
Factors Influencing Credit Spread
 Industry sector and credit quality:
 Credit spreads between corporate bond sectors with different ratings will
often have very different sensitivities to the business cycle
 Some industrial sectors are more sensitive to the business cycle than
others. During recession, the spread on the consumer cyclical sector rose
more dramatically than it did for corporate bonds in the consumer noncyclical sector.
 Company-specific factors:
 Issuers that are profitable, have low debt interest payments, and that are not
heavily reliant on debt financing will tend to have a high credit rating
because their ability to pay is commensurately high.
 If this ability declines relative to other issuers in their sector, then the
spread demanded on their debt will rise and their rating may be
lowered by the rating agency.
Factors Influencing Credit Spread
 Sovereign credit risk
 The credit risk embodied in bonds issued by governments in emerging
markets is normally expressed by comparing the yields on these bonds
with the yields on bonds with comparable maturity issued by the US
 Credit premiums have always been an important component of the
expected return on bonds issued by governments in developing or
emerging economies.
 sovereign issuers’ ability to pay and the likelihood that they might
 The category of bonds whose spreads can be expected to widen the
most during an economic downturn are bonds from the:
cyclical sector with low credit ratings.
cyclical sector with high credit ratings.
non-cyclical sector with low credit ratings.
 Correct Answer : A
 Northwest bank’s 20-year bonds are currently yielding 8%. The real risk
free rate is 3%, and expected inflation is 2%, credit spread on
Northwest bank bonds is:
Equal to 3%
Less than 3%
Greater than 3%
 Correct Answer : B
5. Equities and the Equity Risk Premium
𝑃𝑡𝑖 =
𝐸𝑡 𝐶𝐹𝑡+𝑠
1 + 𝑙𝑡,𝑠 + 𝜃𝑡,𝑠 + 𝜋𝑡,𝑠 + 𝛾𝑡,𝑠
+ 𝜿𝑖𝑡,𝑠
 𝑙𝑡,𝑠 + 𝜃𝑡,𝑠 + 𝜋𝑡,𝑠 + 𝛾𝑡,𝑠
is the return that investors require for investing in
credit risky bonds.
 Equity risk premium is equal to 𝛾𝑡,𝑠
+ 𝜅𝑡,𝑠
 𝜿𝑖𝑡,𝑠 is essentially the equity premium relative to credit risky bonds.
 equity premium to be larger than the credit premium.
 The two premiums will tend to be positively correlated over time
 Sharp falls in equity prices are associated with recessions—bad times
 Bad consumption hedge, and we would thus expect the equity risk
premium to be positive.
Equities and the Equity Risk Premium
 Valuation multiples:
 P/E ratio tells investors the price they are paying for the shares as a
multiple of the company’s earnings per share
 if a stock is trading with a low P/E relative to the rest of the market,
it implies that investors are not willing to pay a high price for a
dollar’s worth of the company’s earnings.
 P/Es tend to rise during periods of economic expansion. Holding all
else constant, a relatively high P/E valuation level should be
associated with a lower return premium to bearing equity risk going
 The P/B tells investors the extent to which the value of their shares is
“covered” by the company’s net assets.
 The higher the ratio, the greater the expectations for growth but the
lower the safety margin if things do not turn out as expected.
Investment Strategy
 Investment styles
 Growth stocks
 Strong earnings growth
 High P/E and a very low dividend yield
 Have very low(or no) earnings
 Value stocks
 Operates in more mature markets with a lower earnings growth
 Low P/E and a very high dividend yield
 Value tends to outperform growth investing in the aftermath of a
recession, and that growth stocks tend to outperform value stocks in
times when the economy is expanding.
 Company size
 Generally speaking, one might expect small stocks to underperform large
stocks in bad times.
 Small stock companies will tend to have less diversified businesses and
have more difficulty in raising financing, particularly during recessions.
 One might expect investors to demand a higher equity premium on small.
 Summary
 During economic expansion, by rotating into growth stocks, or small-cap
stocks, a manager can, if correct, outperform a broad equity market index.
6. Commercial Real Estate
 Regular Cash Flow from Commercial Real Estate Investments
 The cash flow is derived from the rents paid by the tenants. These rents
are normally collected net of ownership costs.
 property investment is both bond-like and stock-like
 Most of the asset classes are liquid relative to an investment in commercial
 Other things being equal, illiquidity acts to reduce an asset class’s usefulness
as a hedge against bad consumption outcomes. Because of this, investors
will demand a liquidity risk premium, ϕt,s.
 The pro-cyclical nature of commercial property prices means that investors
will demand a higher risk premium in return for investing in this asset class.
𝑬𝒕 𝑪𝑭𝒊𝒕+𝒔
𝟏 + 𝒍𝒕,𝒔 + 𝜽𝒕,𝒔 + 𝝅𝒕,𝒔 + 𝜸𝒊𝒕,𝒔 + 𝜿𝒊𝒕,𝒔 + 𝝓𝒊𝒕,𝒔
 Which of the following statements relating to commercial real estate is
Rental income from commercial real estate is generally unstable
across business cycles.
Commercial real estate investments generally offer a good hedge
against bad consumption outcomes.
The key difference in the discount rates applied to the cash flows
of equity investments and commercial real estate investments
relate to liquidity.
 Correct Answer : C
Analysis of Active Portfolio Management
1. Value Added
2. Decomposition of value added
3. The Sharpe Ratio & Information Ratio
4. Constructing optimal portfolios
5. Information coefficient & Transfer
6. The basic fundamental law
7. The full fundamental law of active
Value Added
 The value added (active return) is the difference between the return on the
manage portfolio and the return on a passive benchmark portfolio.
𝑅𝐴 = 𝑅𝑃 − 𝑅𝐵
 A risk-adjusted value added (α), often captured by the portfolio’s beta
𝛼𝑝 = 𝑅𝑃 − 𝛽𝑃 𝑅𝐵
 Active weight(∆𝝎𝒊 ) is the differences in managed portfolio weights and
benchmark weights. Individual assets can be overweighed (positive active
weights) or underweighted (negative active weights), the sum of active weights is
 Value added can be the sum product of active weights and active security returns:
𝑅𝐴 =
∆𝜔𝑖 𝑅𝐴𝑖
Decomposition of Value Added
 The common decomposition: value added due to asset allocation and value
added due to security selection.
 The total value added is the difference between the actual portfolio and the
benchmark return:
𝑅𝐴 =
𝜔𝑃,𝑗 𝑅𝑃,𝑗 − 𝑅𝐵,𝑗 +
Security Selection
𝜔𝑃,𝑗 − 𝜔𝐵,𝑗 𝑅𝐵,𝑗
Asset Allocation
Example-Decomposition of Value Added
 Consider the fund returns in 2017 in the following table.
Fund Return (%)
Benchmark Return (%)
Portfolio Return
 Consider an investor who invested in both actively managed funds,
with 68% of the total portfolio in Fidelity and 32% in PIMCO and
assume that the investor’s policy portfolio (strategic asset allocation)
specifies weights of 60% for equities and 40% for bonds.
 Calculate the active return.
Example-Decomposition of Value Added
 Correct Answer:
 Value added from security selection:
𝑗=1 𝜔𝑃,𝑗
𝑅𝑃,𝑗 − 𝑅𝐵,𝑗
=0.68(35.3% – 32.3% ) + 0.32(-1.9% – (-2.0%) ) = 2.1%
 Value added by the active asset allocation
𝜔𝑃,𝑗 − 𝜔𝐵,𝑗 𝑅𝐵,𝑗
=(68% – 60%) (32.3%) + (32% – 40%) (-2.0%) = 2.7%.
 Total value added = 2.1% + 2.7% = 4.8%.
The Sharpe Ratio
 The Sharpe ratio measures reward per unit of risk in absolute returns.
 Sharpe ratio is unaffected by the addition of cash or leverage in a portfolio.
(created by borrowing risk-free cash)
𝑅𝐶 − 𝑅𝐹 𝜔𝑃 𝑅𝑃 − 𝑅𝐹
𝜔𝑃 𝜎𝑃
 Two-fund separation: Investors should form portfolios using the risk-free
asset and risky asset portfolio with the highest Sharpe ratio.
 If the expected volatility of the risky asset portfolio is higher than the
investor prefers, the volatility can be reduced by holding more cash
and less of the risky portfolio.
 If the expected volatility of the risky portfolio is lower than the investor
desires, the volatility and expected return can be increased by
 The current risk-free rate is 2.8%. The forecasted 0.50 Sharpe ratio of
the small-cap portfolio is higher than the 0.47 ratio of the large-cap
portfolio, but suppose the investor does not want the high 21.1%
volatility associated with the small-cap stocks.
Large Cap
Small Cap
Expected return
Expected volatility
Sharpe ratio
1. How much would an investor need to hold in cash (in percentage terms)
to reduce the risk of a portfolio invested in the small-cap portfolio and
cash to the same risk level as that of the large-cap portfolio?
 Correct Answer:
 We want to reduce the 21.1% volatility to 15.2% by adding cash.
The weight of small-cap stocks in the combined portfolio must
therefore be 15.2/21.1 = 72%, leaving a 28% weight in risk-free
 With that amount of cash, the volatility of the combined portfolio
will be 0.72(21.1%) = 15.2%, the same as the large-cap portfolio.
2. Based on your answer to 1, calculate the Sharpe ratio of the small-cap
plus cash portfolio.
 Correct Answer:
 The Sharpe ratio of the combined portfolio is unaffected by the
amount in cash, so it remains 0.50.
3. Compare the expected return of the small-cap plus cash portfolio with
the expected return of the large-cap portfolio.
 Correct Answer:
 expected return = 0.72(13.4%) + 0.28(2.8%) = 10.4%, 40 basis
points (bps) higher than the 10.0% expected return on the largecap portfolio but with the same risk as the large-cap portfolio.
 To reconfirm, the Sharpe ratio of the combined portfolio is (10.4% 2.8%)/15.2% = 0.50, the same as the original 0.50 value.
Information Ratio
 The information ratio measures reward per unit of risk in benchmark relative
𝐼𝑅 =
 The information ratio is affected by the addition of cash or the use of
 if the investor adds cash to a portfolio of risky assets, the information ratio
for the combined portfolio will generally shrink
 The information ratio of an unconstrained portfolio is unaffected by the
aggressiveness of active weights.
 Investor can adjust the active risk of an existing fund by taking positions in
the benchmark portfolio.
𝐼𝑅 =
𝜔𝑅𝑃 + 1 − 𝜔 𝑅𝐵 − 𝑅𝐵
 The blended portfolio is combined by an actively managed portfolio
and benchmark portfolio. Assume the active risk of the actively
managed fund is 5.0%, combining that fund in an 80/20 mix with the
benchmark portfolio (i.e., a benchmark portfolio weight of 20%) will
result in an active risk of the combined portfolio of 0.80(5.0%) = 4.0%,
with a proportional reduction in the active return.
 The investor can short sell the benchmark portfolio and use the
proceeds to invest in the actively managed fund to increase the active
risk and return of blended portfolio. For example, if active risk of a fund
is 10%, an investor seeks to limit active risk to 6%. He can invest 60% in
active portfolio and remaining 40% to benchmark portfolio.
Sharpe Ratio and Information Ratio
 Closet index fund
 A fund that advertises itself as being actively managed but is actually
close to being an index fund.
 The information ratio of a closet index fund will likely be close to zero or
even slightly negative if value added cannot overcome the management
 The sharpe ratio is close to the benchmark because the excess return
and volatility will be similar to the benchmark.
 Market-neutral long-short equity fund
 A fund with offsetting long and short positions that has a beta of zero.
 The Sharpe ratio and the information ratio would be identical if we
consider the benchmark to be the riskless rate.
Constructing Optimal Portfolios
 Given the opportunity to adjust absolute risk and return, the objective is to
find the single risky asset portfolio with the maximum Sharpe ratio.
 Given the opportunity to adjust active risk and return by investing in both
the actively managed and benchmark portfolios, the squared Sharpe ratio of
an actively managed portfolio is equal to the squared Sharpe ratio of the
benchmark plus the information ratio squared:
𝑆𝑅𝑃2 = 𝑆𝑅𝐵2 + 𝐼𝑅 2
 The active portfolio with the highest (squared) information ratio will also
have the highest (squared) Sharpe ratio
Constructing Optimal Portfolios
 For unconstrained portfolios, the level of active risk that leads to the optimal
portfolio is:
𝜎𝑅𝐴 =
 the ratio of expected active return to active return variance of the managed
portfolio is equal to the ratio of expected benchmark excess return to
benchmark return variance
𝐸(𝑅𝐴 ) 𝐸(𝑅𝐵 − 𝑅𝐹 )
 By definition, the total risk of the actively managed portfolio is the sum of
the benchmark return variance and active return variance.
𝜎𝑅2𝑃 = 𝜎𝑅2𝐵 + 𝜎𝑅2𝐴
Example: Constructing Optimal Portfolios
 Suppose that the historical performance of the Fidelity and Vanguard
funds are indicative of the future performance of hypothetical funds
“Fund I” and “Fund II.” In addition, suppose that the historical
performance of the S&P 500 benchmark portfolio is indicative of
expected returns and risk going forward. We use historical values in
this problem for convenience.
Exhibit (based on a risk-free rate of 2.8%)
S&P 500
Fidelity (Fund I) Vanguard (Fund II)
Average annual return
Return standard dev.
Active return
Active risk
Information ratio
Sharpe ratio
 Q1: State which of the two actively managed funds would be
better to combine with the passive benchmark portfolio and why.
 Fund II is better, because Fund II has the higher expected
information ratio: 0.05 compared with –0.25.
 Calculate highest possible Sharpe ratio of the new “Fund III”,
which has an expected IR of 0.20.
 Highest possible Sharpe ratio of the new “Fund III” would be
𝑆𝑅𝐵2 + 𝐼𝑅2 =
0.472 + 0.202 = 0.51
 Determine the weight of the benchmark portfolio required to
create a combined portfolio with the highest possible expected
Sharpe ratio. Suppose Fund III comes with an active risk of 5.0%
 The optimal amount of active risk is (0.20/0.47)15.2% = 6.5%
 The benchmark portfolio weight needed to adjust the active risk in
Fund III is 1 − 6.5%/5.0% = −30%.
Active Security Returns
 The Correlation Triangle
Active Security Returns
 Signal quality is measured by the correlation between the forecasted active
returns, μi, at the top of the triangle, and the realized active returns, RAi, at
the right corner, commonly called the information coefficient (IC).
 Investors with higher IC, or ability to forecast returns, will add more
value over time, but only to the extent that those forecasts are exploited
in the construction of the managed portfolio.
 The correlation between any set of active weights, Δwi, in the left corner, and
forecasted active returns, μi, at the top of the triangle, measures the degree
to which the investor’s forecasts are translated into active weights, called
the transfer coefficient (TC).
Information Coefficient
 Assume IC is the ex ante (i.e., anticipated) cross-sectional correlation
between the N forecasted active returns, μi, and the N realized active returns,
RAi. To be more accurate, IC is the ex ante risk-weighted correlation.
𝜎𝑖 ,
 The transfer coefficient, TC, is basically the cross-sectional correlation
between the forecasted active security returns and actual active weights.
𝜎𝑖 , ∆𝜔𝑖 𝜎𝑖
Size Active Weights
 In addition to employing mean–variance optimization, proofs of the
fundamental law generally assume that active return forecasts are scaled
prior to optimization using the Grinold (1994) rule:
𝜇𝑖 = 𝐼𝐶𝜎𝑖 𝑆𝑖
 IC is the expected information coefficient
 σi is separate for individual securities
 Si represents a set of standardized forecasts of expected returns across
securities, sometime called “scores.” Scores with a cross-sectional
variance of 1 are used to ensure the correct magnitude of the expected
active returns.
Size Active Weights
 mean–variance-optimal active security weights for uncorrelated active
returns, subject to a limit on active portfolio risk, are given by
∆𝑤𝑖∗ =
𝑁 𝜇𝑖
𝑖=1 2
 In addition to employing mean–variance optimization, proofs of the
fundamental law generally assume that active return forecasts are scaled
prior to optimization using the Grinold (1994) rule.
= 2
𝜎𝑖 𝐼𝐶 × 𝐵𝑅
The Basic Fundamental Law
 The anticipated value added for an actively managed portfolio, or
expected active portfolio return, is the sum product of active security
weights and forecasted active security returns:
𝐸 𝑅𝐴 =
∆𝑤𝑖 𝜇𝑖
 Using the optimal active weights and forecasted active security returns , the
expected active portfolio return is:
𝐼𝑅∗ = 𝐼𝐶 × 𝐵𝑅
 BR(breadth) is the number of securities: BR=N
 The actively managed portfolio is contructed from optimal active
security weights
 Consider four individual securities whose active returns are defined to
be uncorrelated with each other. The information about assets are
depicted in the following exhibit:
 The investor with an ex-ante IC of 0.2 and wants to maximize the
expected active return of the portfolio subject to an active risk
constraint of 9.0%.
 If he makes one forecast of each security every year independently.
Calculate the active weights that should be assigned to each of these
securities using the formula:
∆𝑤𝑖∗ = 2
𝜎𝑖 𝐼𝐶 × 𝐵𝑅
 Correct Answer:
 Using the formula 𝜇𝑖 = 𝐼𝐶𝜎𝑖 𝑆𝑖 , the forecasted active return to
Security #1 is 0.20(25.0%)(1.0) = 5.0%.
 The active returns are uncorrelated with each other and the
forecasts are independent from year to year, so the investor has
made four separate decisions and BR = 4.
 The active weight for Security #1 is ∆𝑤𝑖∗ = 0.252 × 0.2×
= 18%
 Similar calculations for the other three securities are shown in the
following exhibit.
Active Return
Expected Active
Active Weight
The Full Fundamental Law of Active Mgt.
 Although we were able to derive an analytic (i.e., formula-based) solution for
the set of unconstrained optimal active weights, a number of practical or
strategic constraints are often imposed in practice. For example,
 If the unconstrained active weight of a particular security is negative and
large, that might lead to short sell of the security. Many investors are
constrained to be long only, either by regulation or costs of short selling.
 For quantitatively oriented investors, there may exist limits on turnover.
The Full Fundamental Law of Active Mgt.
 Including the impact of the transfer coefficient, the full fundamental law is
expressed in the following equation:
 A low TC results from the constraints imposed on the structure of the
 If TC=0, there would be no expectation of value added from active
 Specifically, with constraints and using notation consistent with
expressions in the fundamental law:
𝜎𝐴 = 𝑇𝐶
𝑆𝑅𝑃2 = 𝑆𝑅𝐵2 + 𝑇𝐶
𝐼𝑅 ∗
Ex Post Performance Measurement
 Most of the fundamental law perspectives discussed up to this point relate
to the expected value added through active portfolio management.
 Actual performance in any given period will vary from its expected value in a
range determined by the benchmark tracking risk.
 Expected value added conditional on the realized information
coefficient, ICR, is
 We can represent any difference between the actual active return of the
portfolio and the conditional expected active return with a noise term
𝑅𝐴 = 𝐸 𝑅𝐴 |𝐼𝐶𝑅 + 𝑁𝑜𝑖𝑠𝑒
 an ex post (i.e., realized) decomposition of the portfolio’s active return
variance into two parts: variation due to the realized information
coefficient(TC2) and variation due to constraint-induced noise(1-TC2)
 Consider an active management strategy that includes BR = 100
investment decisions (e.g., 100 individual stocks whose active returns
are uncorrelated, and annual rebalancing), an expected information
coefficient of IC = 0.05, a transfer coefficient of TC = 0.80, and
annualized active risk of σA = 4.0%. Calculate the expected value added
and information ratio according to the fundamental law.
 Correct Answer 1:
𝐵𝑅𝜎𝐴 = 0.80 × 0.05 × 100 × 4.0%=1.6%
𝐵𝑅 = 0.80 × 0.05 × 100 = 0.4
 Suppose that the realized information coefficient in a given period is –
0.10, instead of the expected value of IC = 0.05. In the absence of
constraint-induced noise, what would be the value added that period?
 Correct Answer 2:
 The value added, without including constraint-induced noise
(which has an expected value of zero) is
𝐵𝑅𝜎𝐴 = −3.2%
 In other words, conditional on the actual information coefficient,
the investor should expect an active return that is negative
because the realized information coefficient is negative.
 Suppose that the actual return on the active portfolio was –2.6%. Given
the –0.10 realized information coefficient, how much of the forecasted
active return was offset by the noise component?
 Correct Answer 3:
 The noise portion of the active return is the difference between the
actual active return and the forecasted active return: –2.6 – (–3.2) =
 In other words, the noise component helped offset the negative
value added from poor return forecasting.
 What percentage of the performance variance (i.e., tracking risk
squared) in this strategy over time is attributed to variation in the
realized information coefficient (i.e., forecasting success), and what
percentage of performance variance is attributed to constraint-induced
 Correct Answer 4:
 Given the transfer coefficient of TC = 0.80, TC2 = 64%.
 In that case, 64% of the variation in performance over time is
attributed to the success of the forecasting process, leaving 36%
due to constraint-induced noise.
Applications of The Fundamental Law
 Global Equity Strategy (TC)
 selection of country equity markets in a global equity fund.
 the constraints that are imposed on the portfolio should inform the
decision of how aggressively to apply an active management strategy.
 Fixed-Income Strategy (IC,BR)
 timing of credit and duration exposures in a fixed-income fund.
 The increasing in BR is at the cost of decreasing IC.
Practical Limitations
 Ex Ante Measurement of Skill
 Behaviorally, one might argue that investors tend to overestimate their
own skills as embedded in the assumed IC.
 Forecasting ability probably differs among different asset segments
and varies over time.
 The key impact of accounting for the uncertainty of skill is that actual
information ratios are substantially lower than predicted by an
objective application of the original form of the fundamental law.
Specifically, security (i.e., individual stock) selection strategies are
analytically and empirically confirmed to be 45%–91% of original
estimates using the fundamental law.
Practical Limitations
 Independence of Investment Decisions
𝐵𝑅 =
1+ 𝑁−1 𝜌
 All the stocks in a given industry or all the countries in a given region
because they are responding to similar influences cannot be counted as
completely independent decisions (ρ>0), so breadth in these contexts
is lower than the number of assets.
 Similarly, when fundamental law concepts are applied to hedging
strategies using derivatives or other forms of arbitrage (ρ<0), breadth
can increase well beyond the number of securities.
Trading Costs and Electronic Markets
1. Costs of trading
2. Advantages of Electronic Trading
3. Market Fragmentation
4. The Major Types of Electronic Traders
5. Low latency
6. Impact of Electronic Trading
7. Risks of Electronic Trading
8. Real-Time Surveillance for Abusive
Trading Practices
1. Costs of Trading
 The costs of trading include fixed costs and variable costs.
 Fixed trading costs include the costs of employing buy-side traders,
the costs of equipping them with proper trading tools (electronic
systems and data), and the costs of office space (trading rooms or
 Variable transaction costs : Variable transaction costs arise from
trading activity and consist of explicit and implicit costs.
 Explicit costs are the direct costs of trading, such as broker
commission costs, transaction taxes, stamp duties, and fees paid to
They are costs for which a trader could receive a receipt.
 Implicit costs are indirect costs caused by the market impact of
1. Costs of Trading
 Implicit costs result from the following issues:
 The bid–ask spread is the ask price (the price at which a trader will sell
a specified quantity of a security) minus the bid price (the price at which
a trader will buy a specified quantity of a security).
 Market impact (or price impact) is the effect of the trade on transaction
 Delay costs (also called slippage) arise from the inability to complete
the desired trade immediately.
 Opportunity costs (or unrealized profit/loss) arise from the failure to
execute a trade promptly.
1. Costs of Trading
 Bid–Ask Spreads and Order Books
 Bid–ask spread=Ask price – bid price
 The best bid (inside bid) is the offer to buy with the highest bid price.
 The best ask (best offer or inside ask) is the offer to sell with the
lowest ask price.
 Market bid–ask spread(inside spread)=best ask-best bid
 The market spread is a measure of trade execution costs.
It is how much traders would lose per quantity traded if they
simultaneously submitted buy and sell market orders that
respectively execute at the ask and bid prices.
Given that two trades generated the cost, the cost per trade is
one half of the quoted spread.
 Midquote price = (bid + ask)/2
 For example, suppose that a portfolio manager gives the firm’s trading
desk an order to buy 1,000 shares of Economical Chemical Systems,
Inc. (ECSI). Three dealers (coded A, B, and C) make a market in those
shares. When the trader views the market in ECSI at 10:22 a.m. on his
computer screen, the three dealers have put in the following limit
orders to trade at an exchange market:
Exhibit 1 The Limit Order Book for Economical Chemical Systems, Inc.
Dealer Time
Price Size
Dealer Time
10:21 a.m. 98.85 600
10:21 a.m. 100.49 200
10:21 a.m. 98.84 500
10:21 a.m. 100.51 1,000
10:19 a.m. 98.82 700
10:19 a.m. 100.55 500
Note: The bids are ordered from highest to lowest, while the asks are ordered
from lowest to highest. These orderings are from best bid or ask to worst bid
or ask.
The bid–ask spreads of Dealers A, B, and C are, respectively,
 A: 100.51 – 98.85 = 1.66
 B; 100.55 – 98.84 = 1.71
 C: 100.49 – 98.82 = 1.67
The best bid price, 98.85 by Dealer A, is lower than the best ask price,
100.49 by Dealer C. The market spread is thus 100.49 – 98.85 = 1.64,
which is lower than any of the dealers’ spreads.
The trader might see the quote information organized on his screen as
shown in Exhibit 1. In this display, called a limit order book, the bids
and asks are separately ordered from best to worst with the best at the
top. The trader also notes that the midquote price (halfway between
the market bid and ask prices) is (100.49 + 98.85)/2 = 99.67.
 If the trader on the firm’s trading desk submits a market buy order for
 the trader would purchase 200 shares from Dealer C at 100.49
per share ;
 and 800 shares from Dealer A at 100.51 per share.
 Note that filling the second part of the order cost the trader 0.02 per
share more than the first part because Dealer C’s ask size was
insufficient to fill the entire order.
 Large orders have price impact when they move down the book as
they fill. The price impact of an order depends on its size and the
available liquidity.
1. Costs of Trading
 Transaction Cost Estimates
 To estimate transaction costs, analysts compare trade prices to a
benchmark price.
 Commonly used price benchmarks:
 midquote price at the time of the trade;
 the midquote price at the time of the order submission;
 volume-weighted average price around the time of the trade.
 These three benchmarks, respectively, correspond to the
 Effective spread;
 Implementation shortfall;
1.1 Effective Spreads
 The effective spread is a sensible estimate of transaction costs when orders
are filled in single trades.
 Benchmark price: midquote price at the time the order was entered
 Effective spread transaction cost estimate =
Trade size ⅹ
Trade price – (Bid + Ask) /2
(Bid + Ask) /2 – Trade price
for buy orders
for sell orders
 2 × this midquote price transaction cost estimtate= effective spread
 If an order fills at a price better than the quoted price, the order is said to
receive price improvement and the spread is lower.
 A buy(sell) order fills at a price below(above) the ask(bid) price.
 An order fills at a price outside the quoted spread has an effective spread
that is larger than de quoted spread.
1.1 Effective Spreads
 The effective spread is a poor estimate of transaction costs when traders
split large orders into many parts to fill over time.
 Market impact makes trading expensive especially for the last parts to
fill, but the effective spread will not fully identify this cost if it is
computed separately for each trade.
 Effective spread do not measure delay costs and opportunity cost.
1.1 Effective Spreads
 The effective spread is a poor estimate of transaction costs when traders
split large orders into many parts to fill over time.
 Market impact makes trading expensive especially for the last parts to
fill, but the effective spread will not fully identify this cost if it is
computed separately for each trade.
 Effective spread do not measure delay costs and opportunity cost.
1.2 VWAP Transaction Cost Estimates
 Volume-weighted average price (VWAP)
 Most widely used benchmark prices.
 Use all trades that occurred from the start of the order until the order
was completed. (interval VWAP)
 VWAP = sum of the total dollar value of the trades / total quantity
of the trades.
 VWAP transaction cost estimate =
Trade size ⅹ
Trade VWAP – VWAP benchmark for buy orders
VWAP benchmark – Trade VWAP for sell orders
1.2 VWAP Transaction Cost Estimates
 Limitations of VWAP
 VWAP is problematic when the trades being evaluated are a substantial
fraction of all trades in the VWAP benchmark, or when the trades took
place at the same rate as other trades in the market.
 In both cases, the trade VWAP and the benchmark VWAP will be
nearly equal, suggesting the trades were not costly. But this is
misleading if the trade has substantial price impact.
E.g. A large trader were the only buyer, the VWAP transaction
cost estimate would be zero regardless of the market impact.
 This bias towards zero also helps explain why the measure is so
 VWAP does not consider missed traders.
 Arapahoe Tanager, portfolio manager of a Canadian small-cap equity
mutual fund, and his firm’s chief trader, Lief Schrader, are reviewing the
execution of a ticket to sell 12,000 shares of Alpha Company, limit
C$9.95. The order was traded over the day.
Schrader split the ticket into three orders that executed that day as
A. A market order to sell 2,000 shares executed at a price of C$10.15.
Upon order submission, the market was C$10.12 bid for 3,000 shares,
2,000 shares offered at C$10.24.
A market order to sell 3,000 shares executed at a price of C$10.11.
Upon order submission, the market was C$10.11 bid for 3,000 shares,
2,000 shares offered at C$10.22.
C Toward the end of the trading day, Schrader submitted an order to
sell the remaining 7,000 shares, limit C$9.95. The order executed in
part, with 5,000 shares trading at an average price of C$10.01. Upon
order submission, the market was C$10.05 bid for 3,000 shares,
2,000 shares offered at C$10.19.
This order exceeded the quoted bid size and “walked down” the
limit order book (i.e., after the market bid was filled, the order
continued to buy at lower prices). After the market closed, Schrader
allowed the order to cancel. Tanager did want to sell the 2,000
unfilled shares on the next trading day.
 Only two other trades in Alpha Company occurred on this day: 2,000
shares at C$10.20 and 1,000 shares at C$10.15. The last trade price of
the day was C$9.95; it was C$9.50 on the following day.
For each of the three fund trades, compute the quoted spread. Also,
compute the average quoted spreads prevailing at the times of each
For each of the three fund trades, compute the effective spread (use
the average fill price for the third trade). Also, compute the average
effective spread.
Explain the relative magnitudes of quoted and effective spreads for
each of the three fund trades.
Calculate the VWAP for all 13,000 Alpha Company shares that
traded that day and for the 10,000 shares sold by the mutual fund.
Compute the VWAP transaction cost estimate for the 10,000 shares
 Solution to 1:
The quoted spread is the difference between the ask and bid prices.
First order, C$10.24 – C$10.12 = C$0.12.
Second order = C$0.11
Third order = C$0.14
Average quoted spread is (C$0.12 + C$0.11 + C$0.14)/3 = C$0.1233.
 Solution to 2:
The effective spread for a sell order is 2 × (Midpoint of the market at the
time of order entry – Trade price).
For the first order, midpoint of the market at the time of order entry is
(C$10.12 + C$10.24)/2 = C$10.18, so that the effective spread is 2 ×
(C$10.18 – C$10.15) = C$0.06.
The effective spread for the second order is 2 × [(C$10.11 + C$10.22)/2
– C$10.11] = C$0.11.
The effective spread for the third order is 2 × [(C$10.05 + C$10.19)/2 –
C$10.01] = C$0.22.
Average effective spread is (C$0.06 + C$0.11 + C$0.22)/3 = C$0.13.
 Solution to 3:
 The first trade received price improvement because the shares
sold at a price above the bid price. Therefore, the effective spread is
less than the quoted spread.
 No price improvement occurred for the second trade because the
shares sold at the bid price. Also, the second trade had no price
impact beyond trading at the bid; the entire order traded at the
quoted bid. Accordingly, the effective and quoted spreads are equal.
 The effective spread for the third trade is greater than the quoted
spread because the large order size, which was greater than the bid
size, caused the order to walk down the limit order book. The
average sale price was less than the bid so that the effective spread
was higher than the quoted spread.
 Solution to 4:
The VWAP for the day is the total dollar volume divided by the total
number of shares traded. The dollar volume is 2,000 shares × C$10.15
+ 3,000 shares × C$10.11 + 5,000 shares × C$10.01 + 2,000 shares ×
C$10.20 + 1,000 shares at C$10.15 = C$131,230. Dividing this by the
13,000-share total volume gives a VWAP of C$10.0946.
A similar calculation using only the sales sold by the mutual fund
2,000 shares × C$10.15 + 3,000 shares × C$10.11 + 5,000 shares ×
C$10.01 = C$100,680 Dividing this by the 10,000-share gives a trade
VWAP of C$10.0680.
The VWAP transaction cost estimate for the sale is the difference
multiplied by the 10,000 shares sold: C$266.15 = 10,000 shares ×
(C$10.0946 – C$10.0680) [differences due to rounding].
1.3 Implementation Shortfall
 Implementation shortfall (IS) = values of paper portfolio – values of
actual portfolio
 Paper portfolio: trades could be arranged at the decision price.
 Prevailing price (decision price, arrival price, strike price) = midquote
price at the time of the trade decision.
 IS = market impact costs + delay costs + opportunity costs + explicit
 IS captures all explicit and implicit costs but the computation is more
Delay cost Market impact costs
£10.00 £10.03
Paper portfolio
Opportunity cost
Explicit costs
2. Advantages of Electronic Trading Systems
 Traders use electronic systems to generate the orders that the exchanges
process. The most important electronic traders are dealers, arbitrageurs, and
buy-side institutional traders who use algorithmic trading tools provided by
their brokers to fill their large orders.
 The two types of systems are co-dependent:
 Traders need high-speed order processing and communication
systems to implement their electronic trading strategies;
 The exchanges need electronic exchange systems to process the
vast numbers of orders that these electronic traders produce.
 The electronic market structures of equity, futures, and options
markets have attracted tremendous attention throughout the world.
Much less attention has been given to the market structures of
corporate and municipal bond markets.
2. Advantages of Electronic Trading Systems
 Compared with floor-based trading systems, electronic order-matching
systems enjoy many advantages.
 1. Cost: electronic systems are cheap to operate once built.
 The widespread use of electronic trading systems significantly
decreased trading costs for buy-side traders by allowing a smaller
number of buy-side traders to process more orders and to process
them more efficiently than manual traders.
 Costs fell as exchanges obtained greater cost efficiencies from using
electronic matching systems.
 These technologies also decreased costs and increased efficiencies for
the dealers and arbitrageurs, who provide much of the liquidity offered
at exchanges.
 Competition forced them to pass along many of the benefits of their
new technologies to buy-side traders in the form of narrower
spreads quoted for larger sizes.
2. Advantages of Electronic Trading Systems
 2. Accuracy: Electronic exchange systems do exactly what they are
programmed to do.
 3. Audit trails: Electronic exchange systems can also keep perfect audit
trails so that forensic investigators can determine the exact sequence and
timing of events that may interest them
 4. Fraud prevention: Electronic exchange systems keep hidden orders
perfectly hidden.
 5. Continuous market: Electronic order-matching systems can operate on a
continuous, “around-the-clock” basis, even when bad weather or other
events would likely prevent workers from convening on a floor.
2. Advantages of Electronic Trading Systems
 6. Additionally, computers have come to dominate the implementation of
many trading strategies because they are so efficient compared with
human traders
 Computers have infinite attention spans and a very wide attention
 Their responses are extraordinarily fast.
 Computers are perfectly disciplined and do only what they are
instructed (programmed) to do.
 Computers do not forget any information that their programmers
want to save.
3. Market Fragmentation
 Market fragmentation: trading the same instrument in multiple venues.
 With increasing market fragmentation,
 Increases the potential for price and liquidity disparities across venues.
 Traders filling large orders now adapt their trading strategies to search
for liquidity across multiple venues and across time to control the
market impacts of their trades.
 Electronic algorithmic trading techniques, such as liquidity
aggregation and smart order routing, help traders manage the
challenges and opportunities presented by fragmentation.
 Liquidity aggregators create “super books” that present
liquidity across markets for a given instrument.
 Smart order-routing (SOR) algorithms send orders to the
markets that display the best-quoted prices and sizes.
4. The Major Types of Electronic Traders
 1. Electronic news traders subscribe to high-speed electronic news feeds
that report news releases made by corporations, governments, and other
aggregators of information.
 News traders profit when they can execute against stale orders –
orders that do not yet reflect the new information.
 Besides quantitative data, some traders also using natural languageprocessing techniques, they try to identify the importance of the
information for market valuations.
 2. Electronic dealers make markets by placing bids and offers with the
expectation that they can profit from round trips at favorable net spreads.
 Electronic dealers often monitor electronic news feeds. If the news is
material, they do not want to offer liquidity to news traders to whom
they would lose.
4. The Major Types of Electronic Traders
 3. Electronic arbitrageurs look across markets for arbitrage opportunities in which
they can buy an undervalued instrument and sell a similar overvalued one..
 4. Electronic front runners are low-latency traders, who use artificial intelligence
methods to identify when large traders or many small traders, are trying to fill
orders on the same side of the market ahead of them.
 Some front runners examine the patterns and other events to predict future
 5. Electronic quote matchers try to exploit the option value of the standing limit
 Standing orders are limit orders waiting to be filled.
 The main risk of the quote-matching strategy is that the standing order may be
unavailable when the quote matcher needs it. Standing orders disappear when
filled by another trader or when canceled.
5. Low Latency
 Latency is the elapsed time between the occurrence of an event and a
subsequent action that depends on that event.
 Electronic traders have three needs for speed.
 1) Taking. Electronic traders sometimes want to take a trading
opportunity before others do;
 2) Making. Market events often create attractive opportunities to offer
liquidity, electronic traders must be fast so they can acquire priority when
they want it and before other traders do
 3) Canceling. Traders must quickly cancel orders they no longer want to
 Note that electronic traders do not simply need to be fast to trade effectively.
They must be faster than their competitors.
5. Low Latency
 How to reduce latency
 1. Fast Communications
 locate their computers as close as possible to the exchanges;
place their servers in the rooms where the exchange servers
operate, a practice called collocation.
 Use the fastest communication technologies;
 subscribe to special high-speed data feeds directly from exchanges
and other data vendors.
 2. Fast Computations
 They overclock their processors and use liquid cooling systems;
 They often use simple and specialized operating systems;
 Electronic traders optimize their computer code for speed;
 Use faster developing language for better coding speed;
 Creating contingency tables that contain prearranged action plans.
6. Impact of Electronic Trading
 Buy-side traders often use electronic brokers and their systems to deal with
advanced orders, trading tactics, and algorithms provided by their
electronic brokers to search for liquidity.
 Advanced orders generally are limit orders with limit prices that change
as market conditions change.
 A trading tactic is a plan for executing a simple function that generally
involves the submission of multiple orders to find hidden liquidity.
 Immediate or cancel order (IOC)
 Algorithms (“algos” for short) are programmed strategies for filling
6. Impact of Electronic Trading
 Some characteristics of electronic trading are described below.
 1. Hidden orders are orders that are exposed (or shown) only to the
brokers or exchanges who receive them.
 Traders use IOC(immediate or cancel ) orders to discover hidden
orders that may stand in the spread between a market’s quoted bid
and ask prices.
 2. Leapfrog. When bid–ask spreads are wide, dealers often are willing to
trade at better prices than they quote. When another trader quotes a
better price, dealers often immediately quote an even better price.
 3. Flickering quotes are exposed limit orders that electronic traders
submit and then cancel shortly thereafter, often within a second.
6. Impact of Electronic Trading
 4. Electronic arbitrage
 Take liquidity on both sides
buying an undervalued instrument and selling a similar
overvalued instrument;
Using marketable orders;
 Offer liquidity on one side
When they obtain a fill in one market, they immediately take
liquidity in the other market to complete the construction of
their arbitrage portfolio.
If trade opportunity is unavailable, traders will immediately
cancel his order in first market.
 Offer liquidity on both sides
In this strategy, after the first order to execute fills, the
arbitrageur continues to offer liquidity to complete the second
trade. (much like dealers)
6. Impact of Electronic Trading
 5. Machine learning (data mining): uses advanced statistical methods
to characterize data structures, particularly relations among variables.
 Machine-learning methods produce models based on observed
empirical regularities rather than on theoretical principles identified
by analysts.
 ML is suitable:
In active financial markets, these methods can be powerful
when stable processes generate vast amounts of data.
When the problems repeat regularly.
 Machine-learning systems frequently do not produce useful
information during volatility episodes because these episodes
have few precedents from which the machines can learn.
7. Risks of Electronic Trading
 High-frequency traders (HFT) arms race. Arms race serve as an unfair
entry barrier to small traders and be compromised by introducing delays in
trading at random intervals.
 Systemic Risks of Electronic Trading
 Runaway algorithms produce streams of unintended orders that result
from programming mistakes.
 Fat finger errors occur when a manual trader submits a larger order
than intended.
 Overlarge orders demand more liquidity than the market can provide.
(e.g. 2010/5/6 Flash Crash)
 Malevolent order streams are created deliberately to disrupt the
7. Risks of Electronic Trading
 Solutions for regulators to mitigate systemic risk
 1. Most obviously, traders must test software thoroughly before using it in
live trading.
 2. Rigorous market access controls must ensure that only those orders
coming from approved sources enter electronic order-matching systems.
 3. Rigorous access controls on software developers must ensure that only
authorized developers can change software.
 4. The electronic traders who generate orders and the electronic exchanges
that receive orders must surveil their order flow in real time to ensure that it
conforms to preset parameters that characterize its expected volume, size,
and other characteristics.
 5. Brokers must surveil all client orders that clients introduce into electronic
trading systems to ensure that their clients’ trading is appropriate. Brokers
must not allow their clients to enter orders directly into exchange trading
systems, because it would allow clients to avoid broker oversight.
 6. Some exchanges have adopted price limits and stop trading when prices
move too quickly.
8. Real-Time Surveillance for Abusive Trading Practices
 Front running involves buying in front of anticipated purchases and selling in
front of anticipated sales.
 Market manipulation: produce misleading or false market prices, quotes, or
fundamental information to profit from distorting the normal operation of
 Trading for market impact involves trading to raise or lower prices
 Rumormongering is the dissemination of false information about
fundamental values or about other traders’ trading intentions to alter
investors’ value assessments.
 Wash trading consists of trades arranged among commonly controlled
accounts to create the impression of market activity at a particular price.
 Spoofing, also known as layering, is a trading practice in which traders
place exposed standing limit orders to convey an impression to other
traders that the market is more liquid than it is, or to suggest to other
traders that the security is under- or overvalued.
8. Real-Time Surveillance for Abusive Trading Practices
 Market manipulation strategies include:
 Bluffing involves submitting orders and arranging trades to influence other
traders’ perceptions of value, especially momentum traders.
 Gunning the market is a strategy used by market manipulators to force
traders to do disadvantageous trades.
 Selling quickly to push prices down with the hope of triggering stoploss sell orders.
 Squeezing and cornering. Squeezing, cornering are also schemes that
market manipulators use to force traders to do disadvantageous trades.
 1) The manipulator obtains control over resources necessary to settle
trading contracts.
 2) The manipulator then unexpectedly withdraws those resources from
the market, which causes traders to default on their contracts.
 3) The manipulator profits by providing the resources at high prices or
by closing the contracts at exceptionally high prices.
It’s not the end but just beginning.
Life is short. If there was ever a moment to follow your passion and
do something that matters to you, that moment is now.
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