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Introduction to the Mathematics of Financial Derivatives Chapter 1 SC-3

© ACTEX 2015 SOA QFI Core Exam Study Manual

An Introduction to the Mathematics of Financial Derivatives, 3rd Edition Ali Hirsa, Salih Neftci

Chapter 1 Financial Derivatives - A Brief Introduction

I. Introduction

A. Book discusses logic behind asset pricing

B. Chapter discusses two building blocks of financial derivatives, options and forwards

C. Complicated derivatives can be decomposed into simpler derivatives

D. Study manual author note: Much of the material in this chapter is a review from Exams FM and MFE

II. Definition

A. Derivative security or contingent claim: "A financial contract is a derivative security, or a contingent claim, if its value at expiration date T is determined exactly by the market price of the underlying cash instrument at time T"

B. Notation

1. T: expiration date of derivative

a. After this date the derivative contract no longer exists

2. F(T): price of derivative asset

3. ST: price of the relevant cash instrument, called the "underlying asset", at time T

4. F(t) and F(St, t): price of derivative on underlying St at time t

5. dt : Yield of payout on a derivative

III. Types of Derivatives

A. Introduction

1. Three general types of derivatives

a. Futures and forwards: basic building block

b. Options: basic building block

c. Swap: hybrid security that can be decomposed into set of forwards and options

SC-4 An Introduction to the Mathematics of Financial Derivatives Chapter 1


2. Five main groups of underlying assets

a. Stocks

i. Claims on returns from production of goods/services

b. Currencies

i. Not direct claims on real assets

c. Interest Rates

i. Not really assets, so a notional asset needs to be determined

d. Indexes

i. E. G. S&P 500, FT-SE100

ii. Need notional amount

e. Commodities

i. Not financial assets, can be physically purchased and stored

B. Cash-and-Carry Markets

1. Cash-and-Carry are an alternative to holding a forward/futures contract on commodity

a. Buy directly in cash markets or buy indirectly using forward/futures

i. At expiration, in either case, the long position holds the commodity

2. Examples: gold, silver, currencies, T-bonds

3. Features of Cash-and-Carry Market

a. Borrow at risk-free rates by collateralizing the underlying

b. Buy and store the product

c. Insure it until expiration of derivative contract

4. Additional property

a. Information about future conditions, supply/demand, etc, should not change the spread between cash and futures prices

i. New information changes the price of both by about the same amount



C. Price-Discovery Markets

1. Differences from Cash-and-Carry

a. Commodity is perishable or

b. Commodity does not yet exist, e.g. spring wheat

2. Features

a. Not physically possible to buy commodity and store until expiration date of derivative

b. No cash market yet exists, e.g. crop has not yet been harvested

3. Impact of Information

a. Future supply/demand cannot influence cash price

b. Such information will influence, and thus can be discovered, in the futures market

D. Expiration Date

1. Value of futures contract at expiry should be equal to the value of the underlying

a. F(T) = ST (1.1)

2. Value of F(t), t < T, is not necessarily equal to St

IV. Forwards and Futures

A. Forward

1. Forward: "a forward contract is an obligation to buy (sell) an underlying asset at a specified forward price on a known date"

2. Review of payoff diagrams, pages 4,6

B. Futures

1. Similar to forwards in that they are both obligations for settlement at a specified future date of an underlying

2. Differences between forwards and futures

a. Where Traded

i. Futures: on formal, public exchanges

ii. Forwards: private contracts in the OTC market (over the counter)



b. Credit Risk

i. Futures: cleared through an exchange designed to reduce default risk

ii. Forwards: private contracts so risk depends on the credit quality of the counterparty

c. Mark-to-Market

i. Futures: Daily, effectively settled daily with a new contract created

a) Daily profit/loss recorded

ii. Forwards: no mark-to-market

C. Repos, Reverse Repos, and Flexible Repos

1. Repos

a. Definition

i. "A repurchase agreement, also known as a repo, is a transaction in which one party sells securities to another party in return for cash, with an agreement to repurchase equivalent securities at an agreed upon price and on an agreed upon future date."

a) You could think of this as a loan with the difference between the cash amount and the "agreed upon price" as interest on the loan.

i) Securities are the collateral for the loan

ii) This interest amount creates the "repo rate"

b) Alternatively, think of a repo as a spot sale and a forward contract

ii. For seller, called repo

iii. For buyer of security, called reverse repo

iv. Classified as money market instruments

b. Types of repos

i. Overnight: one-day maturity

ii. Term: specified maturity date other than one-day

iii. Open: no end date

c. Forms of Repo Transactions

i. Specified Delivery



ii. Tri-party

iii. Hold-in-custody

a) Selling party holds security during repo term

2. Flexible Repos

a. A repo with a flexible withdrawal schedule, both in terms of timing and amount

b. Differences from traditional repo

i. "Convexity due to cash withdrawals

ii. "Formal written auction like trade

iii. Additional documentation is necessary for credit protection

iv. Counterparties are typically muni bond issuers

c. Types of flexible repos

i. Secured

a) Municipality receives collateral

i) Treasuries, GNMA, agency MBS, etc

ii) Collateral comes from a reverse repo

b) Average size 10 - 20 million

ii. Unsecured

a) No collateral but likely higher rate

b) Average size - smaller than secured

V. Options

A. General

1. The right, but not an obligation, to do something

2. Call: "A European-type call option on a security St is the right to buy the security at a preset strike price K. This right may be exercised at the expiration date T of the Option. The call option can be purchased for a price of Ct dollars, the call premium, at time t < T"

3. Put: right to sell



4. Exercise Dates

a. European options: only at expiry

b. American options: exercised at any time, not just the expiration date

5. Reasons traders need to know the value of Ct

a. Estimate of price to trade, especially if traded infrequently

b. Evaluate risk

c. Determine any mispricing for an arbitrage opportunity

B. Some Notation

1. Best to find a formula, or closed form solution, for Ct as a function the underlying

2. Only known formula for Ct is at T

a. Out of the money expiry, ST < K (1.2)

i. Implies CT = 0 (1.3)

b. In-the-money expiry, ST > K (1.4)

i. Implies CT = ST - K (1.5)

c. CT = max(ST - K, 0) (1.6)

i. Formula for CT shows options are non-linear

3. Figures 1.3 and 1.4, page 8, graph values of call options at times before expiry

VI. Swaps

A. General

1. Swap: "A swap is the simultaneous selling and purchasing of cash flows involving various currencies, interest rates, and a number of other financial assets"

2. Method to price swaps is to decompose swap into forwards and options, price the forwards and options, sum these prices to get price of swap

B. A Simple Interest Rate Swap

1. Notional principals



2. Two counterparties for interest rate swap

a. Party A to pay fixed on notional, receive floating on notional

b. Party B to received fixed on notional, pay floating on notional

c. Each period, payments are netted, net payment is essentially the interest rate differential times the notional principal

d. Based on comparative advantage, each party should secure lower rates

e. Swap dealer earns a fee for bringing parties together

3. Basket of forward contracts will replicate the cash flow helping to value the swap

C. Cancelable Swaps

1. Definition

a. "A cancelable swap is a swap where one or both parties has the right but not the obligation to cancel the swap before its maturity."

i. Dates to cancel the swap specified in the contract

2. Types of Cancelable Swaps

a. Callable Swap

i. Payer of fixed rate has the option

a) On specified dates

b) At cancelation, payer

i) Pays present value of future payments

ii) Receives a premium

b. Puttable Swap

i. Receiver of fixed rate has the option

a) On specified dates

c. Vanilla Swap can be "closed out"

i. By paying net present value of future payments

3. Uses of Cancelable Swaps

a. Use with callable bonds



b. Use as asset/liability hedges

i. Specifically when liabilities have prepayment options

Quantitative Finance Chapter 9 V-3


Quantitative Finance Paul Wilmott

Chapter 9 Overview of Volatility Modeling

I. Introduction

A. A flaw in Black-Scholes model is volatility

1. "Quite frankly, we do not know what volatility currently is, never mind what it may be in the future."

B. Volatility modeling necessary to correctly price derivatives

II. The Different Types of Volatility

A. Actual Volatility

1. Measure of randomness of return at point in time

2. Difficult to measure

3. What is the right "timescale"?

4. Does volatility vary? Yes

B. Historical (or Realized) Volatility

1. Measure of randomness over a past period

a. Time period and model need to be specified

b. Should the past be used to estimate future?

2. In option pricing, time period for volatility should be over the life of the option

C. Implied Volatility

1. "Volatility which when input into the Black-Scholes option pricing formulae gives the market price of the option"

2. I.e., a "market view"

3. Timescale is option maturity

4. Volatility is also influenced by supply/demand forces

D. Forward Volatility

1. "refers to the volatility (whether actual or implied) over some period in the future"

V-4 Quantitative Finance Chapter 9


III. Volatility Estimation by Statistical Means

A. Constant Volatility/Moving Window

1. If Volatility is constant or slowly changing

a. ∑ , where , the return on the ith day

i. Issue

a) Returns equally weighted creates "plateauing effect" if 1-day has a large return

i) Volatility estimate stays high for N days until the large return dropped from calculation

B. Incorporating Mean Reversion

1. ARCH Autoregressive Conditional Heteroscedasticity

2. Time varying volatility, rather than constant or slowly changing

a. Consider σn the volatility on the nth day

b. σlongtermmeanvolatility,volatilitytendstovaryaroundthislongtermaverage,then

i. 1 ∑ ,

c. Should each of the n returns have the same weight?

d. Should experience that is more recent have more weight than older experience?

C. Exponentially Weighted Moving Average

1. Consider this estimate for volatility

a. 1 ∑ ,

i. 0 < λ < 1

ii. The more recent the return, the greater the weight

iii. Sum includes all data available

iv. Coefficient (1-λ ensuresweightsaddto1

b. Simplifies to: 1



D. Simple GARCH Model

1. GARCH Generalized Autoregressive Conditional Heteroscedasticity

2. Putting the mean-revering model together with the ARCH model

a. 1 1

E. Expected Future Volatility

1. What is expected variance k days in the future?

2. Ways to estimate future volatility

a. Expected Future Volatility: EWMA

i. Using above ARCH model and taking expectations of all terms in the formula, the result is expected future variance is equal to current variance

ii.

b. Expected Future Volatility: GARCH

i. GARCH uses a mean reverting volatility, the expectation is

a) mean reverting volatility

b) plus a factor based on

i) the difference between mean reverting volatility and the most recent volatility

ii) how far into the future you are estimating

ii. 1

a) Where υ = α/[1-(1-α)*(1-λ)]

F. Range-Based Estimation of Volatility

1. Problems:

a. Lots of data needed to avoid sampling errors

b. Parameter varies so more data can lead to incorrect results

2. Potential Resolution: Use intra-day data

a. Problem: Intra-day returns not normally distributed and variance might not be finite



3. Potential Measures

a. Traditional close-to-close measure

i. With small drift (returns), 1/ ∑ log ,

a) Ci is the closing price on the i-th day

b) Variance of the log of price returns

ii. To adjust for drift

a) ∑ log log / 1 ,

b) While not quite, this formula looks like, or is similar to, the equally weighted sums of the difference in the squares of the log of daily returns less the square of the log of average return over the period (log(Cn/C0))

c) If Cn = Co, then the subtractive term is zero and the two formulas are almost the same

iii. Recognize if data is daily, then variance is daily

a) Annualize result by taking square root of number of trading days in a year

b) Some take the view that there are 260 trading days per year (52 * 5), others as 250 days (less holidays)

b. Parkinson (1980)

i. This method uses extreme high, H, and extreme low, L for a day

ii. ∗

∑ log ,

iii. Five times more efficient than close-to-close method (variance 1/5 of close-to-close variance)

iv. Not independent of drift

c. Garman & Klass (1980)

i. 7.4 times more efficient than close-to-close method

ii. Oi is opening price

iii. Looks like an empirical formula

iv. ∑ .511 log . 019 log log 2log log ,

v. Not independent of drift

d. Rogers & Satchell (1991)

i. Independent of drift

ii. ∑ log log log log ,



IV. Maximum Likelihood Estimation (MLE)

A. Motivating Example

1. Probability of Event (getting into a taxi with a particular number) a question of assumptions and statistical methodology

a. Assumptions

i. Numbers are strictly positive

ii. Numbering starts at one

iii. No number is repeated

iv. No number is skipped

b. Statistical Methodology

i. N is the taxi number

ii. Then the probability of getting into any specific taxi is 1/N

iii. Answers the question, "Which N maximizes the probability of getting into taxi 20,922?"

a) N = 20922, due to any higher N will make the probability lower

2. Concept of MLE: "choose parameters that maximize the probability of the outcome actually happening"

B. Three hats

1. Hats contain normally distributed random numbers, each hat has a different mean and standard deviation of the numbers included

2. Given a number is picked, what is the most likely hat it was drawn from?

3. Example

a. Draw number

b. Figure out the probability (actually probability density function value ) of the number for each hat

i. Which hat has highest density function value?

c. Draw a second number from the same hat

d. Figure out the density function value of the number for each hat

e. Figure out the joint value (multiply the two density values)

f. Highest value is the MLE for the hat the numbers were drawn from



4. Find the standard deviation

a. Drawing N numbers, ϕi from the hat where the standard deviation is unknown, how to find this standard deviation

b. Wilmott sets the first derivative of the joint "probability" with respect to σ of the = 0 to maximize the probability

c. The result is the familiar equation for variance, ∑

5. Quant Salaries example

V. Skews and Smiles

A. Description

1. For a series of options that all expire on the same date, on the same underlying, graph the value of implied volatility (vertical axis) versus strike price (horizontal axis)

a. If actual volatility were constant and if Black-Scholes were correct and if options were priced correctly, then the graph would be flat

i. That is all implied volatilities would be the same

ii. "Of course, none of those assumptions is correct, and so there is plenty of scope for that plot to be curved, or skewed"

b. Skew - graph of implied volatility that is curved

i. Negative skew - downward sloping

ii. Positive skew - upward sloping

iii. Smile - curve with minimum in the middle

a) Implication from Wilmott FAQ 39, "a smile means that out-of-the-money puts and out-of-the-money call both have higher implied volatilities than at-the-money options"

c. Implication of skew is market view that Black-Scholes model or assumptions are incorrect

i. Or that market "does not care" about result of formula

a) Example out of money options are cheap, and investor needs protection so investor does not care about "implied volatility" level

d. Skew does provide information about

i. Future volatility

ii. Expectations of future volatility

iii. Level of demand for hedging "how desperately they need to hedge"

a) Higher demand implies high price implies higher implied volatility



B. How to speculate

1. On implied volatility, use straddle

a. Straddle = long call and long put with same strike and expiration

b. Put-call parity ensures both have same implied volatility

c. Portfolio monotonic in volatility

i. Meaning increase volatility and value of portfolio goes up

a) Think implied volatility will increase, buy straddle

b) Think implied volatility will decrease, sell straddle

2. On skew, use risk reversal

a. Risk reversal = long call and short put, with call having higher strike than the put

b. Two strikes so implied volatility skew matters

c. Figure 9.8, page 215 shows three skew curves, positive skew above no skew, which is above negative skew.

d. If have view on the direction of skew, can use risk reversal trade to speculate

VI. Different Approaches to Modeling Volatility

A. PWOQF2 would appear to be "Paul Wilmott on Quantitative Finance", Second Edition

B. To Calibrate or Not?

1. A discussion but he does not answer the question

2. Is implied volatility an estimate of future volatility?

a. Wilmott does not believe market is statistically sophisticated enough to reach this conclusion that implied volatility is an estimate of future volatility

b. Supply/demand issues

i. "out-of-the-money puts are expensive, due to demand, otherwise known as 'fear'"

ii. "out-of-the-money calls are cheap, due to supply, otherwise known as 'greed'"

3. Should your option-pricing model, producing theoretical option prices, match exactly quoted market prices?



C. Deterministic Volatility Surfaces

1. Simple model consistent with Black-Scholes has volatility as function of stock price and time, σ(S ,t)

a. σ(S,t) is coefficient in gamma term (Γ ∂2V/∂S2

b. However, generally no closed-form solutions now

2. σ(S,t) provides enough "room" to make theoretical values consistent with market prices

3. An "inverse" problem

a. Implied volatility are an average volatility over the life of the option

b. Actual volatility, σ(S,t), is the volatility for a particular forward time period

c. Going from implied volatility, actual small changes in implied can have large change on σ(S,t)

i. Issues:

a) Financial world is too complicated to have a simple σ(S,t)

b) σ(S,t) change depending on measurement period

D. Stochastic Volatility Surfaces

1. Impossible to forecast future volatility implies volatility should be treated as being random

2. Issues with stochastic volatility models

a. What model to use?

b. What is the volatility of volatility?

c. Hedging now has two sources of randomness, stock price and volatility, creating a two unknowns in one equation

E. Uncertain Parameters

1. Random - "you have a probabilistic description, perhaps even a probability density function, for the random variable"

2. Uncertainty "is when you have no such concept", and therefore is "vague"

3. Simple model - volatility will be within a range and therefore option value will be within a range

a. Pricing becomes non-linear



F. Static Hedging

1. Volatility risk - risk that the volatility used in option pricing model is incorrect

2. In practice, both dynamically hedging and statically hedging are used together

a. Static hedge with vanilla options to remove most of risk, then dynamically hedge the remaining risk

i. Also, allows you to know the value of the exotic part of the option

G. Stochastic Volatility and Mean-Variance Analysis

1. Governing equation with stochastic volatility model and you can assume dynamic hedging producing market price of volatility risk, however, such value is unreliable in practice

H. Asymptotic Analysis of Volatility

1. Use either

a. Easy-to-compute model that might not be so accurate, but fast,

b. Alternatively, a more precise model that is slow to calculate?

2. Most practitioners pick former

3. Asymptotic analysis - "finding approximate solutions to differential equations…by exploiting the relative largeness or smallness of a parameter in the model". Maybe some terms when multiplied become so small they can be ignored, leaving a simpler equation that can be solved

VII. The Choice of Volatility Models

A. Constant volatility

1. σ is constant

2. Black-Scholes formula

3. Very popular, especially for vanilla options

B. Deterministic volatility

1. σ(S,t),

2. Black-Scholes Partial Differential Equation

3. Very popular for exotics



C. Stochastic volatility

1. dσ = …higher dimension

2. Transforms

3. Very popular for exotics

D. Jump diffusion

1. Poisson processes

2. Jumps in stock and/or volatility

3. Becoming increasingly popular

E. Uncertain volatility

1. σ - ≤ σ ≤ σ+

2. Non-linear partial differential equation

3. Not currently popular

F. Stochastic volatility and mean-variance

1. dσ = …higher dimension

2. Non-linearity

3. Not currently popular

VIII. Appendix: How to Derive Black-Scholes PDE, Minimum Fuss

A. Brownian motion model for stock price dynamics yields a linear diffusion equation for financial assets including options

B. V(S,t) is option value , 0

1. If V is empty portfolio, if V has nothing in it, it's value should be zero

2. If V is a zero coupon bond, V = e-r(T-t)

a. Then

i. ∂V/∂S = 0

ii. ∂2V/∂S2 = 0

iii. ∂V/∂t = -r*(-1)* e-r(T-t) = +rV

b. Substituting into the general equation above, + rV + 0 + 0 + cV = 0

i. c = -r



3. If V is a the stock S, V = S

a. Then

i. ∂V/∂S = 1

ii. ∂2V/∂S2 = 0

iii. ∂V/∂t = 0

b. Substituting into the general equation above, + 0 + 0 + b(1) - rS = 0

i. b = rS

4. Remaining coefficient "a" is left to be determined

a. The coefficient of ∂2V/∂S2, the term based on change in change in stock price, or a measure of volatility

b. "a" is the only parameter that needs to be "calibrated"

Analysis of Financial Time Series Chapter 1 V-15


Analysis of Financial Time Series Ruey Tsay

Chapter 1 Financial Time Series and Their Characteristics

Link to errata for text, http://faculty.chicagobooth.edu/ruey.tsay/teaching/fts3/errata.pdf

Note: this chapter is included in the SOA syllabus for background only.

I. Introduction

A. "Financial time series analysis is concerned with the theory and practice of asset valuation over time"

1. Financial time series analysis

a. Empirical discipline

b. Elements of uncertainty

c. Asset volatility not directly observable

II. Asset Returns

A. Note: Tsay's vocabulary, use of terms, is different than that of other authors in the SOA syllabus

B. This section should be a review from prior exams or VEE Finance courses

C. Notation

1. Pt = the price of an asset at time t

2. 1 + Rt = simple gross return

3. Rt = simple net return or simple return

4. k number of periods between dates (t-k) and t

5. exp(x) = exponential function

6. ln(x) = natural log of x, x a positive number

7. rt = continuously compounded return or log return

8. pt = ln(Pt)

9. Dt = dividend payment of an asset between dates t-1 and t

V-16 Analysis of Financial Time Series Chapter 1


D. One-Period Simple Return

1. Simple gross return 1+ Rt = Pt/Pt-1 (1.1)

2. Simple net return = Rt = Pt/Pt-1 - 1 (1.2)

E. Multiperiod Simple Return

1. k-period simple gross return = 1 ∏ 1

a. Product of k one-period simple returns, called a compound return

2. Annualized {Rt[k]} ∏ 1 1

3. Geometric mean Annualized {Rt[k]} exp ∏ 1 1

4. If Rt are small, Annualized {Rt[k]} ∑ (1.3)

F. Continuous Compounding

1. If C is initial capital and A is a future value, n number of years, r a continuous compounded rate

a. A= C exp(r*n) (1.4)

b. C = A exp(-r*n) (1.5)

G. Continuously Compounded Return

1. Natural log of simple gross return is the continuously compounded return or log return

a. ln 1 (1.7)

i. Also 1

b. ln 1 ln ∏ 1 ∑ ln 1 ∑

c. Continuously compounded returns are "more tractable" to work with

H. Portfolio Return

1. A weighted average of the returns of the components of the portfolio

a. Weights equal percentage of portfolio in that component

2. Simple return of p at time t, , ∑

a. N total assets

b. wi weight of i-th asset or its percent of total portfolio value

c. Rit is simple return of asset i at time t



3. Continuously compounded portfolio return is not so simple

a. If Rit are all small, the continuously compounded portfolio return at time t is

i. , ∑

I. Dividend Payment

1. Need to reflect dividend payment in return calculations

a. Rt = (Pt +Dt)/Pt-1 - 1

b. ln lnP

J. Excess Return

1. Difference between an asset's return and the return on a reference asset, most often the risk-free asset

2. R0t, r0t are the returns of the reference asset

3. Simple Excess return = Zt = Rt - R0t (1.7)

4. Log Excess return = zt = rt - r0t (1.7)

K. Remark

1. Long: owning an asset

2. Short: borrowing and selling an asset to be repaid later by buying the asset

a. Short seller is responsible to pay any dividends paid on the underlying stock to the lender

L. Summary of Relationship

1. Nothing new

III. Distributional Properties of Returns

A. Notation

1. Rk is a k-dimensional Euclidean space

2. Point in Rk is denoted x ∈ Rk

3. Random vectors X = (X1, …Xk), Y = (Y1, …Yk)

4. P(X∈A,Y∈B probabilityXisinsubspaceA⊂RkandYisinsubspaceB∈Rq

5. θaparameter



B. Review of Statistical Distributions and Their Moments

1. Joint Distribution

a. Function FX,Y(x, y; θ) = P(X≤x, Y≤y; θ)

i. Inequality ≤ is a component-by-component operation

ii. FX,Y is a joint distribution function

b. If joint probability density function fx,y(x, y; θ) of X and Y exists, then

i. FX,Y(x, y; θ) = , , ;

ii. And X and Y are continuous random vectors

2. Marginal Distribution

a. Marginal distribution of X, FX(x; θ) = FX,Y(x, ∞…∞; θ)

i. Obtained by integrating out of Y

b. Marginal distribution of Y found similarly

c. If k=1, X is a scalar random variable and distribution function becomes

i. FX(x) = P(X ≤ x;θ)

a) FX(x) is also known as the cumulative distribution function (CDF) of X

i) FX(-∞) = 0, FX(∞) = 1

3. Conditional Distribution

a. Conditional distribution of X given Y ≤ y is | ;, ;

;

b. Conditional density of X given Y = y is | ; , , ;

; (1.8)

c. Marginal density fy(y;θ) = . , ; dx

d. Relation among the three is fx,y(x, y; θ) = fx|y(x; θ) times fy(y;θ) (1.9)

i. Identity is used frequently in time series analysis

e. X and Y are independent if and only if fx|y(x; θ) = fx(x; θ)



4. Moments of a Random Variable

a. The lth moment of a continuous random variable X is

i.

a) E is expectation

b) f(x) is probability density function of X

b. First moment is also known as the mean or expectation of X

i. A measure of central location of X

ii. Notation: μX

c. The lth central moment of a continuous random variable X is

i.

a) Second central moment, , measures variability of X, called variance

b) , the positive square root is standard deviation

d. Third central moment measures symmetry with respect to mean

i. Called skewness, which summarizes asymmetry

a)

e. Fourth central moment measures the tail behavior of X

i. Called kurtosis, measures tail thickness

a)

b) For normal distribution K(x) = 3

ii. Excess kurtosis = K(x) - 3

a) Positive excess kurtosis has "heavy tails"

i) More mass in tails than normal distribution

ii) Implies more extreme values

iii) Called Leptokurtic



b) Negative excess kurtosis has "short tails"

i) Less mass in tails than normal distribution

ii) Implies fewer extreme values

iii) Called Platykurtic

f. Random sample of X with T observations

i. Sample mean ∑ (1.10)

ii. Sample variance ∑ (1.11)

iii. Sample skewness ∑ (1.12)

iv. Sample kurtosis ∑ (1.13)

g. Tests for Normality of Returns

i. Under assumption of normality, Ŝ(x) and K(x) - 3 have asymptotic distributions with mean 0 and variance 6/T for skew and 24/T for excess kurtosis

ii. Test for Skew: Null Hypothesis H0: S(r) = 0 and Ha: S(r) ≠ 0

a) t-ratio statistic of sample skewness is /

b) Decision rule: reject null hypothesis at α significance if |t| > Zα/2, the upper 100(α/2)th quantile of the standard normal distribution

c) Alternative decision rule: calculate p value of the statistic and reject null hypothesis if and only if p < α

iii. Test for Excess Kurtosis: Null Hypothesis H0: K(r) - 3 = 0 and Ha: K(r) - 3 ≠ 0

a) t-ratio statistic, asymptotically a standard normal random variable, is

/

b) Decision rule: reject null hypothesis if and only if p < α

C. Distributions of Returns

1. General

a. General model for log returns Fr(r11, …,rN1; r12…rn2; r1T…rNT; Y; θ)

b. Joint distribution function of returns, Y, a state vector of economic environmental variables, and θ, a vector of parameters that determine the distribution function



c. Main issue is specification of conditional distribution, given the economic environment Y

d. Different distributional specifications lead to different theories

i. One version of conditional distribution of a random walk is equal to its marginal distribution. Returns are then independent and not predictable

e. While conditional distributions are important, it is easier to estimate marginal distributions using past returns

i. When there is weak serial correlation, then marginal and conditional distributions are close

f. Proposals in finance literature for marginal distributions of asset returns

i. Normal, lognormal, stable, and scale mixture of normal

2. Normal Distribution

a. Traditional assumption: simple returns, Ri, are independent and identically distributed, have a normal distribution with constant mean and variance

b. Problems with traditional assumption

i. Returns have a minimum value of -1

a) Normal distribution has range from -∞ to +∞

ii. If Rit is normally distributed, then multiperiod Rit[k] is not normal as the product of one-period returns

iii. Empirical studies show asset returns have positive excess kurtosis

3. Lognormal Distribution

a. Traditional assumption: log returns, ri, are independent and identically distributed, have a normal distribution with mean, μ and variance σ2

i. exp 1 (1.17)

ii. exp 2 exp 1 (1.17)

b. If m1, m2 are mean and variance of single return Rt, which is lognormally distributed, then

i. Mean rt = ln

ii. Variance rt = ln 1



c. Under the assumption that rt is normal, rt[k] is also normally distributed

d. Stock returns exhibiting positive excess kurtosis, implies a lognormal assumption is not consistent with historical returns

4. Stable Distribution

a. Distribution is stable under addition

i. Good with continuously compounded returns rt (as you sum them over time)

b. Distribution capable of capturing excess kurtosis

i. Good as historical stock returns show positive excess kurtosis

c. However, non-normal stable distributions do not have finite variance

i. Bad as result conflicts with finance theory

ii. Example: Cauchy distribution

a) Symmetric with respect to mean but variance is infinite

5. Scale Mixture of Normal Distributions

a. Studies sometimes use combination of distributions

b. Example rt assumed normally distributed, mean, μ and variance σ2, rt ~ N(μ, σ2)

i. Mixture could be ~ 1 , ,

a) X a Bernoulli random variable

ii. Advantage of mixtures of normal

a) Maintain tractability of normal

b) Have finite higher order moments

c) Can capture excess kurtosis

iii. Disadvantage of mixtures of normal

a) Can be hard to estimate mixture parameters

D. Multivariate Returns

1. Multivariate normal distribution is often used for log return rt = (r1t,….,rNt)'

2. Review of multivariate

a. Random vector X = (X1,…Xp)



b. Mean vector E(X) = μX = [E(X1),…,E(Xp)]'

c. Covariance Matrix Cov(X) = Σx = E[(X - μX)(X - μX)']

d. Sample mean vector ∑

e. Sample covariance matrix ∑ ′

E. Likelihood Function of Returns

1. "If conditional distribution f(rt| rt-1, …r1, θ) is normal with mean μt and variance , then θconsists of the parameters in μt and , and the likelihood function of the data is

f(r1,……rT;θ) = ; ∏√

exp (1.18)

a. where f(r1;Θ) is the marginal density function of the first observation r1"

2. Maximum-likelihood estimate (MLE) is the value of θthat maximizes the likelihood function

3. Log function is monotone and its MLE is f(r1,……rT;θ) = ln ; ∑ ln 2

ln

F. Empirical Properties of Returns

1. Time plot: graph of data on y-axis and time on x-axis

2. IBM stock return date 1926 to 2008

a. Basic patterns of simple returns and log returns are similar

b. Shows IBM returns are not normal, higher peak and fatter tails

3. Observations of selected US market indexes and individual stock

a. Daily returns tend to have high excess kurtosis

b. Monthly return for indexes have higher excess kurtosis than for individual stock

c. Mean of daily returns is close to zero

d. Mean of monthly returns is slightly higher

e. Monthly returns have higher standard deviations than daily returns

f. Among daily returns, indexes have lower standard deviation than individual stocks

g. Skewness not a serious problem

h. Descriptive statistics show difference simple and log returns not large



IV. Processes Considered

A. Volatility and Extreme Returns

1. Variability of returns varies over time and appears in clusters

2. Volatility used in derivative pricing

3. Extremes of return series: large positive or negative returns

a. Extreme negative returns important for risk management

b. Extreme positive returns important for the holding of short positions

B. Other financial time series

1. US Treasury 1-year and 10-year rates

a. Generally move together although 1-year are more volatile

2. Interest rate series

a. Sample means are proportional to time to maturity

b. Sample standard deviation inversely proportional to time to maturity

3. Bond return series

a. Sample standard deviations are positively related to time to maturity

b. Sample means are stable for all maturities

4. Most series considered exhibit positive excess kurtosis



Analysis of Financial Time Series Ruey Tsay

Chapter 2 Linear Time Series Analysis and Its Applications

Note: this chapter is included in the SOA syllabus for background only.

I. Introduction

A. A time series {rt} is created if rt, log of a stock's return, is treated as a set of random variable over time

B. Linear time series: a way to study financial time series

C. Theories discussed in this chapter include

1. Stationary, Dynamic dependence, Autocorrelation function, Modeling and Forecasting

2. Econometric models include

a. Simple autoregressive (AR), Simple moving average (MA), Mixed autoregressive moving average (ARMA), seasonal, unit-root nonstationarity, regression models with time series errors, and fractionally differenced models for long-range dependence

D. General model for log returns Fr(r11, …,rN1; r12…rn2; r1T…rNT; Y; Θ)

1. Random Y determines economic environment

2. rii historical returns

3. Correlations between historical data of the variable of interest,

a. the tools for studying time series include

i. Serial correlations

ii. Autocorrelations

II. Stationarity

A. Strictly stationary

1. If joint distribution of , … , and , … , are identical

a. Time invariant

b. k: an arbitrary positive number, (t1,…,tk) set of positive numbers

2. Strictly stationary is hard to verify empirically



B. Weakly stationary

1. Mean of rt and covariance of rt and rt-l are time invariant, independent of time period chosen

a. Arbitrary integer l

2. E(rt) = μ, a constant, and Cov(rt, rt-l) = γl, which depends only on l

3. Time plot of data would show values fluctuate around fixed level

4. Can make inferences about future values

C. Moments of rt

1. Weak stationary: Its first two moments are finite

2. If rt strong stationary and first two moments are finite, then have weak stationary

a. Converse is not true

3. If rt is normally distributed and is weakly stationary, then rt is strongly stationary

4. Covariance

a. γl = Cov(rt, rt-l) called "lag-l autocovariance of rt"

i. Properties of lag-l autocovariance

a) γ0 = Var(rt)

i) covariance of something with itself (0-lag) is its variance

b) γ-l = γl

i) Cov(rt, rt-(-l)) = Cov(rt-(-l), rt) = Cov(rt+l, rt) = Cov(rt, rt-l)

D. Finance Application

1. Common to assume asset returns weakly stationary

2. Can be empirically tested

III. Correlation and Autocorrelation Function

A. Correlation between random variables X, Y

1. Populations

a. ,,

i. Where μx and μy are mean of X and Y, variances assumed to exist

ii. ‐1 ρx,y 1, ρx,y ρy,x



iii. Uncorrelatedifρx,y 0

iv. X,Ynormalrandomvariable,thenρx,y 0ifandonlyifXandYareindependent

2. SamplesofX,Y,xt,yt

a. ,

b. Means ∑ and ∑

B. AutocorrelationFunction ACF

1. When linear dependence of rt and rt-l is of interest, for a weakly stationary series

a. Correlation coefficient called lag-l autocorrelation of rt, ρl

i. , ,

(2.1)

ii. By definition, ρo = 1, ρl = ρ-l

b. rt not serially correlated if and only if ρl = 0 for all l > 0

2. Sample returns rt

a. Lag-1 sample autocorrelation ∑

∑

i. Note the summations ranges are different

ii. Generalize to l other than 1, by changing the numerator summation to start at t=l+1

b. a consistent estimate of ρ1 under some general conditions

i. "If {rt} is independent and identically distributed (iid) sequence and ∞, then is asymptotically normal with mean zero and variance 1/T"

a) "An asymptotically normal estimator is a consistent estimator whose distribution around the true parameter θ approaches a normal distribution

with standard deviation shrinking in proportion to √

as the sample size n

grows" (http://en.wikipedia.org/wiki/Estimator)

b) Use to test null hypothesis H0: ρ1 = 0 versus Ha: ρ1 ≠ 0

c) Test statistic = √



3. More Generally

a. If rt weakly stationary, ∑

i. ψ0 = 1, {aj} is iid random variable with mean = 0

b. Then is asymptotically normal with mean zero and variance 1 2∑ /

i. Known as Bartlett's formula

C. Testing Individual ACF

1. General test statistic, t-ratio = ∑ /

2. If {rt} is stationary Gaussian (normally distributed) series with ρj = 0 for all j > l, the t-ratio is asymptotically distributed as a normal random variable

3. Decision rule: reject H0 if |t-ratio| > Zα/2 , Zα/2 = 100(1-α/2)th percentile of standard normal distribution

4. Bias in estimator

a. In finite samples a biased estimator, bias on order of 1/T

i. Not good for small samples

ii. Financial data typically has large T, so generally not an issue

D. Portmanteau Test

1. Testing jointly to see if several autocorrelations of rt are zero

a. Portmanteau statistic, a test statistic ∗ ∑

i. Null H0: ρ1 … ρm= 0 versus Ha: ρi ≠ 0 for some i ∈ {1, …, m}

ii. Assumption rt is independent and identically distributed with certain moment conditions, Q*(m) is asymptotically a chi-squared random variable with m degrees of freedom

b. Modified statistic to increase power of test, Q 2 ∑ (2.3)

i. Decision rule: reject H0 if Q(m) > , the 100(1 - α) percentile of chi-squared distribution with m degrees of freedom



E. Sample Autocorrelation Function

1. Statistics , , … called sample autocorrelation function (ACF) of rt

a. Is the return series correlated with a 1-period lag return series? With a 2-period lag?, etc

2. Examples

a. IBM, Figure 2.1, page 33, ACF function all within confidence level (parallel dotted lines) desired, so suggest likelihood of serial correlation is low

b. US Market index, Figure 2.2, page 34, values of function cross the dotted lines suggesting serial correlation

F. Capital Asset Pricing Model (CAPM)

1. Theory {rt} is not predictable with no autocorrelation

2. Efficient capital markets would have zero autocorrelation, thus a test for market efficiency is that the autocorrelations of time series of returns would be close to zero.

3. Method for calculating prices and index returns might introduce autocorrelations

IV. White Noise and Linear Time Series

A. White Noise

1. Definition: "a time series rt is called a white noise if {rt} is a sequence of independent and identically distributed random variables with finite mean and variance"

a. Gaussian white noise: {rt} ~ N(0, σ2)

2. Autocorrelation function for white noise series = 0

a. Sample ACF are close to 0

3. Some asset returns show series dependence that must be modeled

B. Linear Time Series

1. Definition: a time series rt is said to be linear if it can be written as

a. ∑ (2.4)

b. where μ is the mean of rt,

c. ψ0 = 1, ψi are the weights of rt

d. {at} is a white noise series



e. at is "new information" at time t, innovation or shock

f. In this text, at is typically continuous random variable

2. If rt is weakly stationary, and independence of at,

a. E(rt) = μ, ∑ (2.5)

b. As weakly stationary, variance of rt is finite, implying is convergent

3. Lag-l autocovariance γl = Cov(rt, rt-l) = ∑ ∑∑ (2.6)

a. Relation of weights to autocorrelation

i. ∑

∑, for l ≥ 0 (2.7)

b. For weakly stationary series

c. ψi → 0 as i → ∞, therefore ρl → 0 as l increases

d. "the linear dependence of current return rt on the remote past return rt-l diminishes for large l"

V. Simple AR Models

A. Introduction

1. If return series has significant lag-1 autocorrelation, then rt-1 might be helpful in forecasting rt

2. Simple model rt = ϕ0 ϕ1rt-1 + at (2.8)

a. {at} is white noise mean 0 variance

b. Called an autoregressive (AR) model of order 1 or AR(1)

c. Implications

i. E(rt |rt-1) = ϕ0 ϕ1rt-1

ii. Var(rt |rt-1) =Var(at) =

d. Not the same as linear regression

3. Generalization of AR(1) to AR(p)

a. rt = ϕ0 ϕ1rt-1 + …+ ϕprt-p + at (2.9)

b. Saying past data jointly determines conditional expectation of rt



B. Properties of AR Models

1. AR(1) Model

a. Assume weak stationarity

i. E(rt) = μ, a constant, Var(rt) = γ0 and Cov(rt, rt-j) = γj, E(at) = 0

b. E[rt] = E[ϕ0 E ϕ1rt-1] + E[at] = ϕ0 ϕ1E rt-1] + 0

i. Under stationarity,

a) E[rt] = E rt-1] = μ

b) Algebra, E[rt] = μ =

c) Expectation exists if ϕ1 ≠ 1

d) Mean = 0 if ϕ0 = 0

c. Rewrite AR(1), model rt - μ = ϕ1 rt-1 - μ)+ at (2.10)

i. Repeated applications, ∑ (2.11)

d. Other results

i. rt - μ is linear in at-i

ii. Using independence of at, E[(rt - μ)at+1] = 0

a) Careful to watch the time indexes, r and a have index off by 1 here

iii. By stationarity, Cov(rt-1, at) = 0

iv.

v. By stationarity,

a) But only if 1, and |ϕ1| < 1

vi. Mean, variance of rt are finite and time invariant, therefore AR(1) is weakly stationary

e. Another version of the formula, rt = (1- ϕ1 μ ϕ1rt-1 + at

i. A weighted average (ϕ1) of the mean and the last return, plus an error term



2. Autocorrelation Function of an AR(1) Model

a. E[at(rt-u)]

i. Note: time index are the same

b. Substituting, we can get E[at(rt - μ)] = ϕ1E at rt-1 - μ)]+ E[(at)2] = 0 +

c. Other relations

i. Correlation γl= ϕ1γ1 + if l = 0

ii. Correlation γl= ϕ1γ1-1 if l > 0

iii. Var(rt) = γ0 =

iv. With ρ0 = 1, ρl =

d. Some results

i. Series decays exponentially with rate ϕ1, starting value ρ 0 =1

ii. For ϕ1 > 0, shows exponential decay

iii. For ϕ1 < 0, consists of alternative exponential decays with rate

3. AR(2) Model

a. rt = ϕ0 ϕ1rt-1 + ϕ2rt-2 + at (2.12)

b. , ϕ1+ ϕ2 ≠ 1

c. Can rewrite model as rt - μ = ϕ1(rt-1 - μ) + ϕ2(rt-2 - μ) + at

d. E[(rt-l - μ)at] = 0, l > 0

e. Moment equation of stationary AR(2) model γl = ϕ1 γl-1 + ϕ2 γl-2

f. ACF of rt ρl = ϕ1 ρl-1 + ϕ2 ρl-2, l > 0 (2.13)

i. For lag-1, ρ1 = ϕ1 ρ0 + ϕ2 ρ-1 = ϕ1 + ϕ2 ρ1

g. ACF of stationary AR(2) satisfies (1 - ϕ1B - ϕ2B2)ρl = 0

i. B = Back-shift operator, Bρl = ρl-1

ii. Determines properties of the time series

iii. Determines behavior of the forecasts of rt

Modern Investment Management Chapter 10 IP-25


Modern Investment Management: An Equilibrium Approach

Bob Litterman

QFIC-106-13 Chapter 10: Strategic Asset Allocation in the Presence of Uncertain Liabilities

Ronald Howard and Yoel Lax

I. Introduction

A. Much research on proper investment strategies ignores the presence of liabilities

B. This chapter focuses on strategic asset allocation in the presence of liabilities

C. This means in general terms that the ideal allocation may no longer be one that maximizes expected asset returns, but rather, one whose goal incorporates the protection of surplus or other metrics that are liability-focused

D. The primary issues investigated are:

1. Equity/bond split

2. Level of diversification

3. Duration of the bond portfolio

E. General conclusions are:

1. Overfunded plans benefit far less from high equity allocations than underfunded plans

2. Overfunded plans benefit from global equity diversification, while underfunded plans do not

3. Benefits of duration matching the bond portfolio with the liabilities is much greater for underfunded plans

F. Analysis is done on both a single- and multi-period basis

II. Modeling Liabilities

A. Sources of uncertainty in projected future liability benefits:

1. Mortality tables can be inaccurate

2. Future salary growth must be estimated

3. Employee demographics

IP-26 Modern Investment Management Chapter 10


B. It makes sense to model the value of the liabilities as a bond with fixed cash flows (best guess) plus a noise term that reflects uncertainty:

, , , ,L t f t B t f t tR R R R (10.1)

where:

a. ,L tR is the total return on the liability index at t

b. ,f tR is the risk-free rate of return

c. ,B tR is the total return on a bond index, and

d. t is a noise term

C. The parameter is used to duration-match the liability and bond indices (it is just the ratio of

liability duration to asset duration), while the noise term has a volatility of and is assumed to be uncorrelated with the bond index

III. Evaluating Investment Decisions in the Presence of Liabilities

A. In the absence of liabilities, alternative investment structures are compared in terms of Sharpe ratios:

Re i ft

i

RExpected Excess turnSR

Risk

(10.2)

B. Shortcomings of the Sharpe ratio approach for liability-based investing:

1. Considers asset risk and return only, not surplus risk and return

2. Sharpe ratio maximizes utility only under the assumptions of quadratic utility and terminal utility optimization (i.e. if consumption matters during the period, as opposed to at the end of the period only, then the Shape ratio no longer presents an optimal solution)

a. Pension funds generally expected to remain in business indefinitely, so choosing a specific future date seems unreasonable

b. Pension funds must be able to fund intermediate cash flows

IV. Static Analysis

A. Surplus is defined as the difference between the value of assets and the value of liabilities:

t t tS A L (10.3)



B. The funding ratio is defined as tt

t

AF

L (10.4)

C. The primary focus will be on the absolute dollar return of the surplus portfolio (since a percentage return on surplus is difficult to conceptualize if surplus is zero (infinite returns) or if surplus is negative (positive gains lead to “negative” returns)

D. Several quantities of interest for the pension fund:

1. Expected change in surplus

2. Volatility of the change in surplus

3. Risk/reward tradeoff in the surplus return

E. Define the risk-adjusted change in surplus (RACS) to be

1

1

1t t t ft

t t

E S S RRACS

S

(10.5)

This is nothing more than a Sharpe ratio analogy for pension funds; i.e. the expected excess change in surplus divided by the volatility of the surplus

F. There is no completely risk-free strategy for pension funds since the liability cash flows are always uncertain

1. The least risky strategy for this fund is to purchase a portfolio of bonds that represents the best guess about future liabilities and to invest the remainder into the risk-free asset

2. The volatility of this strategy will just equal the volatility of the noise term

3. It makes sense to evaluate other strategies relative to this “lowest possible risk strategy”

G. Illustration of Static Model

1. Hypothetical example:

a. Bond index duration = 10.5

b. Pension fund liabilities duration = 12

c. 12 /10.5 1.14

d. .02

e. The complete data is shown in Table 10.1

2. Preliminary observations:

a. Since the duration of the pension fund liabilities is greater than the duration of the bond index, its excess return is also greater; this means that a 100% allocation to bonds is a losing strategy in the long run



b. However, the liabilities are much more highly correlated with the bond index than with the equity index, and therefore this would suggest that bonds are a better hedge

c. This creates a trade-off between allocating to equities

3. Surplus Risk, Expected Change, and the RACS

a. Figure 10.1 plots the volatility of surplus as a percentage of the value of assets for given equity allocations from 0% to 100%, assuming various levels of initial funding

b. In the appendix it is shown that, for a given funding ratio, the allocation to equities that minimizes surplus risk is equal to

2

2 2

1

2

tB B E

t

E B B E

LA

(10.9)

c. Notice in the graph that for most of the lines there is no inflection point (i.e. the allocation to equities that minimizes surplus risk is actually negative, and therefore not shown on the graph)

d. This expression is:

i. Independent of the noise volatility

ii. An increasing function of the initial funding ratio (overfunded plans can duration match liabilities and then allocate a portion of excess assets to equities to realize correlation benefits, while underfunded plans cannot afford that luxury)

e. In the appendix it is shown that the minimum equity allocation needed to prevent the surplus from shrinking (really should say, to prevent the expected future surplus from becoming negative) is

1 1t t

t tB f

E B

L LA A R

(10.10)

f. Other notes:

i. If the expected return on equities is greater than bonds, then the expected change in surplus is linearly increasing in the equity allocation

g. Figure 10.3 shows (similar to an efficient frontier analysis) the tradeoff of expected dollar change in surplus versus dollar volatility of surplus for three funding levels

i. Nothing too surprising here; underfunded plans face considerable surplus risk

h. Figure 10.4 graphs the RACS for different funding levels

i. Steeper slope for underfunded plan shows that underfunded plans get rewarded more on a risk-adjusted basis for taking equity risk

ii. For overfunded plans, RACS increases up to a point and then decreases afterward, indicating the presence of a ”sweet spot” for equities which is efficient (beyond that point, additional risk is being taken suboptimally)



i. When noise is introduced (Figure 10.5):

i. Optimal equity allocations increase as noise volatility increases (since the additional uncertainty of the liabilities voids some of the benefits of hedging with bonds)

j. Overarching conclusion of this section: (the more underfunded a plan is, and the more uncertain future liabilities are, the more attractive equity appears relative to fixed income)

4. Global Diversification

a. Conclusions from the analysis regarding the benefit of global diversification:

i. Overfunded plans benefit on a risk-adjusted basis but exactly funded and underfunded plans are harmed

ii. In the presence noise with large volatility, the ability of bonds to fully hedge liabilities is voided, and in those cases it will be suboptimal for an overfunded plan to allocate to global equities)

iii. Finally, it is demonstrated that allocations to global fixed income reduce RACS for all funding levels; but this was by design, since liabilities were modeled with respect to a global bond index

5. Choosing the Right Duration of the Bond Portfolio

a. Conclusions from the analysis regarding the duration of the bond portfolio:

i. All funds will lose as a result of investing in a bond index with a duration different from that of the liability index

ii. The one caveat is that differences in liquidity may drive the attractiveness of shorter duration bonds versus longer duration bonds, and the pros/cons of investing “shorter” will need to be considered on a plan-specific basis

V. Dynamic Analysis

A. Here we extend the analysis to multi-period and assume the fund pays out a fixed fraction p of the liabilities at the end of each period

B. Therefore, we can now write the assets and liabilities in terms of the recursion:

1 , 1 , 1

1 , 1

1 1

1 1

t t A t t L t

t t L t

A A R pL R

L L R p

C. Other assumptions are the same as in the earlier section

D. The metric focused on most here is the funding ratio



E. The analysis will attempt to answer two questions:

1. For an underfunded plan, what return on assets in excess of the return on liabilities is necessary to (a) retain the original funding ratio and (b) reach fully funded status over a given horizon?

2. For an initial funding ratio, payout policy, and bond/equity split, how does the probability of being underfunded vary with the horizon?

F. Required Returns

a. Define ,,

,

11

1A t

x tL t

RR

R

as the return on assets in excess of the return on liabilities

b. The expected excess return required to keep the funding ratio constant

is 1

, 1

(1 ) (1 )t t tt x t

t t

E F p p F p pE R

F F

c. For underfunded plans, the required average excess return is:

i. An increasing function of the payout

ii. A decreasing function of the funding ratio

d. For underfunded plans, the required average excess return is:

i. A decreasing function of the payout

ii. A decreasing function of the funding ratio

e. The expected funding ratio at any time is given by

0 0

1111

1

xt

tx

tx

E RpE R

E F F pp E R p

(10.14)

f. Fixing the left-hand side at a level of 1 and the time horizon at t we can solve (numerically) for the required average excess return

G. Funding Probabilities

1. Through carrying out Monte Carlo simulation, it is possible to estimate the probability of being underfunded at any given time

2. Unsurprisingly the conclusions are:

a. Plans with higher initial funding ratios increase the probability of being underfunded by allocating significantly to equities

b. Underfunded plans can greatly decrease the probability of being underfunded with modest equity allocations



VI. Conclusions

A. The main findings were:

1. Underfunded plans benefit more from higher equity allocations than do overfunded plans for which the RACS often decreases after a certain equity allocation is reached

2. Matching the duration of the bond portfolio to that of liabilities is important for all plans, with underfunded plans benefiting the most

3. Global equity diversification is an attractive opportunity for overfunded plans, which can benefit from the higher Sharpe ratio of global equity, but underfunded plans will typically find domestic equities more attractive due to higher correlation with the liabilities

4. Fixed income diversification is not attractive for any of the plans studied

5. Underfunded plans must take more equity risk to improve their funding status

Practice Problems PP-31


Practice Problem 10 Source: FETE 2010-11, Q6 (8 points) Investment assets U and V follow geometric Brownian motions with volatilities σU and σV respectively; they are correlated with correlation coefficient ρ. Consider a contract which pays the greater of U or V at time T. (a) (2 points) Derive the value of this contract at issue; define all symbols. You are given:

• The current price of gold is $1,250 per ounce with a volatility of 20%. • The current price of platinum is $1,600 per ounce with a volatility of 25%. • The correlation coefficient between the two asset prices is ρ = 0.7. • Storage costs are assumed to be zero.

(b)

(i) (3 points) Calculate the price of a 1-year European option which pays the price of 4 ounces of gold or 3 ounces of platinum, whichever is greater.

(ii) (1 point) Describe the effect on the option price if the assets require storage at non-zero

cost (without further calculation). Assume the option given in (b)(i) is available through a trader; however you believe that the correlation coefficient between these two assets is higher than that implied by the trader’s asking price. (c) (2 points) Describe how you could make money.

PP-32 Practice Problems


Solution to Practice Problem 10 Source: FETE 2010-11, Q6 Commentary on Question: This question tested the pricing an exotic option involving asset exchange. The candidates needed to demonstrate they knew how the price is affected by (i) the presence of storage cost, and (ii) the change in correlation coefficient between the two underlying assets. Most candidates did not get that the option value at time 0 is U0 e-quT + V0e-qvT N(d1) - U0 e-quT N(d2). Instead many candidates derived the option value at time 0 using U0 + V0 e-qvT N(d1) - U0e-quT N(d2), which is only right when assuming qu = 0. In general part (b)(i) was very straight forward and the candidates did well. Some common mistakes included: � Only calculating the value of the exchange option in part (b)(i), but did not give the final value of the contract � Using 1600 and 1250 as initial value, rather than 4800 and 5000 � Used 4Su and 3Sv instead of Su and Sv In part (b)(ii), most candidates knew that storage cost is similar to dividend, but only a few candidates knew that it’s actually negative (i.e.: qu < 0, qv < 0 ). Many candidates discuss the effects of storage cost in terms of e-quT, but very few candidates discussed how the relative value of qu, qv affects N(d1) and N(d2). In general, candidates did very well in part (c).



Practice Problem 11 Source: FETE 2010-11, Q7 (a) (1 point) Describe and critique the Comparative Advantage argument to explain the popularity of

interest rate swaps in capital financing. Malbec Corp. prefers to pay a floating rate of interest. Valpolicella Corp. prefers to pay a fixed rate of interest.

Current Terms for Borrowing Fixed (semiannual compounding) Floating

Malbec 8.0% 6-month LIBOR + 1.30% Valpolicella 9.7% 6-month LIBOR + 1.91%

(b) (3 points) Show that a swap rate of 7.36% between Malbec and Valpolicella would lower the

borrowing costs for both companies, and calculate the rate reduction for each company. Malbec and Valpolicella entered into a 3-year interest rate swap. After one year, the current bid and offer rates in the fixed swap market are (with semiannual compounding):

Maturity (years) Bid Offer 2 8.15% 8.19%

Assume:

• The swap has a $10 million notional, and 4 future swap payments will take place • Both sides of the swap pay semiannually • Valpolicella has invested in a $10 million project with the proceeds from their borrowing. The present value of the project’s cash flows has decreased in value by $192,000 due to interest rate moves • The 0.5, 1, and 1.5 year continuously compounded LIBOR zero rates are 7.9%, 8.0%, and 8.1% respectively.

(c) (3 points) Evaluate whether or not the swap has proved to be an effective asset liability

management tool for Valpolicella.



Solution to Practice Problem 11 Source: FETE 2010-11, Q7 Commentary on Question: The question tested the candidates’ knowledge of an interest rate swap transaction and also the correct theoretical valuation of the swap after issue. The question also demanded the candidates understand the interest rate swap transaction and whether or not it would be an effective asset liability management tool. For part (a), few candidates received full credit for this part. Frequently answers were vague and didn't provide sufficient explanation of a company borrowing where it has comparative advantage (e.g., floating) and then swapping to pay the kind of rate (e.g., fixed) that it really wants. Also, some candidates did not answer the question as asked but answered based on currency swaps (sometimes points could be awarded, though). For part (b), many candidates received full credit. A common error, however, was assuming that the gain from the swap had to be split evenly, and so an incorrect swap fixed rate or incorrect swap floating rate (or both) would be used to force an evenly split gain. A diagram of the transactions would have been helpful but was not necessary for full credit. Partial credit was awarded for correctly calculating total gain even if the answer was otherwise incorrect. It was not uncommon for candidates to get the net effect correct for each party but then not calculate rate reduction for each company (as asked in the question). For part (c), the reading provides two approaches to valuing the swap. One of them, using a forward rate agreement approach, is untenable because it requires knowing the actual 6-month LIBOR rate at the valuation date and that information is not provided. The appropriate valuation approach is “Valuation in Term of Bond Prices.” Nearly every candidate failed to recognize that the value of the floating piece of the swap was equal to the notional at the valuation date (and so required no calculations). Few understood how to use the swap rate to determine the 2-yr zero rate, which is needed for correct discounting. Also, a common error in valuing B(fix) was to not use the fixed rate of the swap entered into (i.e., 7.68%) as the fixed payment, but instead use the swap rate (or bid rate or offer rate) at the valuation date. Most candidates knew that the V(swap) = B(float) – B(fix) , and most candidates demonstrated correctly how to use the result of the V(swap) as calculated. (a) Describe and critique the Comparative Advantage argument to explain the popularity of interest

rate swaps in capital financing.

The argument is called the comparative-advantage argument. Some companies have a comparative advantage when borrowing in fixed-rate markets. Others have a comparative advantage in floating rate markets. To obtain a new loan, it makes sense for a company to go to the market where it has a comparative advantage. As a result, the company may borrow floating (fixed) when it wants fixed (floating). A swap is then used to transform a fixed (floating) rate loan into a floating (fixed) rate loan.

Criticism: The terms of floating rates bonds are likely to be renegotiated in the event that a credit rating downgrade occurs. In this way, the company paying the fixed rate in the swap doesn’t necessarily lock in their cost of funds.



(b) Show that a swap rate of 7.36% between Malbec and Valpolicella would lower the borrowing costs for both companies, and calculate the rate reduction for each company.

Malbec should borrow fixed at 8.00% because it has comparative advantage (CA) at fixed rate

Enter into a swap to receive fixed at 7.36% Pay floating rate of LIBOR (=L) Malbec’s net financing cost is to pay floating = 8.00% - 7.36% + L = L + 0.64%

Malbec’s rate reduction using swap = Usual floating borrowing cost – Floating cost w/ swap = (L + 1.30%) – (L + 0.64%) = 0.66%

Valpolicella should borrow floating at L + 1.91% because it has CA at floating rate

Enter into a swap to pay fixed 7.36%, and Receive floating of L Valpolicella’s net financing cost is to pay fixed = L + 1.91% +7.36% - L = 9.27%

Valpolicella’s rate reduction using swap = Usual fixed borrowing cost – Fixed cost w/ swap = 9.70% - 9.27% = 0.43% (Note, as a check: 0.66% + 0.43% = 1.09% = (9.70% - 8.00%) – [(L+1.91%) – (L+1.30%)].)

(c) Evaluate whether or not the swap has proved to be an effective asset liability management tool for Valpolicella. The swap is effective if the gain in the swap offsets the loss in the project. Gain in swap = (value of swap at t=1) – (value of swap at t=0) At time 0, the V(swap) = 0. At time 1, B(Floating) = Notional = 10,000,000 , because a payment was just made At time 1, B(Fix) = discounting of remaining 4 fixed payments Need 2 yr zero rate to value B(Fix): The two year swap rate is the average of the bid and offer rates, 8.17% = (8.15% + 8.19%)/2 The two year swap rate is equal to the two year par yield, thus, for 100 notional, 4.085*[e^(-.5*7.9%) + e^(-1*8%) + e^(-1.5*8.1%)] + 104.085*e^(-2*2 yr zero) = 100 So 2 yr zero = 8.01% At time 1, B(Fix) = 10,000,000*[3.68% (e^(-.5*7.9%)+e^(-1*8.0%)+e^(-1.5*8.1%)+e^(-2*8.01%))+e^(-2*8.01%)]

B(Fix) = 9,852,612.64 V(swap) = B(Floating) - B(Fixed) = 147,388.36 V(swap) offsets about 77% of loss in project value, therefore the swap was mostly effective

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