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1 P U B L I C A T I O N S The Experts In Actuarial Career Advancement Product Preview For More Information: Support@ActexMadRiver.com or call 1(800)

2 SOA Learning Objectives and Learning Outcomes AOP - 1 Topic 1: Advanced Option Pricing SOA Learning Objectives and Learning Outcomes I. Learning Objectives A. The candidate will understand the standard yield curve models, including: 1. One and two-factor short rate models 2. LIBOR market models II. Learning Outcomes A. The Candidate will be able to: 1. Identify and differentiate the features of the classic short rate models including the Vasicek and the Cox-Ingersoll-Ross (CIR) models. 2. Understand and explain the terms Time Homogeneous Models, Affine Term Structure Models, and Affine Coefficient models and explain their significance in the context of short rate interest models. 3. Explain the dynamics of and motivation for the Hull-White extension of the Vasicek model. 4. Explain the features of the Black-Karasinski model 5. Understand and explain the relationship between market-quoted caplet volatilities and model volatilities. 6. Explain how deterministic shifts can be used to fit any given interest rate term structure and demonstrate an understanding of the CIR++ model. 7. Understand and explain the features of the G2++ model, including: The motivation for more than one factor, calibration approaches, the pricing of bonds and options, and the model s relationship to the two-factor Hull-White model 8. Explain the set up and motivation of the lognormal Forward LIBOR Model (LFM) 9. Describe the calibration of the LFM to Cap and Floor prices 10. Explain the LFM drift terms and their dependence on the calibration and choice of numeraire 11. Define and explain the concept of volatility smile and some arguments for its existence 12. Calculate the hedge ratio for a call option given the dependency of the Black-Scholes volatility on the underlying 13. Compare and contrast floating and sticky smiles

3 AOP - 2 SOA Learning Objectives and Outcomes 14. Calculate the risk-neutral density given call option prices 15. Identify several stylized empirical facts about smiles in a variety of options markets 16. Describe and contrast several approaches for modeling smiles, including: Stochastic Volatility, local-volatility, jump-diffusions, variance-gamma and mixture models.

4 SOA Learning Objectives and Learning Outcomes AOP - 3 Interest Rate Models - Theory and Practice Damiano Brigo and Fabio Mercurio Chapter 1 Definitions and Notation This chapter is for background only. Note: there is a nice abbreviation and notation section before the table of contents I. Introduction A. Government rates - interest rates on bonds issued by governments B. Interbank rates - interest rates on deposits exchanges between banks or swap transactions between banks 1. LIBOR - most important interbank rate - London InterBank Offered Rate C. Zero-coupon rates can be developed from either government rates or interbank rates, but create two different curves II. The Bank Account and the Short Rate A. Money market account - "represents a (locally) risk-less investments, where the profit is accrued instantaneously at the risk-free rate prevailing in the market at every instant" B. Definition Bank account (Money-market account) 1. "B(t) to be the value of a bank account at time t 0. We assume B(0) = 1 and that the bank account evolves according to the following differential equation a. db(t) = r t B(t)dt, B(0) = 1, (1.1) b. Where r t is a positive function of time. As a consequence, c. exp " (1.2) 2. Instantaneous rate or instantaneous spot rate or short rate, r t 3. First order expansion in Δt, B(t + Δt) = B(t)(1 + r(t)δt) (1.3) 4. Value of 1 payable at time T when viewed at time t, B(t)/B(T) C. Definition Stochastic discount factor 1. "The (stochastic) discount factor D(t, T) between two time instants t and T is the amount at time t that is "equivalent" to one unit of currency payable at time T, and is given by a., exp " (1.4) 2. r often considered to be constant, e.g., Black-Scholes formula and FX market

5 AOP - 4 Interest Rate Models - Theory and Practice Chapter 1 3. However, text about interest rate products and variability of interest rates is important, must consider stochastic interest rates, r t a. Thus, formula (1.2) and (1.4) are stochastic III. Zero-Coupon Bonds and Spot Interest Rate A. Definition Zero-coupon bond 1. "A T-maturity zero-coupon bond (pure discount bond) is a contract that guarantees its holder the payment of one unit of currency at time T, with no intermediate payments. The contract value at time t < T is denoted by P(t,T). Clearly, P(T,T) = 1 for all T" 2. Relationship between discount factor D and zero-coupon bond price P? a. If r is deterministic, then D is deterministic and D(t,T) = P(t, T) for all t and T b. If r is stochastic, then D is stochastic, but P(t, T) has to be known at time t and is therefore deterministic i. Under a particular probability measure, to be seen later, P(t,T) = E[D(t,T)], the expectation of D(t, T) B. Definition Time to Maturity 1. "The time to maturity T - t is the amount of time (in years) from the present time t to the maturity time T > t" a. T - t is clear if both are real numbers, but not if both are dates C. Definitions Year fraction, Day-count convention 1. Study manual author note: unclear why text author makes a distinction in numbering between "definition" and "definitions" both of which have a "1.2.1". This distinction is used through the text. 2. "We denote by τ(t, T) the chosen time measure between t and T, which is usually referred to as year fraction between dates t and T. When t and T are less than one-day distant (typically when dealing with limit quantities involving time to maturity tending to zero), τ(t, T) is to be interpreted as the time difference T - t (in years)" 3. Day-count convention - "The particular choice that is made to measure the time between two dates" a. Actual /365 - Year is 365 days long and count actual dates between two dates i. D 2 - D 1 the actual number of days between two dates ii. Year fraction = (D 2 - D 1 )/365

6 SOA Learning Objectives and Learning Outcomes AOP - 5 b. Actual/360 - Year is 360 days long and count actual dates between two dates i. D 2 - D 1 the actual number of days between two dates ii. Year fraction = (D 2 - D 1 )/360 c. 30/360 - all months 30 days long, year is 360 days i. D 1 = (d 1, m 1, y 1 ), D 2 = (d 2, m 2, y 2 ), ii. Year fraction = [max(30 - d 1, 0) + min(d 2, 30) + 360(y 2 - y 1 ) + 30(m 2 - m 1-1)] / 360 d. Holidays - typically if D 2 is a holiday, change D 2 to be the next business day D. Zero-coupon bond prices - the basic quantities in interest rate theory 1. Interest rates can be converted to zero-coupon bond prices 2. Zero-coupon bond prices can be converted to interest rates 3. To do conversions, need to know: compounding type and day-count convention E. Definition Continuously-compounded spot interest rate 1. "The continuously-compounded spot interest rate prevailing at time t for the maturity T is denoted by R(t, T) and is the constant rate at which an investment of P(t, T) units of currency at time t accrues continuously to yield a unit amount of currency at maturity T" 2. R(t, T) = - ln P(t, T) / τ(t, T) (1.5) 3. e R(t,T)τ(t, T) P(t, T) = 1 (1.6) 4. P(t, T) = e -R(t,T)τ(t, T) (1.7) F. Definition Simply-compounded spot interest rate 1. "The simply-compounded spot interest rate prevailing at time t for the maturity T is denoted by L(t, T) and is the constant rate at which an investment has to be made to produce an amount of one unit of currency at maturity, starting from P(t, T) units of currency at time t, when accruing occurs proportionally to the investment time" 2. L(t, T) = [1 - P(t, T)] / [τ(t, T) P(t, T)] (1.8) 3. LIBOR rates are simply-compounded 4. P(t, T)(1 + L(t, T)τ(t, T)) = 1 (1.9) 5. P(t, T) = 1 / [1 + L(t, T) τ(t, T)] 6. Study manual author note: The authors of the text are from Italy. In the US, we would call this method "simple interest". It is unclear what is being compounded.

7 AOP - 6 Interest Rate Models - Theory and Practice Chapter 1 G. Definition Annually-compounded spot interest rate 1. "The annually-compounded spot interest rate prevailing at time t for the maturity T is denoted by Y(t, T) and is the constant rate at which an investment has to be made to produce an amount of one unit of currency at maturity, starting from P(t, T) units of currency at time t, when reinvesting the obtained amounts once a year" 2.,,, 1 (1.10) 3. P(t, T)(1+Y(t, T)) τ(t, T) = 1 (1.11) 4.,,, (1.12) H. Definition k-times-per-year compounded spot interest rate 1. "The k-times-per-year compounded spot interest rate prevailing at time t for the maturity T is denoted by Y k (t, T) and is the constant rate (referred to a one-year period) at which an investment has to be made to produce an amount of one unit of currency at maturity, starting from P(t, T) units of currency at time t, when reinvesting the obtained amounts k times a year" 2.,,, (1.13) 3., 1,, 1 (1.14) 4., (1.15),, 5. Continuously compounded rates as limit of k-times-per-year compounded as k I. Short rate as limit as T t of the other rates 1. lim, lim, lim, lim, for each k IV. Fundamental Interest-Rate Curves A. Definition Zero-coupon curve 1. Obtained from market data of interest rates 2. "The zero-coupon curve (sometimes referred to as the 'yield curve') at time t is the graph of the function" a. T L(t, T) for t < T t+1 years b. T Y(t, T) for T > t+1 years c. Also known as the "term structure of interest rates" at time t

8 SOA Learning Objectives and Learning Outcomes AOP - 7 d. Simple rates for a year, then annual compound rate thereafter 3. Note: in this text "yield curve" means the zero-coupon curve, unless specifically mentioned a. Terminology can be different in different texts 4. Curve can take many shapes, positively sloped, inverted, flat, etc B. Definition Zero-bond curve V. Forward Rates 1. "The zero-bond curve at time t is the graph of the function T P(t, T) for T > t, which because of the positivity of interest rates is a T-decreasing function starting from P(t, t) = 1." 2. Term structure of discount factors - the above curve 3. Curve is monotonic, unlike yield curve that can take on various shapes A. Section Introduction 1. Forward rates - "Interest rates that can be locked in today for an investment in a future time period" 2. Characterization of forward rates have three time instances a. Time t the time at which the rate is considered b. Time T the "expiry" i. Study manual author note: curious term "expiration" as it is the time when the rate starts. Term is clear when you think at T as the expiry date of a forward rate contract for a rate between T and S c. Time S the maturity, or time rate ends, with t T S 3. Forward Rate Agreement (FRA) a. Involves the three times, t T S b. "Contract gives its holder an interest-rate payment for the period between T and S. At the maturity S, a fixed payment based on a fixed rate K is exchanged against a floating payment based on the spot rate L(T, S) resetting in T and with maturity S" c. Value of contract at time S i. N - contract nominal value ii. At time S, holder receives τ(t, S)KN units of currency

9 AOP - 8 Interest Rate Models - Theory and Practice Chapter 1 iii. At time S, holder pays τ(t, S)L(T, S)N iv. Value at time S = N τ(t, S)[K - L(T, S)] (1.17) a) Assumes both rates have the same day-count convention v. Rewrite (1.17) using (1.8) for L a) Value at time S = N[τ(T, S)K - 1/P(T, S) +1] (1.18) d. Value of FRA at time t, discount (1.18) by multiplying by P(t, S), the price of the zero-coupon bond i. Note, P(t, S)/P(T, S) = P(t, T) ii. FRA(t, T, S, τ(t,s), N,K) = N [ P(t, S)τ(T, S)K - P(t, T) + P(t, S)] (1.19) a) Only one K, the fixed rate, creating a fair contract, b) Fair contract is one with value 0 at time t B. Definition Simply-compounded forward interest rate 1. "The simply-compounded forward interest rate prevailing at time t for expiry T > t and maturity S > T is denoted by F(t; T, S) and is defined by a. ";,,,, 1 (1.20) b. It is that value of the fixed rate in a prototypical FRA with expiry T and maturity S that renders the FRA a fair contract at time t" 2. Substitute (1.20) in (1.19) and the result is (1.21) a. FRA(t, T, S, τ(t,s), N,K) = NP(t, S)τ(T, S)[K - F(t; T S)] (1.21) b. Valuation of FRA, present value of (1.17) where use F(t; T, S) rather than L(T, S) c. F(t; T, S) = E[L(T, S)] based on the market at time t 3. Instantaneous forward rate, When S T +, a. lim ;, C. Definition Instantaneous forward interest rate 1. "The instantaneous forward interest rate prevailing at time t for the maturity T > t is denoted by f(t, T) and is defined as ln, a., lim ;, (1.23)

10 SOA Learning Objectives and Learning Outcomes AOP - 9 b. So that we also have, exp, " 2. Smoothness of zero-coupon price curve is assumed 3. Instantaneous forward rate is approximately the forward rate with very little time to maturity a. f(t, T) F(t; T, T + ΔT), ΔT is small D. "Instantaneous forward rates are fundamental quantities in the theory of interest rates" E. "Fairness" of an interest rate model - model with the "absence of arbitrage opportunities" VI. Interest-Rate Swaps and Forward Swap Rates A. Section introduction 1. Interest rate swap (IRS) a generalization of the FRA contract, swap being a series of FRAs 2. Forward Start, Payer Interest Rate Swap (PFS) a. Description/Notation i. Payments start at a future date ii. Payments of each leg use different index iii. Payments dates, T α+1, T α+2,,t β iv. Fixed leg pays Nτ i K a) τ i - year fraction b) K - fixed interest rate c) N nominal or notional value v. Floating leg pays Nτ i L(T i-1, T i ) a) Rate resets occur at T α,t α+1,,t β-1 for payment the following period vi. Vectors a) { T α,t α+1,,t β } b) τ {τ α+1,,τ β } 3. Legs a. When fixed leg is paid, called Payer swap (PFS)

11 AOP - 10 Interest Rate Models - Theory and Practice Chapter 1 b. When floating leg is paid, called Receiver swap (RFS) c. Fixed leg can be thought of as fixed rate bond d. Floating leg can be thought of as floating rate bond i. Swap of principal at maturity of fixed and floating legs cancel 4. Discounted payoff of PFS, t < T α a.,, 5. Discounted payoff of RFS, t < T α a.,, 6. RFS as portfolio of FRAs a.,,,,,,,,,, ;,,,, (1.24) B. Definition Prototypical coupon-bearing bond 1. "A prototypical coupon-bearing bond is a contract that ensures the payment at future times T α+1,, T β of the deterministic amounts for currency (cash-flows) c {c α+1,,c β }. Typically, the cash flows are defined as c i = Nτ i K for i < β and c β = Nτ i K + N, where K is a fixed interest rate and N is bond nominal value. The last cash flow includes the reimbursement of the notional value of the bond" 2. With K = 0, a zero-coupon bond 3. Value of bond,,,, C. Definition Prototypical floating-rate note 1. "A prototypical floating-rate note is a contract that ensuring the payment at future times T α+1,, T β of the LIBOR rates that reset at the previous instances T α,,t β-1. Moreover, the note pays a last cash flows consisting of the reimbursement of the notional value of the note at final time T β " 2. Value of note RFS(t,,τ, N, 0) + NP(t, T β ) = NP(t, T α ) a. Value at all reset dates is N 3. "Fairness" i. "floating-rate note always trades at par" a. Forward rates are derived from "fair" FRAs

12 SOA Learning Objectives and Learning Outcomes AOP - 11 b. "Fair" IRS defined such that initial value is 0, find rate K that makes this so D. Definition Forward Swap Rate 1. "The forward swap rate S α,β (t) at time t for the sets of times and year fractions τ is the rate in the fixed leg of the above IRS that makes the IRS a fair contract at the present time, i.e., it is the fixed rate K for which RFS(t,,τ, N, K) = 0. We easily obtain a.,,,, (1.25)" 2. Notice a.,,,, i. F j (t) = F(t; T j-1,t j ) 3. And swap rates can be expressed in terms of forward rates a., VII. Interest-Rate Caps/Floors and Swaptions A. Interest-Rate Caps/Floors 1. Cap = "a contract that can be viewed as a payer IRS where each exchange payment is executed only if it has positive value" a.,, + i. Positive-part operator, + b. Example, company has LIBOR debt and is concerned that rates will increase i. Floating debt plus cap, so company pays min(l, K) ii. Company pays at most K and has "capped" interest expense at rate K c. Caplet - each individual potential payment of the cap series of payments, with value i.,, ii. Cap price = sum of caplet prices, that is cap price can be decomposed into individual pieces

13 AOP - 12 Interest Rate Models - Theory and Practice Chapter 1 d. Black's formula (time zero) i. 0,,,,,, 0,, 0,,,,1 (1.26) a) Bl(K, F, v, ω) = FωΦωd 1 K, F, v K FωΦωd 2 K, F, v i) ω is either 1 or 1, if excluded it is 1 b) Φ the cumulative normal distribution c) d 1 K, F, v / d) d 2 K, F, v / e), f), - the volatility parameter 2. Floor = "equivalent to a receiver IRS where each exchange payment is executed only if it has positive value" a.,, + b. Floorlet - each individual potential floor c. Black's formula (time zero) B. Definition i. 0,,,,,, 0,, 0,,,,1 (1.27) 1. "Consider a cap (floor) with payment times T α+1,, T β, associated year fractions τ α+1,,τ β and strike K. The cap (floor) is said to be at-the-money (ATM) if and only if a., 0,,. The cap is instead said to be in-the-money, (ITM) if K < K ATM, and out-of-the-money (OTM) if K > K ATM, with the converse holding for a floor." 2. Cap - Floor = Forward-start swap a. (L - K) + - (K - L) + = L - K b. Cap (floor) is ATM if and only if price = corresponding floor (cap)

14 SOA Learning Objectives and Learning Outcomes AOP Caplet C. Swaptions a. ATM caplet strike, K ATM = F(0, T α,t α+1 ) b. ITM if K < F(0, T α,t α+1 ) 1. Swap options also known as swaptions 2. European payer swaption - "option giving the right (and no obligation) to enter a payer IRS at a given future time, the swaption maturity" a. Swaption maturity coincides with first reset date of underlying IRS b. Tenor of swaption - the length of underlying IRS, T β - T α c. Tenor structure - set of reset and payment dates 3. Discounted payoff of payer swaption = value of underlying swap at first reset date T α a. Formula (1.24) but changing the sign, and only if positive i.,, ;, b. Cannot be decomposed into more elementary, or individual, pieces, unlike cap i. Positive part operator is outside the, unlike cap where operator is inside the ii. Swaption formula is piece-wise linear and convex iii. Swaption price cap price iv., ;,, ;, 4. Valuing and managing swaptions a. Joint action of different rates important in contract payoff i. Consider correlation of rates, not changes in rates b. Payer Swaption (PS) value i. 0,,,,,,,, 0,,,1 a) σ α,β different from volatility parameter for caps/floors 0, (1.28)

15 AOP - 14 Interest Rate Models - Theory and Practice Chapter 1 c. Receiver Swaption (RS) value i. 0,,,,,,,, 0,,,1 5. Definition , (1.29) a. "Consider a payer (respectively receiver) swaption with strike K giving the holder the right to enter at time T α a payer (receiver) IRS with payment dates T α+1,, T β and associated year fractions τ α+1,,τ β. The swaption (either payer or receiver) is then said to be at-the-money (ATM) if and only if i., 0,,., ii. The payer swaption is instead said to be in-the-money (ITM) if K < K ATM, and out-of-the-money (OTM) if K > K ATM. The receiver swaption is ITM if K > K ATM, and out-of-the-money (OTM) if K < K ATM " b. "Difference between a payer swaption and the corresponding receive swaption is equivalent to a forward-start swap" i. Payer swaption is ATM if and only if price = price corresponding receiver swaption c. Alternative Expression for discounted payer-swaption at t = 0 i. 0,,, ii., 0 is strictly positive, so if, 0, then in-the-money

16 QFIA QT - 41 QFIA CAIA Level II, Advanced Core Topics in Alternative Investment Chapter16: Unsmoothing of Appraisal-Based Returns I. Introduction A. "Critical task of empirical analysis of real estate returns: data unsmoothing" 1. Smoothed data also impact hedge funds and private equity B. Delayed response of real estate prices to major market moves and or changes in accompanying changes in economic conditions 1. Resulting prices and return series derived from such prices form a "smoothed series" C. Arbitrage 1. "Tradable prices that are smoothed can be arbitraged if transaction costs are relatively small" 2. Appraisals are not tradable, so cannot be arbitraged a. Smoothing can (and does) exist and "may be more pronounced and permanent" D. Risk management and other financial analysis needs unsmoothed data as smoothing tends to dampen the volatility of the returns series and resulting correlations are suspect II. Smoothed Pricing A. Section Introduction 1. Exhibit 16.1 illustrates market returns (10%), and three other series earning 8%, unsmoothed, lightly smoothed and strongly smoothed 2. The two smoothed series show lower volatility than the unsmoothed (true?) series, along with lower correlations and lower β B. Price Smoothing and Arbitrage in a Perfect Market 1. If prices are smoothed, arbitrage can exist buying/selling smoothed assets and hedging with unsmoothed assets a. Arbitrageurs will buy/sell smoothed assets in anticipation of subsequent price moves 2. "In perfect market, competition between arbitrageurs will force prices to respond fully and immediately in the absence of transaction costs"

17 QT - 42 QFIA C. Persistence in Price Smoothing 1. Reasons why no arbitrage a. Return series based on appraisals are not true trading prices i. Appraisals are not buy/sell offers ii. Appraisals are used for accounting purposes, are estimated, not market, prices b. Timing and Transaction cost can be substantial i. It can take a long time to close a real estate transaction ii. Transaction costs include: sales commissions, transfer taxes, and the costs of legal, financing, search, inspection, etc c. Other barriers to trading assets with smoothed pricing i. Stale prices, e.g. international open-end equity mutual fund a) Arbitrageurs profited due to stale pricing b) Redemption fees imposed to increase cost, deter arbitrage 2. "The need to unsmooth prices tends to be greatest for nontradable prices and assets with high transaction costs or trading barriers" D. Problems Resulting from Price Smoothing 1. The two smoothed series show lower volatility than the unsmoothed (true?) series, along with lower correlations and lower β 2. Problem with smoothed return series is understatement of volatility and correlation a. Problem in asset allocation as smooth series asset classes will have inappropriately high allocations b. Underestimated correlation will interfere with hedge ratios used in risk management 3. However, long-term smoothed return series can produce reasonable mean returns a. Sharpe ratio distorted due to understated volatility III. Models of Price and Return Smoothing A. Section Introduction 1. Need to "form a belief with regard to [the] particular nature" of smoothing 2. Model to "detect, correct, or exploit smoothing"

18 QFIA QT - 43 B. Reported Prices as Lags of True Prices 1. Notation a. - the reported or smoothed price of an asset at time t i. Example: appraisal price b. - the true price of the asset at time t i. True price - "the best indication of the market price at which the asset would trade with ready buyers and sellers" 2. Consider Real Estate Index a. One or both of two things can happen i. Prices are lagged market values ii. Prices are appraisals with opinion based partially on lagged market values iii. Therefore current prices are function of lagged market values 3. A General Moving Average Model a.. (16.1) b. Simpler model i. 1 1 (16.2) a) 0 < α 1 b) Decay function with lesser weights on older observations i) Example α =.5, current observation 50%, prior observation, 25%, next prior 12.5%, etc c. Factoring to get True Price i. a) True price as function of observable reported prices d. Rearrange again (16.3) i. )] (16.4) a) True price is prior reported price plus constant (greater than 1) times change in reported prices i) Example: α =.5 and change in reported price of $5 implies true price $10 greater than last reported price

19 QT - 44 QFIA Importance of formula (16.4) a. Given α, can create a true return series from smoothed series of prices b. Fisher (2005) estimate α =.4 for US private unleveraged real estate i. Implication is that true price change is 2.5 times reported price change a) Note: smoothed prices lag whether prices are increasing or decreasing C. Modeling Lagged Returns Rather Than Prices 1. Rewriting (16.1) as returns rather than prices a.,,,, (16.5) b. R t, reported - return on reported prices in period t c. R t, true - return on unobserved but true prices in period t D. Estimating the Parameter for First-Order Autocorrelation 1. Equation 16.3, α "determined extent to which the reported price (or return) in a particular time period is determined or driven by the value of the true price (or return) in the same period" 2. Let α = 1 - ρ 3. From (16.4) a. 1 (16.6) 4. Alternatively as change in prices or as returns a. 1 (16.7) b., 1,, (16.8) c. ρ "is the first-order autocorrelation coefficient, given the assumption that the reported price series (or return series) is autoregressive of order one" d. "The use of ρ in place of α is to place emphasis on a statistical interpretation of the relationships" e. In (16.8), the reported return is based partially on new market information in the true return and partially on smoothed return of previous period f. Higher values of ρ imply higher amounts of smoothing E. Four Reasons for Smoothed Prices and Delayed Price Changes

20 QFIA QT Index is being based on observed prices of the most recent transactions and old or stale prices are being used in components with no recent trades 2. Appraiser might generate a series of smoothed prices a. "Appraiser observes price changes on delayed basis and only on those properties that have transacted" b. Exhibits behavioral anchoring - "observed tendency of humans to give disproportionate weight or reliability to previous observations" 3. Even efficient market current transaction prices may signal lagged price responses if nature of the transactions has a bias a. Two types in index, equally weighted i. True return 25% for one, 5% for other, 15% for index ii. If more transactions in 5% type (they have relatively lower prices) and no/less transactions in 25% type, then reported index will increase less than 15% 4. Time delay between setting price, closing of transaction and reporting of transaction to appraisers, maybe many months IV. Unsmoothing a Price or Return Series A. Section Introduction 1. Unsmoothing - "the process of estimating a true but unobserved price or return series from an observable but smoothed price or return series" B. Unsmoothing First-Order Autocorrelation Given ρ 1. Solving (16.8) for the true return a. R t,true = (R t,reported - ρr t-1,reported )/(1- ρ) (16.9) 2. Example shows how using ρ =.4 and reported returns reproduces the initial example true return a. Exact reproduction as the example was set-up to be a first-order autocorrelation process without error term b. In practice, things are not so exact i. ρ has estimation error 3. May want to use a fourth-order autocorrelation model if using smoothed quarterly returns

21 QT - 46 QFIA C. The Three Steps of Unsmoothing 1. Specify the form of autocorrelation, what is the order 2. Estimate the parameter(s) of the assumed autocorrelation process a. First-order process ρ = corr(r t,reported - R t-1,reported ) (16.10) b. Based on covariance and standard deviation i. ρ i,j = σ i,j /(σ i σ j ) (16.11) a) σ i,j - covariance between variables for i,j b) σ j - standard deviation for j c) Example shows ρ i,j =.037, far from true value (by design) of.4 i) Difference due to small sample size in example, plus estimation error 3. Used estimate correlation coefficient in formula 16.9 in place of ρ a. Using.037 provides unsmoothed returns closer to true but they are not a good result as.037 is a poor estimate of the true ρ of.4 i. With larger sample size estimate of ρ should be better 4. "Success of the unsmoothing therefore depends on the proper specification of the autocorrelation scheme and especially the accurate estimation of the parameter(s)" D. Unsmoothing Using Prices Rather Than Returns 1. Equation 16.8 is an approximation if true relationship is based on prices 2. Six steps to unsmoothing a return index based on a model of smoothed prices a. "Convert the returns to a price index using a cumulative wealth index that includes compounding b. "Convert the price index to a series of price changes using subtraction c. "Apply Equation to estimate the correlation between the price change series and its lagged value d. "Apply Equation 16.9, substituting price changes for returns e. "Use the unsmoothed price changes to form a price index f. "Convert the unsmoothed price index back into returns"

22 QFIA QT - 47 E. Unsmoothing Returns with More than First-Order Autocorrelation 1. k-order autocorrelation a. (16.12) b. Impact of true return captured in intercept and error term as true returns do not have autocorrelation 2. Distinction between k-order and first-order autocorrelation a. First order reduces all the coefficients to a single ρ b. k-order a more general or flexible formula as it does not constrain the relationships between the β i 3. Equation and k=1 a. Reduces to or is Equivalent to first-order formula 16.9 V. An Illustration of Unsmoothing A. The Smoothed Data and the Market Data quarters of NCREIF NPI and REIT indexes are used a. NCREIF NPI, a private real estate index which is likely smoothed b. NAREIT index is based on traded REIT stocks and should not have autocorrelation 2. Observations a. Both series have similar mean return b. Standard deviation of NCREIF NPI is lower i. Part of difference is leverage is used in NAREIT, increasing volatility c. 4Q2008, 1Q2009 NAREIT has large negative returns while NCREIF NPI has modest negative returns i. Indication of lagged pricing d. 2Q2009, 3Q2009 NAREIT has large gains while NCREIF NPI continues with negative returns i. Unable to short-sell appraised prices to take advantage of smoothed returns B. Estimating the First-Order Autocorrelation Coefficient 1. NCREIF NPI estimated autocorrelation coefficient of.831

23 QT - 48 QFIA NAREIT estimated autocorrelation coefficient of Sample period "covers the highly unusual real estate market collapse that coincided with the financial crisis that began in 2007" a. Observations not reflective of normal economic conditions and have "outliers" C. Unsmoothing the Smoothed Return Series Using Rho (ρ) 1. Table 16.2 page Observations a. NCREIF NPI i. 4Q2008 smoothed -8.29% becomes unsmoothed -48.3% ii. 1Q2009 smoothed -7.33% becomes unsmoothed -2.60% iii. "The unsmoothing technique captures the likelihood that the" -7.33% "was a lagged reaction to the events of the fourth quarter of 2008 due to smoothing" D. Interpreting the Results of Unsmoothing 1. More observations a. Standard deviations i. NCREIF NPI smoothed 4.01% becomes unsmoothed 13.38%, three times higher a) Smoothed is less than 1/4 of NAREIT b) Unsmoothed is 3/4 of NAREIT i) NAREIT uses leverage so should be higher ii. NAREIT original 17.32% becomes unsmoothed 21.81% b. Autocorrelation in NAREIT is possibly due to turmoil in market in financial crisis c. Asset allocations based on NCREIF NPI smoothed would over-weight this index d. Could improve unsmoothed if used higher than first-order autocorrelation 2. "Smoothed returns generate dangerous perceptions of risk if the returns are not unsmoothed and if the short-term volatility of the returns is used without adjustment to estimate longer-term risk" 3. Smoothed returns understate correlations 4. Unsmoothed NCREIF NPI returns have three times the volatility and twice the correlation to other asset classes as the smoothed returns

24 QFIA QT Unsmoothed returns lead to lower weights for real estate in optimal portfolio 6. Smoothed returns overstate Sharpe ratio a. Explains the "so-called real estate risk premium puzzle", why the very high returns relative to risk 7. "The lesson is clear. Autocorrelation of returns can provide deceptive indications of long-term risk relative to short-term risk" 8. Unsmoothing an advanced technique a. Provides better indication of risk b. Provides better asset allocation

25 QT - 50 QFIA

26 QFIA QT - 51 Principal Component Analysis on Term Structure of Interest Rates Helsinki University of Technology QFIA I. Introduction A. Term Structure 1. Involves lots of data 2. Yet, there are commonalities amount the different points on the yield curve 3. Can data be simplified while retaining common behavior? B. Principal component analysis (PCA) 1. Powerful tool of identifying patterns in data of high dimension (such as a term structure) 2. "PCA is a statistical technique in which the original variables are replaced by a smaller number of artificial variables that preserve as much as possible of the variability of the original variables" C. PCA Objectives of data simplification: 1. Reduce the number of variables and 2. Detect a structure in the relationships between variables. D. This Paper 1. Looks at term structure of four currencies, EUR, USD, JPY and GBP 2. Applied PCA to each currency individually and again looking at all four together II. Principal Component Analysis A. Section Introduction 1. PCA - a statistical technique a. Used to simplify data by reducing the number of variables b. Performs linear transformation of data to create "orthogonal coordinate system" i. Orthogonal - axes are independent ii. Axes ordered by variance, or explanatory power a) First axis explains the most, etc

27 QT - 52 QFIA c. "PCA s basis vectors are not fixed but depend on the data set." d. Principal components - the artificial variables of the transformed data e. Given reducing order of explanatory power, user can choose the number of principal components needed to achieve desired degree of accuracy 2. Comparison to Factor Analysis a. "PCA attempts to find a series of independent linear combinations of the original variables that provide the best possible explanation of diagonal terms of the matrix analysed" b. "Factor analyses focuses on the off-diagonal elements of the correlation matrix (Jorion, 2002)." c. Statistical model i. PCA not based on any particular statistical model ii. Factor analysis is based on a particular model. B. Mathematical Formulation 1. Setup a. Random vector x, components x 1,, x n b. Mean of zero, E(x) = 0 c. Non-singular covariance matrix, Cov(x) = Σ 0 2. First Principal Component a. Objective: "Find the linear combination of random variables x 1,, x n that contains as much of the variability of the random variables x 1,, x n as possible b. Notation i. β vector of weights of x ii. λi - i-th largest eigenvalue of the covariance matrix Σ iii. i - eigenvector corresponding to the i-th largest eigenvalue of the covariance matrix Σ iv. y 1 - the first principal component c. Formulation of First Principal Component i. For a linear combination β T x =

28 QFIA QT - 53 ii. Maximize its variance D 2 (β T x) iii. Subject to the constraint, the norming condition, β 2 β T β 1 iv. Then max a) Max subject to constraints v., the first principal component 3. Second Principal Component a. Second step to find a linear combination of the random variables that is uncorrelated with the first principal component b. Same formulation as for first principal component with one additional constraint c. Formulation of Second Principal Component i. For a linear combination β T x = ii. Maximize its variance D 2 (β T x) iii. Subject to the constraint, the norming condition, β 2 β T β 1 iv. Additional constraint Cov(y 1, β T x) = 0 v. Then max a) Max subject to constraints vi., the second principal component 4. Remaining Principal Components a. Follow above procedure to find the remaining of the n principal components of x b. Properties of Principal Components i. "The variance of the linear combinations is the largest possible" ii. "The linear combination is uncorrelated with the previously identified linear combinations" iii. "These linear combinations form the principal components of random vector x." c. Eigenvalue Decomposition i. Notation a) D = diag(λ 1, λ 2,, λ n ), diagonal matrix of the eigenvalues

29 QT - 54 QFIA b) B - a matrix with the eigenvectors as the columns of the matrix, an orthogonal matrix i) B T B = I ii. Principal Components developed from eigenvalue decomposition of the covariance matrix a) Σ = BDB T b) y i = = 1i x Ni x N 5. "The eigenvectors with the largest eigenvalues correspond to the dimensions that have the strongest correlation in the dataset. This means that the more correlated the data, the bigger share of the total variation is explained by the first principal component." 6. Original data can be recovered from the principal components a. x i = β i y= i1 y in y N i. Study manual author note: the text has B i1 rather than i1, a typo I presume. Also an issue below in the approximation for x i C. Choosing the Number of Principal Components 1. No single criteria a. Case to case differences b. Explanatory power of components different for different cases 2. Some basic rules a. Threshold guideline i. Select as many components, K of N, so cumulative sum exceeds desired explanatory power a) b. Large Eigenvalues i. Sort eigenvalues in size order ii. Determine "large", "small" split iii. Just use components corresponding to large eigenvalues

30 QFIA QT - 55 c. With standardized variables i. "If the original variables are standardized for PCA (see section 2.5), each of them will have a unit variance. ii. "Therefore any PC with an eigenvalue of at least 1 explains more of the total variance than any of the original variables. iii. "Thus a simple heuristic would be select those PCs that have an eigenvalue of at least 1 (Clustan, 2006)." 3. Approximate value a. If keep the first r components i. x i = β i y i1 y ir y r D. Interpretation of Analysis 1. For each principal component a. Eigenvalues are represent variance b. Eigenvectors are "loadings" 2. Compute factor structure, an n by r principal component matrix 3. PC Matrix elements represent correlation between original data and principal components if PC calculation is done using standardized data, that is from the correlation matrix 4. Principal component scores, value of each PC with respect to each observation a. y j = B r x j, j = 1,, n i. where the i-th component of y ji, i = 1,, r ii. y j is an r-vector (y j1,, y jr ) iii. "is thus the value of i.th principal component y at observation j=1,2,,n, i i=1,2,,r "

31 QT - 56 QFIA A. Geometrical Interpretation 1. "The j-dimensional linear subspace spanned by the first j principal components gives the best possible fit to the data points as measured by the sum of squared perpendicular distances from each data point to the subspace." 2. Consider the function f(x) = x T Σ -1 x = c B. Discussion a. c is a constant b. The function "determines an ellipsoid of n-dimensional space as a function of variable x" c. Principal components are the main axis of the ellipsoid d. "The main axis of the ellipsoid coincide with the directions of the eigenvectors of the covariance matrix and the lengths of the main axis relate to each other as numbers" square root of eigenvalues of the covariance matrix" 1. Considerations in performing PCA a. Not statistical analysis in that there is no probability distribution b. PCA represents data in simpler form c. True economic interpretation of principal components is "difficult, if not impossible" 2. For PCA to work exactly, use standardized variables a. "This is because it is often the case that the scales of the original variables are not comparable and that (those) variable (variables) with high absolute variance will dominate the first principal component." b. Standardized so mean = 0, variance = 1 i. z i = (x i - μ x ) / σ x ii. Then apply analysis to z i c. This additional transformation makes economic interpretation even more difficult IV. Modeling Term Structure of Interest Rates with PCA A. Term Structure 1. Yield curve of zero coupon bonds 2. Challenges: How to a. Estimate term structure

32 QFIA QT - 57 b. Evaluate implicit interest rates in future (forward rates) c. Explain the shape and the movements of the curve B. Literature Review of Principal Component Analysis on Term Structure 1. PCA has been applied to short interest rates, long interest rates, in different currencies, equities, FX, option volatility 2. For interest rate levels and differences, typically, first three principal components provide sufficient explanation of the term structure 3. Equities and FX can have similar reduction 4. Option volatility and credit spreads sometimes need more than three components 5. Usability of results depends on analysis setup a. "Choosing to perform PCA on levels rather than differences forgoes the opportunity to construct meaningful and informative market scenarios that for many end users might be the real value of PCA" C. Applications in Finance 1. Simpler and Faster VaR Processes a. PCA can result in simplification of risk model i. Study manual author note: My former employer used PCA for Economic Capital b. VaR needs correlations between assets, the number of which can get very large as the number of assets increases i. Issues with large portfolios a) Accuracy of measurement of a large number of correlations b) Computation time can be very long c) Can have problems in getting a positive-definite correlation matrix, especially if many assets are correlation, typical of fixed income portfolios ii. PCA analysis advantages a) Number of principal components much smaller than the number of assets b) Correlation matrix is positive definite c) Variance of portfolio = sum of squared exposures times variance of each principal component

33 QT - 58 QFIA iii. Caution of using PCA for VaR a) "To perform efficient PCA on a portfolio, one should select the PCs not based on how much they explain of the total variability of the data but on how much of the variability of the particular portfolio" b) Hull example of portfolio with little exposure to first principal component using only first principal component when portfolio has significant exposure to the second principal component i) Similar to duration matching ignoring convexity mismatch 2. More Efficient Hedge Strategies a. Duration hedging based on parallel shift in yield curve i. Does not hedge non-parallel shifts b. "Identifying first few principal components allows achieving a better-hedged position because the components explain almost all of the return variability across the whole spectrum of maturities" i. Easily computation of expected return and variance can help create portfolio with zero sensitivity 3. Macroeconomic Analysis a. Empirical research shows reasonable macroeconomic interpretation can be provided to PCA components b. PCA overcomes computational problems i. Linear regression often used in macroeconomic analysis but can have problem of multicollinearity ii. Multicollinearity not a problem with PCA as components are independent c. Combining PCA with MLR is called Principal Component Regression V. Implementation and Results A. Section applies PCA to the term structure of interest rates of four different currency zones: Euro zone (EUR), United States (USD), Japan (JPY) and Great Britain (GBP). 1. Time series of 3 months is used, from April 7, 2006 to July 7, 2006

34 QFIA QT - 59 B. Data Preparation 1. Data are first differences of Euribor and LIBOR rates for the four currencies a. Widely used and liquid rates b. Most influential economic zones 2. Data standardization not done a. But modified so mean = 0 b. Evidence shows with interest rates of comparable variance, standardization and notstandardizing show similar results 3. Advantage of analyzing first differences a. Different interest rate scenarios can be created 4. Disadvantage of analyzing first differences a. Tends to increase noise in the data, decreasing efficiency of PCA 5. Analysis used two different setups a. Each currency zone term structures independently and compared the correlations between PCs specific to different currency zones b. One aggregate analysis C. Preliminary Analysis 1. EUR a. Long rates tended upward during observation period i. Tended to move together implying few principal components b. Short rates 2. USD i. Showed different characteristics ii. Need for "special considerations" a. Similar behavior to EUR but with difference b. Twist in curve so that 2Y, 3Y and 4Y rates lower than 12 month in middle of observation period

35 QT - 60 QFIA c. "This hints about expectations of market participants about near future development of interest rates and thus the monetary policy moves." 3. JPY a. Similar behavior to EUR but with difference b. Rates are very low with short rate about zero c. "Long rates are much higher implying that contractionary monetary policy actions are expected in the near future" 4. GBP a. General trend similar to EUR and USD rates, but curve is different b. Hump-shaped curve with short and long rates lower than medium term rates c. "This indicates that monetary policy, future expectations and even economic situation were quite different in UK when compared to other currency zones analysed during the observation period" D. Problem Formulation 1. For each currency zone separately a. x = (x 1, x 2,, x 18 ), - the vector of the 18 zero-mean random variables representing absolute daily changes of interest rates of each maturity b. Σ - the corresponding covariance matrix c. Σ BDB T - the eigenvalue decomposition d. the 18 eigenvectors from the decomposition e. D = (λ 1, λ 2,, λ 18 ) the 18 eigenvalues from the decomposition f.,,, the 18 principal components completely explaining the variability of the initial variables g. Aim of problem: Explain at least 90% of the total variability with as few PCs as possible 2. Perform similar analysis for all currencies combined a. 4 (currency zones) * 12 (maturities) variables and corresponding 72 principal components

36 QFIA QT - 61 E. Individual Currency Zone Results 1. Explanatory power of PCs a. Figure 4 plots cumulative percentage of total variance data explained by principal components i. First 3 PC show 99%+ for USD and EUR ii. Need 5 PC for 99% for GPB and JPY a) During observation period, these two have more complicated behavior b. Figure 5 plots variance by point on the yield curve i. In order to achieve 90% explanatory power across all 12 points on the yield curve, 5 PC are needed for each currency a) Model for each currency x i = β i y = B i1 y 1 + B i2 y 2 + B i3 y 3 + B i4 y 4 + B i5 y 5 ii. Observations 2. PC Coefficients a) First PC explains i) Minimal impact on short rate, most of intermediate rates, much of long rates b) Second PC explains i) Much of short rate, minimal of intermediate rates, much of long rates c) Third PC explains i) Some of short rate, minimal of intermediate, some of long rates ii) But pattern is different than that of Second PC d) Fourth and Fifth PC i) Unusual that these PCs are needed to get to 90% explanatory power ii) Mostly describe the short rate a. Figure 6 shows the coefficients of the five principal components for each currency i. Coefficients are the "loadings" a) "The distinction of loadings to curves in figure 4 is that loadings only describe what effect each PC has on each maturity without considering the

37 QT - 62 QFIA Scenarios absolute magnitude of the effect, whereas in figure 4 the curves represent the amount of total variation explained by each component." ii. Descriptions of behavior a) First PC - shift in rates (level of rates) b) Second PC - twist of rates (slope of rates) c) Third PC - bow of rates (curvature of rates) d) Fourth and Fifth PC - "important for short-term interest rates, but their rational interpretation is more difficult" a. Shocks or changes in the principal component loadings would cause changes in the underlying term structure b. Different loadings can be used to create market shock scenarios c. Creation of scenarios i. Can assume principal components move up/down by some amount, maybe some standard deviation amount a) Could create 2 n scenarios from n principal components ii. Author chose Monte Carlo instead a) "create random vectors whose elements are drawn from multinormal distribution with zero mean and covariance matrix equal to diagonal matrix of chosen eigenvalues (that correspond to variances of chosen PCs) and multiply this vector with PCs to get different linear combinations representing daily interest rate movement scenarios." b) Advantages of Monte Carlo i) Arbitrarily choose the number of scenarios ii) Chose from the five chosen principal components rather than the 18 total principal components iii) "examine the correlations between scenarios and original data" d. Figure 7 shows 5th percentile and 95th percentiles along with daily observations i. Most daily observations within the band a) "This suggests that the market model with the chosen PCs performs quite well in estimating the daily changes."

38 QFIA QT Levels versus Differences a. Study based on differences of rates i. Note that differences can introduce "noise" b. So, also studied levels of rates using PCA i. Results show a need for only three principal components ii. Illustrates the disadvantage of using differences c. "trade-off between efficient estimation (low dimensionality of model) and applicability of results (scenarios and other applications where standard deviation of PCs is required)" F. Combined Currency Zone Results 1. Evidence correlations among currencies exist, so possibility of further reducing the dimensionality of the problem 2. Correlations a. Looking at correlations of five principal components of the four currencies (using differencing data) b. First Principal Component i. Correlations between pairs of currencies are all "quite strong" a) Low 53% USD/JPY and GPB/JPY b) High 79% EUR/GBP c. Second Principal Component i. 33% for USD/EUR a) Study Manual author note: There appear to be some typos here as the chart including 33% is labeled "3rd Principal Component" ii. Others are much smaller iii. "This suggests that the twist effect (and therefore possibly monetary policy actions) has been quite inconsistent between the currency zones during the observation period." d. Third Principal Component i. Small correlations, with some being slightly negative