SOLUTIONS 913,

Size: px
Start display at page:

Download "SOLUTIONS 913,"

Transcription

1 Illinois State University, Mathematics 483, Fall 2014 Test No. 3, Tuesday, December 2, 2014 SOLUTIONS 1. Spring 2013 Casualty Actuarial Society Course 9 Examination, Problem No. 7 Given the following information on a currency swap: The swap has a remaining life of 15 months. The swap involves exchanging interest at 9% on 20 million British pounds for interest at 5% on $30 million at the end of each year. Principal amounts will be exchanged when the swap term expires. The term structure of interest rates in both the United Kingdom and the United States is currently flat and if the swap were negotiated today, the interest rates exchanged would be 6% in British pounds and 4% in dollars. The current exchange rate (dollars per British pound) is All interest rates are quoted with annual compounding. Viewing the swap as a portfolio of forward contracts, calculate the current value of the swap to the party paying British pounds, such value calculated in British pounds. There are two payment times remaining on the swap: 3 months from now and 15 months from now. In 3 months, the party paying British pounds will need to pay 9% on 20 million British pounds, or 1.8 million pounds, and will receive 5% on $30 million, or $1.5 million. This is equivalent to a forward contract for purchase of $1.5 million in three months by paying 1.8 million pounds for it. Remember that we are doing calculations in pounds. Current forward price (in pounds) for purchase of $1.5 million in 3 months is (in pounds) 12 $1,500, , Spot price in pounds Accumulated with interest in British pounds 3 Contract price without dollar interest income, i.e., "ex-dividend" Instead, 1.8 million pounds will be paid. This amounts to 913, ,800, = 886, pounds in three months, and in present value 886, , pounds In 15 months, the party paying pounds will pay 1.8 million pounds plus 20 million principal, and receive $1.5 million plus $30 million principal, effectively purchasing $31.5 million for 21.8 million pounds. Current forward price for $31.5 million is

2 12 $31,500, ,550, pounds Spot price in pounds Accumulated with interest in British pounds 15 Contract price without dollar interest income, i.e., "ex-dividend" Instead, 21.8 million pounds will be paid. The value of such transaction is 19,550, ,800, = 2,249, pounds in 15 months, and in today s money, this is 2,249, ,091, pounds Total value, in pounds, to the party paying British pounds, is ( 873,749.38) + ( 2,091,086.40) = 2,964, The problem specifically asks for the value in British pounds, but if you are interested in the value in U.S. dollars, that value is calculated by multiplying the above by 1.65, and it equals approximately 4,891, You are given that the duration of a standard five-year discrete decreasing annuity immediate (paying 5 at time 1, 4 at time 2, 3 at time 3, 2 at time 4, and 1 at time 6), ( ) 5, is 35 whose present value is Da, while the duration of a standard five-year discrete 15 increasing annuity immediate (paying 1 at time 1, 2 at time 2, 3 at time 3, 4 at time 4 and ( ) 5, is 55 5 at time 5), whose present value is Ia. The interest rate is i. Given that 15 information, find the duration of a standard level five-year annuity immediate (paying 1 a times 1, 2, 3, 4 and 5), whose present value is a 5. With zero interest rate Macaulay duration (and duration, as well) of a standard decreasing annuity is = and Macaulay duration (as well as duration) of increasing annuity is = Therefore, the interest rate is 0, and the Macaulay duration (as well as duration) of a unit five year annuity immediate is = ( Ia) 5 i=0% = = 3. 5 a 5 i=0% 5

3 3. You are the investment actuary of an insurance firm, which is subject to new regulatory requirements in Freedonia. Your company has 10 million freebies (currency of Freedonia) worth of assets, half of which is invested in stocks and half in bonds. You are given that the one year effective rate of return on stocks follows the normal distribution with mean 0.10 and standard deviation 0.20, while the one year effective rate of return on bonds follows the normal distribution with mean 0.05 and standard deviation The new regulation requires you to find the 1-st percentile of the joint distribution of the value of your asset portfolio of stocks and bonds assuming that the joint distribution is formed by applying the Gaussian copula to the distributions of stocks and bonds value at the end of the period. That 1-st percentile is the maximum amount of premium that the company is allowed to collect from business sold next year, if it ever exceeds it, it must stop sales for the year. The correlation of stocks and bonds returns is Find that crucial first percentile. You should know immediately that applying the Gaussian copula to two normal distributions will result in a joint bivariate normal distribution, with the prescribed correlation parameter. The value of stocks in the portfolio at year end will have normal distribution with mean 5 million times 1.10, i.e., 5.5 million, and standard deviation of 5 million times 0.20, i.e., 1 million. The value of bonds in the portfolio at year end will have normal distribution with mean 5 million times 1.05, i.e., 5.25 million, and standard deviation of 5 million times 0.02, i.e., 0.1 million. Since the joint distribution is bivariate normal, their sum (i.e., the whole portfolio) will have a normal distribution with mean = million and standard deviation = Since the 99-th percentile of the standard normal distribution is approximately 2.33 (from the table), the 1-st percentile sought is = You are given the following with respect to a one-period securities market model: (i) The time 0 prices of the three securities in the market are S( 0) = [ ]. (ii) The time 1 payoffs of the three securities are described in the matrix: S( 1) = Find the risk neutral probabilities of the three states of the future, or show that the riskneutral probabilities do not exist. Let us write [ ψ 1 ψ 2 ψ 3 ]

4 for the state price vector, if one exists. We see that = so that 0.8 = 1 ψ ψ ψ 3. Similarly, 6 = 14 ψ ψ ψ 3, and 1 = 0 ψ ψ ψ 3. Thus we must have ψ 1 + ψ 2 + ψ 3 = 0.8, 14ψ 1 + 6ψ 2 + 7ψ 3 = 6, 5ψ 3 = 1. Therefore ψ 3 = 0.2, and ψ 1 + ψ = 0.8, 14ψ 1 + 6ψ = 6. Consequently, ψ 2 = 0.6 ψ 1, so that 14ψ ψ 1 ψ 1 = 0.125, ψ 2 = We get [ ψ 1 ψ 2 ψ 3 ] = [ ] = Note that the risk-free interest rate is determined from the relationship 0.8 ( 1 + i) = 1, so that 1+ i = = 1.25, and i = 25%. The risk-neutral probabilities are q 1 = ψ 1 ( 1 + i) = = 5 32, q 2 = ψ 2 ( 1 + i) = = 19 32, q 3 = ψ 3 ( 1 + i) = = 1 4 = ( ) = 6, and we get 5. Consider a position consisting of $1,000,000 investment in asset X and $1,000,000 investment in asset Y. Assume that the daily volatilities of both assets are 0.1% and that the correlation coefficient between their returns is What is the 5-day 99% Value at Risk for this portfolio, assuming a parametric model with zero expected return? The 99-th percentile of the standard normal distribution is The standard deviation of the daily dollar change in the value of each asset is $1,000. The variance of the portfolio s daily change, using the formula:

5 Var( U +V ) = Var( U ) + Var( V ) + 2ρ U,V σ U σ V is: = 2,600,000. The standard deviation of the portfolio s daily change in value is the square root of 2,600,000, i.e., $1, The standard deviation of the five-day change in the portfolio value is: $1, = $3, The 99-th percentile of the standard normal distribution is Therefore (assuming zero mean), the five-day 99% Value at Risk is: $3, $ Regulated Insurance is a company domiciled in Freedonia, a country that implemented a risk-based capital requirement system for regulation of its insurance companies. Riskbased capital requirement is the amount equal to the 95-th percentile of random amount of the loss that the company experiences in one year. The company has liabilities of 1,000,000 freebies (the currency of Freedonia) in life annuities reserves, and assets of $1,200,000 freebies, invested 40% in stocks and 60% in bonds. The liabilities of the company will increase by 2% over the next year. The effective annual rate of return on bonds follows a normal distribution with mean 4% and standard deviation of 2%, while the effective annual rate of return on stocks follows a normal distribution with mean 9% and standard deviation of 12%. The correlation of returns of stocks and bonds is 0.6, and the joint distribution of those returns is bivariate normal. Find the ratio of the capital held by Regulated Insurance to the capital required by regulation in Freedonia. The 95-th percentile of the standard normal distribution is The capital held by Regulated Insurance is equal to the excess of its assets over its liabilities, i.e., 1,200,000 1,000,000 = 200,000 freebies. Let X be the random rate of return of stocks, and Y be the random rate of return on bonds. The loss of Regulated Insurance in one year is X Y = = X Y. This is a normal random variable with mean E( X Y ) = E( X) E( Y ) = = = = 52000, and variance Var X Y ( ) = = Var( X Y ) = = Var( X) + Var( Y ) + 2ρ X,Y Var( X) Var( Y ) = = = = = = The 95-th percentile of the distribution of the loss is

6 The ratio sought is You are in investment manager and you purchase a catastrophe bond, which pays oneyear LIBOR plus 2.5% every year in which Truerisk Catastrophe Index of insured losses is below $3 billion. However, if Truerisk Catastrophe Index of insured losses exceeds $3 billion, the payment is reduced by ten basis point for each billion dollars of losses in excess of $3 billion, and if that reduction results in a negative value, bondholders s principal is appropriately reduced (e.g., payment of negative 1% means that principal is reduced by that 1%). If the current one-year LIBOR is 0.25%, what level of Truerisk Catastrophe Index of insured losses would result in payment to bondholder dropping to zero? With LIBOR at 0.25%, non-reduced payment is 2.5% % = 2.75%. The difference between 0% and 2.75% is 275 basis points, so the Truerisk Catastrophe Index would have to exceed $3 billion by $27.5 billion, and reach the level of $30.5 billion. 8. In the country of Softwareland, there are now only two securities traded: risk-free government bond earning 5% a year, and the stock of the only corporation in that nation: Megasoft, currently trading at 100, and in a year possibly trading at one of these three prices: 90, 105 and 120. You are the Minister of Finance for that nation, and propose to complete the market in Softwareland by adding a third security, paying 1 when Megasoft trades below 100, and 0 otherwise. Will adding that security make the market complete? If the security is issued, it will only be issue if it completes the market, and it will be issued by the government through an initial public offering of 1 billion units of that security with 5% of proceeds paid to brokers marketing the security, and the rest collected by the Treasury of the government of Softwareland. Estimate the proceeds to the Treasury if the security is issued. The current market is S( 0) = [ 1 100], S( 1) = It is, clearly incomplete. Let us see how it looks with respect to being arbitrage-free. If the state price vector ψ 1 ψ 2 ψ 3 exists then it must satisfy the equations: 1.05ψ ψ ψ 3 = 1, 90ψ ψ ψ 3 = 100.

7 Hence ψ 3 = ψ 1 ψ 2 = ψ 1 ψ 2 and or 0.9ψ ψ ψ 1 ψ 2 = 1, = 1.2ψ +1.2ψ 0.9ψ 1.05ψ, ψ 2 = ψ. 1 If the asset proposed is added to the market, its price must be ψ 1, and we get the market as follows: ( ) = ψ 1 S 0, S 1 ( ) = The matrix S(1) is of rank 3 because its determinant is ( ) = The conditions that must be satisfied by the parameters ψ 1, ψ 2, and ψ 3 are ψ 1 > 0, ψ 2 = ψ 1 > 0, ψ 3 = ψ 1 ψ 2 = ψ ψ 1 =ψ 1 > 0. This means that 0 <ψ 1 < We can choose any number between 0 and for the price of the new security, with being the unobtainable maximum. With 1 billion units issues, the total proceeds will be 10 1 billion 476,190, But the Treasury will receive only 95% of these proceeds, or billion 452,380, You are in charge of hedging the property-casualty loss exposure at your company using the Property Claims Service spreads, which consist of a long call of lower exercise price, and short call of a higher exercise price. The premium for the spreads is quoted in points, and each point is worth $200. Suppose that you are given that the 200/250 spread provides protection for the underlying index values at the rate of $ per point, and the spread acts like a long 200 points call combined with short 250 points call, with

8 the actual payment equal to number of points times $200. You have bought 500 contracts, and the index settled at $25 billion. What is the payoff to your company? $25 billion corresponds to $ = 250 points. $ You have a 200 call, which is now worth 50 points, or 50 times $200 = $10000, and you are short a 250 call, which expires worthless. Thus your payoff is $10000 per contract, or $ for 500 contracts. 10. The market price of a security is $50. Assume that Capital Asset Pricing Model holds. The expected rate of return of this security is 14%. The risk-free rate is 6%, and the market risk premium (over the risk-free rate) is 8.50%. What will be the market price of the security if the covariance of its returns with the market doubles, while no other parameters are changed? Assume that the security is a perpetuity of a constant dividend. Basic CAPM formula is E r where ( ) ( ) r f = β E( r M ) r f ( ) β = Cov r,r M Var( r M ). When the covariance doubles, beta doubles. We need to find the original beta first. It must satisfy 14% 6% = β ( 8.50% ) so that β = = After beta doubles, β = Therefore, new rate of return is r new = 6% + 32 ( % ) = 22%. What is the dividend D of this security? At 14% expected rate, perpetuity of D is worth D $50, and this must equal so that D = $7. At 22% expected return, perpetuity of $7 0.14, is worth $31.82.

9 11. You are the investment actuary for a life insurance company. Your company has of a bond portfolio with the value of $3 billion dollars. The portfolio is invested only in two securities: a 3-year bond with annual coupons of 5% and a 10-year zero coupon bond, with % invested in the first security, and the rest in the second security. The continuously compounded risk-free interest rate is 4%. Calculate the 99% 10 day Value at Risk (VaR) of this portfolio, assuming a standard parametric model, and using only duration of the portfolio, and given that the annual yield volatility is 0.5% (yield volatility defined here as the standard deviation of Δy, where y is the yield). Assume 252 days in a year for the calculation, i.e., only use trading days, and you can also assume that the 99-th percentile of the standard normal distribution is The Macaulay duration of the zero coupon bond is its maturity, i.e. 10 years. The coupon paying bond has a Macaulay duration of 0.05e e e e e e Using the allocations given, this produces the portfolio Macaulay duration of ( ) , and duration of approximately , e 0.04 The daily volatility of yield is and the daily volatility of the portfolio is therefore 252 estimated as σ P = D P σ y ,000,000, Since the 99-th percentile of the standard normal distribution is 2.33, the 10-day 99% Value at Risk is estimated as σ P , or approximately million (that s about 1.78% of the portfolio). 12. You are given the following with respect to an option-free bond portfolio worth ten million dollars and held by the insurance company in which you work as an investment actuary: The value of the bond portfolio using the current interest rate of 2% is 800,000, The value of the bond portfolio using the current interest rate plus 20 basis points is 788,000, The value of the bond portfolio using the current interest rate minus 20 basis points is 813,000. You are also given that your company s only liability is a single payment of 800,000 three years from now, with the present value of 753, Calculate the convexity of your company s surplus.

10 The convexity estimate for the asset portfolio is C P( i Δi) 2P( i) + P( i + Δi) = P( i) ( Δi) 2 813, , ,000 1 = = 800, = On the other hand, the liability has convexity of t( t +1) ( 1+ i) = Therefore, the convexity of your company s portfolio is estimated as 800, , , , , ,857.87

SOLUTIONS. Solution. The liabilities are deterministic and their value in one year will be $ = $3.542 billion dollars.

SOLUTIONS. Solution. The liabilities are deterministic and their value in one year will be $ = $3.542 billion dollars. Illinois State University, Mathematics 483, Fall 2014 Test No. 1, Tuesday, September 23, 2014 SOLUTIONS 1. You are the investment actuary for a life insurance company. Your company s assets are invested

More information

Mathematics of Finance Final Preparation December 19. To be thoroughly prepared for the final exam, you should

Mathematics of Finance Final Preparation December 19. To be thoroughly prepared for the final exam, you should Mathematics of Finance Final Preparation December 19 To be thoroughly prepared for the final exam, you should 1. know how to do the homework problems. 2. be able to provide (correct and complete!) definitions

More information

B6302 Sample Placement Exam Academic Year

B6302 Sample Placement Exam Academic Year Revised June 011 B630 Sample Placement Exam Academic Year 011-01 Part 1: Multiple Choice Question 1 Consider the following information on three mutual funds (all information is in annualized units). Fund

More information

Mathematics of Time Value

Mathematics of Time Value CHAPTER 8A Mathematics of Time Value The general expression for computing the present value of future cash flows is as follows: PV t C t (1 rt ) t (8.1A) This expression allows for variations in cash flows

More information

B6302 B7302 Sample Placement Exam Answer Sheet (answers are indicated in bold)

B6302 B7302 Sample Placement Exam Answer Sheet (answers are indicated in bold) B6302 B7302 Sample Placement Exam Answer Sheet (answers are indicated in bold) Part 1: Multiple Choice Question 1 Consider the following information on three mutual funds (all information is in annualized

More information

Portfolio Management

Portfolio Management Portfolio Management 010-011 1. Consider the following prices (calculated under the assumption of absence of arbitrage) corresponding to three sets of options on the Dow Jones index. Each point of the

More information

The exam will be closed book and notes; only the following calculators will be permitted: TI-30X IIS, TI-30X IIB, TI-30Xa.

The exam will be closed book and notes; only the following calculators will be permitted: TI-30X IIS, TI-30X IIB, TI-30Xa. 21-270 Introduction to Mathematical Finance D. Handron Exam #1 Review The exam will be closed book and notes; only the following calculators will be permitted: TI-30X IIS, TI-30X IIB, TI-30Xa. 1. (25 points)

More information

Solutions to Further Problems. Risk Management and Financial Institutions

Solutions to Further Problems. Risk Management and Financial Institutions Solutions to Further Problems Risk Management and Financial Institutions Third Edition John C. Hull 1 Preface This manual contains answers to all the Further Questions at the ends of the chapters. A separate

More information

Derivatives Questions Question 1 Explain carefully the difference between hedging, speculation, and arbitrage.

Derivatives Questions Question 1 Explain carefully the difference between hedging, speculation, and arbitrage. Derivatives Questions Question 1 Explain carefully the difference between hedging, speculation, and arbitrage. Question 2 What is the difference between entering into a long forward contract when the forward

More information

EXAMINATION II: Fixed Income Valuation and Analysis. Derivatives Valuation and Analysis. Portfolio Management

EXAMINATION II: Fixed Income Valuation and Analysis. Derivatives Valuation and Analysis. Portfolio Management EXAMINATION II: Fixed Income Valuation and Analysis Derivatives Valuation and Analysis Portfolio Management Questions Final Examination March 2016 Question 1: Fixed Income Valuation and Analysis / Fixed

More information

John Hull, Risk Management and Financial Institutions, 4th Edition

John Hull, Risk Management and Financial Institutions, 4th Edition P1.T2. Quantitative Analysis John Hull, Risk Management and Financial Institutions, 4th Edition Bionic Turtle FRM Video Tutorials By David Harper, CFA FRM 1 Chapter 10: Volatility (Learning objectives)

More information

Financial Markets & Risk

Financial Markets & Risk Financial Markets & Risk Dr Cesario MATEUS Senior Lecturer in Finance and Banking Room QA259 Department of Accounting and Finance c.mateus@greenwich.ac.uk www.cesariomateus.com Session 3 Derivatives Binomial

More information

Interest Rate Markets

Interest Rate Markets Interest Rate Markets 5. Chapter 5 5. Types of Rates Treasury rates LIBOR rates Repo rates 5.3 Zero Rates A zero rate (or spot rate) for maturity T is the rate of interest earned on an investment with

More information

The University of Nottingham

The University of Nottingham The University of Nottingham BUSINESS SCHOOL A LEVEL 2 MODULE, SPRING SEMESTER 2010 2011 COMPUTATIONAL FINANCE Time allowed TWO hours Candidates may complete the front cover of their answer book and sign

More information

2. Futures and Forward Markets 2.1. Institutions

2. Futures and Forward Markets 2.1. Institutions 2. Futures and Forward Markets 2.1. Institutions 1. (Hull 2.3) Suppose that you enter into a short futures contract to sell July silver for $5.20 per ounce on the New York Commodity Exchange. The size

More information

FIN FINANCIAL INSTRUMENTS SPRING 2008

FIN FINANCIAL INSTRUMENTS SPRING 2008 FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 OPTION RISK Introduction In these notes we consider the risk of an option and relate it to the standard capital asset pricing model. If we are simply interested

More information

Final Exam. 5. (21 points) Short Questions. Parts (i)-(v) are multiple choice: in each case, only one answer is correct.

Final Exam. 5. (21 points) Short Questions. Parts (i)-(v) are multiple choice: in each case, only one answer is correct. Final Exam Spring 016 Econ 180-367 Closed Book. Formula Sheet Provided. Calculators OK. Time Allowed: 3 hours Please write your answers on the page below each question 1. (10 points) What is the duration

More information

SAMPLE FINAL QUESTIONS. William L. Silber

SAMPLE FINAL QUESTIONS. William L. Silber SAMPLE FINAL QUESTIONS William L. Silber HOW TO PREPARE FOR THE FINAL: 1. Study in a group 2. Review the concept questions in the Before and After book 3. When you review the questions listed below, make

More information

Final Exam. 5. (24 points) Multiple choice questions: in each case, only one answer is correct.

Final Exam. 5. (24 points) Multiple choice questions: in each case, only one answer is correct. Final Exam Fall 06 Econ 80-367 Closed Book. Formula Sheet Provided. Calculators OK. Time Allowed: 3 hours Please write your answers on the page below each question. (0 points) A stock trades for $50. After

More information

Market risk measurement in practice

Market risk measurement in practice Lecture notes on risk management, public policy, and the financial system Allan M. Malz Columbia University 2018 Allan M. Malz Last updated: October 23, 2018 2/32 Outline Nonlinearity in market risk Market

More information

Homework. Due Monday 11/2/2009 at beginning of class Chapter 6: 2 Additional Homework: Download data for

Homework. Due Monday 11/2/2009 at beginning of class Chapter 6: 2 Additional Homework: Download data for Data Code Go to http://stonybrook.datacodeinc.com User: SUNYSB Password: STONYBROOK11794 Download software for WorldwatchInsight and Marketlink and corresponding manuals Login using your personal login

More information

NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MATHEMATICS SEMESTER 2 EXAMINATION Investment Instruments: Theory and Computation

NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MATHEMATICS SEMESTER 2 EXAMINATION Investment Instruments: Theory and Computation NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MATHEMATICS SEMESTER 2 EXAMINATION 2012-2013 Investment Instruments: Theory and Computation April/May 2013 Time allowed : 2 hours INSTRUCTIONS TO CANDIDATES

More information

Futures and Forward Markets

Futures and Forward Markets Futures and Forward Markets (Text reference: Chapters 19, 21.4) background hedging and speculation optimal hedge ratio forward and futures prices futures prices and expected spot prices stock index futures

More information

Financial Risk Measurement/Management

Financial Risk Measurement/Management 550.446 Financial Risk Measurement/Management Week of September 23, 2013 Interest Rate Risk & Value at Risk (VaR) 3.1 Where we are Last week: Introduction continued; Insurance company and Investment company

More information

Financial Derivatives Section 1

Financial Derivatives Section 1 Financial Derivatives Section 1 Forwards & Futures Michail Anthropelos anthropel@unipi.gr http://web.xrh.unipi.gr/faculty/anthropelos/ University of Piraeus Spring 2018 M. Anthropelos (Un. of Piraeus)

More information

Random Variables and Applications OPRE 6301

Random Variables and Applications OPRE 6301 Random Variables and Applications OPRE 6301 Random Variables... As noted earlier, variability is omnipresent in the business world. To model variability probabilistically, we need the concept of a random

More information

Forward and Futures Contracts

Forward and Futures Contracts FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 Forward and Futures Contracts These notes explore forward and futures contracts, what they are and how they are used. We will learn how to price forward contracts

More information

Final Exam. Indications

Final Exam. Indications 2012 RISK MANAGEMENT & GOVERNANCE LASTNAME : STUDENT ID : FIRSTNAME : Final Exam Problems Please follow these indications: Indications 1. The exam lasts 2.5 hours in total but was designed to be answered

More information

FINALTERM EXAMINATION Spring 2009 MGT201- Financial Management (Session - 2) Question No: 1 ( Marks: 1 ) - Please choose one What is the long-run objective of financial management? Maximize earnings per

More information

Financial Risk Measurement/Management

Financial Risk Measurement/Management 550.446 Financial Risk Measurement/Management Week of September 23, 2013 Interest Rate Risk & Value at Risk (VaR) 3.1 Where we are Last week: Introduction continued; Insurance company and Investment company

More information

INTEREST RATE FORWARDS AND FUTURES

INTEREST RATE FORWARDS AND FUTURES INTEREST RATE FORWARDS AND FUTURES FORWARD RATES The forward rate is the future zero rate implied by today s term structure of interest rates BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 1 0 /4/2009 2 IMPLIED FORWARD

More information

Jaime Frade Dr. Niu Interest rate modeling

Jaime Frade Dr. Niu Interest rate modeling Interest rate modeling Abstract In this paper, three models were used to forecast short term interest rates for the 3 month LIBOR. Each of the models, regression time series, GARCH, and Cox, Ingersoll,

More information

Consumption- Savings, Portfolio Choice, and Asset Pricing

Consumption- Savings, Portfolio Choice, and Asset Pricing Finance 400 A. Penati - G. Pennacchi Consumption- Savings, Portfolio Choice, and Asset Pricing I. The Consumption - Portfolio Choice Problem We have studied the portfolio choice problem of an individual

More information

UNIVERSITY OF TORONTO Joseph L. Rotman School of Management. RSM332 FINAL EXAMINATION Geoffrey/Wang SOLUTIONS. (1 + r m ) r m

UNIVERSITY OF TORONTO Joseph L. Rotman School of Management. RSM332 FINAL EXAMINATION Geoffrey/Wang SOLUTIONS. (1 + r m ) r m UNIVERSITY OF TORONTO Joseph L. Rotman School of Management Dec. 9, 206 Burke/Corhay/Kan RSM332 FINAL EXAMINATION Geoffrey/Wang SOLUTIONS. (a) We first figure out the effective monthly interest rate, r

More information

Valuing Stock Options: The Black-Scholes-Merton Model. Chapter 13

Valuing Stock Options: The Black-Scholes-Merton Model. Chapter 13 Valuing Stock Options: The Black-Scholes-Merton Model Chapter 13 1 The Black-Scholes-Merton Random Walk Assumption l Consider a stock whose price is S l In a short period of time of length t the return

More information

Answer choice A in Problem 26, Practice Examination 8, should be 2.

Answer choice A in Problem 26, Practice Examination 8, should be 2. Course FM Manual by Dr. Krzysztof Ostaszewski, FSA, CFA, MAAA December 008 Edition Errata Posted May 8, 009 The beginning of Problem 3 in Practice Examination 3 should read: You are given the following

More information

Math 441 Mathematics of Finance Fall Midterm October 24, 2006

Math 441 Mathematics of Finance Fall Midterm October 24, 2006 Math 441 Mathematics of Finance Fall 2006 Name: Midterm October 24, 2006 Instructions: Show all your work for full credit, and box your answers when appropriate. There are 5 questions: the first 4 are

More information

Port(A,B) is a combination of two stocks, A and B, with standard deviations A and B. A,B = correlation (A,B) = 0.

Port(A,B) is a combination of two stocks, A and B, with standard deviations A and B. A,B = correlation (A,B) = 0. Corporate Finance, Module 6: Risk, Return, and Cost of Capital Practice Problems (The attached PDF file has better formatting.) Updated: July 19, 2007 Exercise 6.1: Minimum Variance Portfolio Port(A,B)

More information

Sample Final Exam Fall Some Useful Formulas

Sample Final Exam Fall Some Useful Formulas 15.401 Sample Final Exam Fall 2008 Please make sure that your copy of the examination contains 25 pages (including this one). Write your name and MIT ID number on every page. You are allowed two 8 1 11

More information

LECTURE NOTES 3 ARIEL M. VIALE

LECTURE NOTES 3 ARIEL M. VIALE LECTURE NOTES 3 ARIEL M VIALE I Markowitz-Tobin Mean-Variance Portfolio Analysis Assumption Mean-Variance preferences Markowitz 95 Quadratic utility function E [ w b w ] { = E [ w] b V ar w + E [ w] }

More information

, U.S.A. URL:

, U.S.A. URL: Dr. Krzysztof Ostaszewski, FSA, CFA. MAAA Professor of Mathematics and Actuarial Program Director Illinois State University, Normal, IL 61790-4520, U.S.A. URL: http://www.math.ilstu.edu/krzysio/ E-mail:

More information

Midterm Review. P resent value = P V =

Midterm Review. P resent value = P V = JEM034 Corporate Finance Winter Semester 2017/2018 Instructor: Olga Bychkova Midterm Review F uture value of $100 = $100 (1 + r) t Suppose that you will receive a cash flow of C t dollars at the end of

More information

November Course 8V

November Course 8V November 2000 Course 8V Society of Actuaries COURSE 8: Investment - 1 - GO ON TO NEXT PAGE November 2000 Morning Session ** BEGINNING OF EXAMINATION ** MORNING SESSION Questions 1 3 pertain to the Case

More information

M339W/M389W Financial Mathematics for Actuarial Applications University of Texas at Austin In-Term Exam I Instructor: Milica Čudina

M339W/M389W Financial Mathematics for Actuarial Applications University of Texas at Austin In-Term Exam I Instructor: Milica Čudina M339W/M389W Financial Mathematics for Actuarial Applications University of Texas at Austin In-Term Exam I Instructor: Milica Čudina Notes: This is a closed book and closed notes exam. Time: 50 minutes

More information

When we model expected returns, we implicitly model expected prices

When we model expected returns, we implicitly model expected prices Week 1: Risk and Return Securities: why do we buy them? To take advantage of future cash flows (in the form of dividends or selling a security for a higher price). How much should we pay for this, considering

More information

Mean-Variance Portfolio Theory

Mean-Variance Portfolio Theory Mean-Variance Portfolio Theory Lakehead University Winter 2005 Outline Measures of Location Risk of a Single Asset Risk and Return of Financial Securities Risk of a Portfolio The Capital Asset Pricing

More information

Paper 2.6 Fixed Income Dealing

Paper 2.6 Fixed Income Dealing CHARTERED INSTITUTE OF STOCKBROKERS September 2018 Specialised Certification Examination Paper 2.6 Fixed Income Dealing 2 Question 2 - Fixed Income Valuation and Analysis 2a) i) Why are many bonds callable?

More information

Appendix A Financial Calculations

Appendix A Financial Calculations Derivatives Demystified: A Step-by-Step Guide to Forwards, Futures, Swaps and Options, Second Edition By Andrew M. Chisholm 010 John Wiley & Sons, Ltd. Appendix A Financial Calculations TIME VALUE OF MONEY

More information

1 Asset Pricing: Replicating portfolios

1 Asset Pricing: Replicating portfolios Alberto Bisin Corporate Finance: Lecture Notes Class 1: Valuation updated November 17th, 2002 1 Asset Pricing: Replicating portfolios Consider an economy with two states of nature {s 1, s 2 } and with

More information

SOCIETY OF ACTUARIES FINANCIAL MATHEMATICS. EXAM FM SAMPLE SOLUTIONS Financial Economics

SOCIETY OF ACTUARIES FINANCIAL MATHEMATICS. EXAM FM SAMPLE SOLUTIONS Financial Economics SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE SOLUTIONS Financial Economics June 2014 changes Questions 1-30 are from the prior version of this document. They have been edited to conform

More information

Solution to Problem Set 2

Solution to Problem Set 2 M.I.T. Spring 1999 Sloan School of Management 15.15 Solution to Problem Set 1. The correct statements are (c) and (d). We have seen in class how to obtain bond prices and forward rates given the current

More information

Show that the column rank and the row rank of A are both equal to 3.

Show that the column rank and the row rank of A are both equal to 3. hapter Vectors and matrices.. Exercises. Let A 2 5 4 3 2 4 2 2 3 5 4 2 4 3 Show that the column rank and the row rank of A are both equal to 3. 2. Let x and y be column vectors of size n, andleti be the

More information

Overview of Concepts and Notation

Overview of Concepts and Notation Overview of Concepts and Notation (BUSFIN 4221: Investments) - Fall 2016 1 Main Concepts This section provides a list of questions you should be able to answer. The main concepts you need to know are embedded

More information

Actuarial Society of India

Actuarial Society of India Actuarial Society of India EXAMINATIONS June 005 CT1 Financial Mathematics Indicative Solution Question 1 a. Rate of interest over and above the rate of inflation is called real rate of interest. b. Real

More information

Notes on: J. David Cummins, Allocation of Capital in the Insurance Industry Risk Management and Insurance Review, 3, 2000, pp

Notes on: J. David Cummins, Allocation of Capital in the Insurance Industry Risk Management and Insurance Review, 3, 2000, pp Notes on: J. David Cummins Allocation of Capital in the Insurance Industry Risk Management and Insurance Review 3 2000 pp. 7-27. This reading addresses the standard management problem of allocating capital

More information

CHAPTER 10 SOME LESSONS FROM CAPITAL MARKET HISTORY

CHAPTER 10 SOME LESSONS FROM CAPITAL MARKET HISTORY CHAPTER 10 SOME LESSONS FROM CAPITAL MARKET HISTORY Answers to Concepts Review and Critical Thinking Questions 3. No, stocks are riskier. Some investors are highly risk averse, and the extra possible return

More information

RISKMETRICS. Dr Philip Symes

RISKMETRICS. Dr Philip Symes 1 RISKMETRICS Dr Philip Symes 1. Introduction 2 RiskMetrics is JP Morgan's risk management methodology. It was released in 1994 This was to standardise risk analysis in the industry. Scenarios are generated

More information

MFE8825 Quantitative Management of Bond Portfolios

MFE8825 Quantitative Management of Bond Portfolios MFE8825 Quantitative Management of Bond Portfolios William C. H. Leon Nanyang Business School March 18, 2018 1 / 150 William C. H. Leon MFE8825 Quantitative Management of Bond Portfolios 1 Overview 2 /

More information

Derivatives Options on Bonds and Interest Rates. Professor André Farber Solvay Business School Université Libre de Bruxelles

Derivatives Options on Bonds and Interest Rates. Professor André Farber Solvay Business School Université Libre de Bruxelles Derivatives Options on Bonds and Interest Rates Professor André Farber Solvay Business School Université Libre de Bruxelles Caps Floors Swaption Options on IR futures Options on Government bond futures

More information

FINA 695 Assignment 1 Simon Foucher

FINA 695 Assignment 1 Simon Foucher Answer the following questions. Show your work. Due in the class on March 29. (postponed 1 week) You are expected to do the assignment on your own. Please do not take help from others. 1. (a) The current

More information

Financial Markets and Products

Financial Markets and Products Financial Markets and Products 1. Eric sold a call option on a stock trading at $40 and having a strike of $35 for $7. What is the profit of the Eric from the transaction if at expiry the stock is trading

More information

Final Examination. ACTU 363- Actuarial Mathematics Lab (1) (10/ H, Time 3H) (5 pages)

Final Examination. ACTU 363- Actuarial Mathematics Lab (1) (10/ H, Time 3H) (5 pages) King Saud University Department of Mathematics Exercise 1. [4] Final Examination ACTU 363- Actuarial Mathematics Lab (1) (10/411 438 H, Time 3H) (5 pages) A 30 year annuity is arranged to pay off a loan

More information

Derivatives Revisions 3 Questions. Hedging Strategies Using Futures

Derivatives Revisions 3 Questions. Hedging Strategies Using Futures Derivatives Revisions 3 Questions Hedging Strategies Using Futures 1. Under what circumstances are a. a short hedge and b. a long hedge appropriate? A short hedge is appropriate when a company owns an

More information

CHAPTER 14 BOND PORTFOLIOS

CHAPTER 14 BOND PORTFOLIOS CHAPTER 14 BOND PORTFOLIOS Chapter Overview This chapter describes the international bond market and examines the return and risk properties of international bond portfolios from an investor s perspective.

More information

Optimizing Portfolios

Optimizing Portfolios Optimizing Portfolios An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Introduction Investors may wish to adjust the allocation of financial resources including a mixture

More information

More Actuarial tutorial at 1. An insurance company earned a simple rate of interest of 8% over the last calendar year

More Actuarial tutorial at   1. An insurance company earned a simple rate of interest of 8% over the last calendar year Exam FM November 2005 1. An insurance company earned a simple rate of interest of 8% over the last calendar year based on the following information: Assets, beginning of year 25,000,000 Sales revenue X

More information

MORNING SESSION. Date: Wednesday, April 30, 2014 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES

MORNING SESSION. Date: Wednesday, April 30, 2014 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES SOCIETY OF ACTUARIES Quantitative Finance and Investment Core Exam QFICORE MORNING SESSION Date: Wednesday, April 30, 2014 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES General Instructions 1.

More information

Random Variables and Probability Distributions

Random Variables and Probability Distributions Chapter 3 Random Variables and Probability Distributions Chapter Three Random Variables and Probability Distributions 3. Introduction An event is defined as the possible outcome of an experiment. In engineering

More information

PAPER 2 : STRATEGIC FINANCIAL MANAGEMENT

PAPER 2 : STRATEGIC FINANCIAL MANAGEMENT Question 1 PAPER 2 : STRATEGIC FINANCIAL MANAGEMENT Question No.1 is compulsory. Attempt any five questions from the remaining six questions Working notes should form par t of the answer (a) Amal Ltd.

More information

SOCIETY OF ACTUARIES Advanced Topics in General Insurance. Exam GIADV. Date: Thursday, May 1, 2014 Time: 2:00 p.m. 4:15 p.m.

SOCIETY OF ACTUARIES Advanced Topics in General Insurance. Exam GIADV. Date: Thursday, May 1, 2014 Time: 2:00 p.m. 4:15 p.m. SOCIETY OF ACTUARIES Exam GIADV Date: Thursday, May 1, 014 Time: :00 p.m. 4:15 p.m. INSTRUCTIONS TO CANDIDATES General Instructions 1. This examination has a total of 40 points. This exam consists of 8

More information

Lecture 2. Agenda: Basic descriptions for derivatives. 1. Standard derivatives Forward Futures Options

Lecture 2. Agenda: Basic descriptions for derivatives. 1. Standard derivatives Forward Futures Options Lecture 2 Basic descriptions for derivatives Agenda: 1. Standard derivatives Forward Futures Options 2. Nonstandard derivatives ICON Range forward contract 1. Standard derivatives ~ Forward contracts:

More information

On the Use of Stock Index Returns from Economic Scenario Generators in ERM Modeling

On the Use of Stock Index Returns from Economic Scenario Generators in ERM Modeling On the Use of Stock Index Returns from Economic Scenario Generators in ERM Modeling Michael G. Wacek, FCAS, CERA, MAAA Abstract The modeling of insurance company enterprise risks requires correlated forecasts

More information

For each of the questions 1-6, check one of the response alternatives A, B, C, D, E with a cross in the table below:

For each of the questions 1-6, check one of the response alternatives A, B, C, D, E with a cross in the table below: November 2016 Page 1 of (6) Multiple Choice Questions (3 points per question) For each of the questions 1-6, check one of the response alternatives A, B, C, D, E with a cross in the table below: Question

More information

MFE8812 Bond Portfolio Management

MFE8812 Bond Portfolio Management MFE8812 Bond Portfolio Management William C. H. Leon Nanyang Business School January 16, 2018 1 / 63 William C. H. Leon MFE8812 Bond Portfolio Management 1 Overview Value of Cash Flows Value of a Bond

More information

ECON FINANCIAL ECONOMICS

ECON FINANCIAL ECONOMICS ECON 337901 FINANCIAL ECONOMICS Peter Ireland Boston College Spring 2018 These lecture notes by Peter Ireland are licensed under a Creative Commons Attribution-NonCommerical-ShareAlike 4.0 International

More information

Corporate Finance, Module 21: Option Valuation. Practice Problems. (The attached PDF file has better formatting.) Updated: July 7, 2005

Corporate Finance, Module 21: Option Valuation. Practice Problems. (The attached PDF file has better formatting.) Updated: July 7, 2005 Corporate Finance, Module 21: Option Valuation Practice Problems (The attached PDF file has better formatting.) Updated: July 7, 2005 {This posting has more information than is needed for the corporate

More information

FINAL EXAMINATION GROUP - III (SYLLABUS 2016)

FINAL EXAMINATION GROUP - III (SYLLABUS 2016) FINAL EXAMINATION GROUP - III (SYLLABUS 016) SUGGESTED ANSWERS TO QUESTIONS DECEMBER - 017 Paper-14 : STRATEGIC FINANCIAL MANAGEMENT Time Allowed : 3 Hours Full Marks : 100 The figures in the margin on

More information

INSTITUTE OF ACTUARIES OF INDIA

INSTITUTE OF ACTUARIES OF INDIA INSTITUTE OF ACTUARIES OF INDIA EXAMINATIONS 06 th November 2015 Subject ST6 Finance and Investment B Time allowed: Three Hours (10.15* 13.30 Hrs) Total Marks: 100 INSTRUCTIONS TO THE CANDIDATES 1. Please

More information

The Multistep Binomial Model

The Multistep Binomial Model Lecture 10 The Multistep Binomial Model Reminder: Mid Term Test Friday 9th March - 12pm Examples Sheet 1 4 (not qu 3 or qu 5 on sheet 4) Lectures 1-9 10.1 A Discrete Model for Stock Price Reminder: The

More information

Super-replicating portfolios

Super-replicating portfolios Super-replicating portfolios 1. Introduction Assume that in one year from now the price for a stock X may take values in the set. Consider four derivative instruments and their payoffs which depends on

More information

INSTITUTE OF ACTUARIES OF INDIA

INSTITUTE OF ACTUARIES OF INDIA INSTITUTE OF ACTUARIES OF INDIA EXAMINATIONS 24 th March 2017 Subject ST6 Finance and Investment B Time allowed: Three Hours (10.15* 13.30 Hours) Total Marks: 100 INSTRUCTIONS TO THE CANDIDATES 1. Please

More information

University of North Carolina at Charlotte Mathematical Finance Program Comprehensive Exam. Spring, 2015

University of North Carolina at Charlotte Mathematical Finance Program Comprehensive Exam. Spring, 2015 University of North Carolina at Charlotte Mathematical Finance Program Comprehensive Exam Spring, 2015 Directions: This exam consists of 6 questions. In order to pass the exam, you must answer each question.

More information

CREDIT RATINGS. Rating Agencies: Moody s and S&P Creditworthiness of corporate bonds

CREDIT RATINGS. Rating Agencies: Moody s and S&P Creditworthiness of corporate bonds CREDIT RISK CREDIT RATINGS Rating Agencies: Moody s and S&P Creditworthiness of corporate bonds In the S&P rating system, AAA is the best rating. After that comes AA, A, BBB, BB, B, and CCC The corresponding

More information

Lecture 8 Foundations of Finance

Lecture 8 Foundations of Finance Lecture 8: Bond Portfolio Management. I. Reading. II. Risks associated with Fixed Income Investments. A. Reinvestment Risk. B. Liquidation Risk. III. Duration. A. Definition. B. Duration can be interpreted

More information

Exercise Session #1 Suggested Solutions

Exercise Session #1 Suggested Solutions JEM034 Corporate Finance Winter Semester 2017/2018 Instructor: Olga Bychkova Date: 3/10/2017 Exercise Session #1 Suggested Solutions Problem 1. 2.10 The continuously compounded interest rate is 12%. a

More information

ERM (Part 1) Measurement and Modeling of Depedencies in Economic Capital. PAK Study Manual

ERM (Part 1) Measurement and Modeling of Depedencies in Economic Capital. PAK Study Manual ERM-101-12 (Part 1) Measurement and Modeling of Depedencies in Economic Capital Related Learning Objectives 2b) Evaluate how risks are correlated, and give examples of risks that are positively correlated

More information

Market Risk VaR: Model- Building Approach. Chapter 15

Market Risk VaR: Model- Building Approach. Chapter 15 Market Risk VaR: Model- Building Approach Chapter 15 Risk Management and Financial Institutions 3e, Chapter 15, Copyright John C. Hull 01 1 The Model-Building Approach The main alternative to historical

More information

Analytical Problem Set

Analytical Problem Set Analytical Problem Set Unless otherwise stated, any coupon payments, cash dividends, or other cash payouts delivered by a security in the following problems should be assume to be distributed at the end

More information

EXAMINATION II: Fixed Income Valuation and Analysis. Derivatives Valuation and Analysis. Portfolio Management

EXAMINATION II: Fixed Income Valuation and Analysis. Derivatives Valuation and Analysis. Portfolio Management EXAMINATION II: Fixed Income Valuation and Analysis Derivatives Valuation and Analysis Portfolio Management Questions Final Examination March 2011 Question 1: Fixed Income Valuation and Analysis (43 points)

More information

Credit Risk in Banking

Credit Risk in Banking Credit Risk in Banking CREDIT RISK MODELS Sebastiano Vitali, 2017/2018 Merton model It consider the financial structure of a company, therefore it belongs to the structural approach models Notation: E

More information

Business Statistics 41000: Probability 3

Business Statistics 41000: Probability 3 Business Statistics 41000: Probability 3 Drew D. Creal University of Chicago, Booth School of Business February 7 and 8, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office: 404

More information

SOLVENCY, CAPITAL ALLOCATION, AND FAIR RATE OF RETURN IN INSURANCE

SOLVENCY, CAPITAL ALLOCATION, AND FAIR RATE OF RETURN IN INSURANCE C The Journal of Risk and Insurance, 2006, Vol. 73, No. 1, 71-96 SOLVENCY, CAPITAL ALLOCATION, AND FAIR RATE OF RETURN IN INSURANCE Michael Sherris INTRODUCTION ABSTRACT In this article, we consider the

More information

Econ 424/CFRM 462 Portfolio Risk Budgeting

Econ 424/CFRM 462 Portfolio Risk Budgeting Econ 424/CFRM 462 Portfolio Risk Budgeting Eric Zivot August 14, 2014 Portfolio Risk Budgeting Idea: Additively decompose a measure of portfolio risk into contributions from the individual assets in the

More information

ELEMENTS OF MATRIX MATHEMATICS

ELEMENTS OF MATRIX MATHEMATICS QRMC07 9/7/0 4:45 PM Page 5 CHAPTER SEVEN ELEMENTS OF MATRIX MATHEMATICS 7. AN INTRODUCTION TO MATRICES Investors frequently encounter situations involving numerous potential outcomes, many discrete periods

More information

DISCLAIMER. The Institute of Chartered Accountants of India

DISCLAIMER. The Institute of Chartered Accountants of India DISCLAIMER The Suggested Answers hosted on the website do not constitute the basis for evaluation of the students answers in the examination. The answers are prepared by the Faculty of the Board of Studies

More information

( 0) ,...,S N ,S 2 ( 0)... S N S 2. N and a portfolio is created that way, the value of the portfolio at time 0 is: (0) N S N ( 1, ) +...

( 0) ,...,S N ,S 2 ( 0)... S N S 2. N and a portfolio is created that way, the value of the portfolio at time 0 is: (0) N S N ( 1, ) +... No-Arbitrage Pricing Theory Single-Period odel There are N securities denoted ( S,S,...,S N ), they can be stocks, bonds, or any securities, we assume they are all traded, and have prices available. Ω

More information

May 2012 Course MLC Examination, Problem No. 1 For a 2-year select and ultimate mortality model, you are given:

May 2012 Course MLC Examination, Problem No. 1 For a 2-year select and ultimate mortality model, you are given: Solutions to the May 2012 Course MLC Examination by Krzysztof Ostaszewski, http://www.krzysio.net, krzysio@krzysio.net Copyright 2012 by Krzysztof Ostaszewski All rights reserved. No reproduction in any

More information

Hedging and Regression. Hedging and Regression

Hedging and Regression. Hedging and Regression Returns The discrete return on a stock is the percentage change: S i S i 1 S i 1. The index i can represent days, weeks, hours etc. What happens if we compute returns at infinitesimally short intervals

More information

Calculating EAR and continuous compounding: Find the EAR in each of the cases below.

Calculating EAR and continuous compounding: Find the EAR in each of the cases below. Problem Set 1: Time Value of Money and Equity Markets. I-III can be started after Lecture 1. IV-VI can be started after Lecture 2. VII can be started after Lecture 3. VIII and IX can be started after Lecture

More information

The Term Structure and Interest Rate Dynamics Cross-Reference to CFA Institute Assigned Topic Review #35

The Term Structure and Interest Rate Dynamics Cross-Reference to CFA Institute Assigned Topic Review #35 Study Sessions 12 & 13 Topic Weight on Exam 10 20% SchweserNotes TM Reference Book 4, Pages 1 105 The Term Structure and Interest Rate Dynamics Cross-Reference to CFA Institute Assigned Topic Review #35

More information