Interest Rate Forwards and Swaps

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Interest Rate Forwards and Swaps"

Transcription

1 Interest Rate Forwards and Swaps Forward rate agreement (FRA) mxn FRA = agreement that fxes desgnated nterest rate coverng a perod of (n-m) months, startng n m months: Example: Depostor wants to fx rate on 3-month depost startng n 3 months enters 3x FRA wth bank. FRA rate s rate that causes nvestor to acheve same return through ether: -month depost, or 3-month depost, followed by renvestment of prncpal and nterest for 3 addtonal months at FRA rate Calculatng FRA rate F = { [+ R L Days L ] [+ R S Days S ] } ( Days F ) R L =3.5% 0 3 MO MO R S =3.0% F=FRA rate Where: R L R S F Days L Days S Days F = spot Lbor for long perod = spot Lbor for short perod = forward Lbor = no. of days n long perod = no. of days n short perod = no. of days n forward perod FRA Mechancs Customer does not need to place funds wth bank quotng the FRA. Customer places funds anywhere she lkes whle FRA s net settled wth bank. Example: FRA rate= 3.948%, 3m-Lbor reset=3.75% and Notonal= $00. Interest earned on depost = $00 x (3.75% x 9/) = $ Under FRA, bank pays $00 x (3.948%-3.75%) x 9/ = $ Sum of 2 amounts gves customer $.0022 annualzed 3.948% for a 9-day perod

2 Calculaton above assumed FRA settles at end of forward perod. In realty, FRA conventon s to settle net payment as soon as Lbor fxng takes place.e. at begnnng of forward perod. Amount of net settlement reflects tme value and thus equals Notonal (Contract rate Settlement rate) (Days (+ Settlement Rate Days ) ) FRA Arbtrage Assume on Jan, 2009 dealer quotes 3x FRA at 4.5% whle m Lbor = 3.5% and 3m Lbor = 3%. To arbtrage these rates: You borrow at 3.5% for months. Invest at 3% for 3 months. Lock n today 4.5% re-nvestment rate under 3 x FRA. Proft per $ at maturty = $0.003 = ( + 3% x 90/) x ( + 4.5% x 9/) ( + 3.5% x 8/) Conversely, f FRA = 3.%: nvest for months at 3.5%, borrow for 3 months at 3%, and lock n 3.% borrowng rate for 3 months n 3 months under the FRA Proft per $ at maturty = $ = ( + 3.5% x 8/) ( + 3% x 90/) x ( + 3.% x 9/) When bd-offer spreads are taken nto consderaton, arbtrage opportunty exsts f ether: ( + R 3m (bd) x 90/) x ( + FRA 3m (bd) x 9/) > ( + R m (offer) x 8/); or ( + R m (bd) x 8/) > ( + R 3m (offer) x 90/) x ( + FRA 3m (offer) x 9/) Man resdual rsks of arbtrage are: Rsk of default by enttes wth whom you nvest the money. Rsk of default by FRA counterparty when FRA s n your favor at settlement. Rsk that settlement payment under FRA cannot be renvested at rate rate used for dscountng under FRA net settlement formula. Interest Rate Exposure Assume on Jan, 2009 a borrower has borrowed $00MM for 3 years at 3m Lbor flat. Lbor spot = 3% frst nterest payment = $0.75MM. Exposure to Lbor for all subsequent perods: Lbor nterest expense and reported ncome. 2

3 Quoted FRA rates: Perod Number Contract Desgnaton Perod Start Date Number of Days n perod Rate for clent Spot Jan % 2 3x Aprl % 3 x9 July % 4 9x Oct % 5 x5 Jan % 5x8 Apr % 7 8x2 July % 8 2x24 Oct % 9 24x27 Jan 90.00% 0 27x30 Apr 9.25% 30x33 July 92.50% 33x3 Oct 92.75% Borrower could hedge each reset usng seres of FRAs Unequal and nterest expenses n each quarter. Clent needs smoother pattern of nterest outflows: nterest rate swap (IRS). Prcng: aggregate PV of CFs under IRS fxed leg (usng sngle rate) = aggregate PV of CFs pad under seres of FRAs. All settlements under IRS occur at end of nterest perods n contrast to FRA conventon Borrower 5.22% Lbor Bank Smple IRS formula F = = Days (Notonal Lbor DF Days (Notonal DF = Equaton explans why swap rate F s often descrbed as tme-weghted average of relevant Lbors. 3

4 Perod Lbor/ FRA Net payment DFs PVNP Adjusted net payment PVANP 3.00% 750, ,47,30,53,29, %,0, ,538,32,08,298, %,50, ,7,,335,598,297, %,23, ,5,088,335,598,28, %,250, ,84,93,30,53,238, %,327, ,24,53,32,08,235, %,405, ,29,78,335,598,232, %,49, ,33,027,335,598,24, %,500, ,343,54,30,53,70, %,579, ,393,8,32,08,4,978.50%,, 0.874,440,895,335,598,58,53.75%,725, ,470,940,335,598,38,890 Totals 4,728,09 4,728,09 PV of Floatng leg PV of Fxed Leg IRS formula ncludng dealer proft margn If dealer proft margn n PV terms = Proft, IRS rate F may be derved from followng equaton: = Days Days (Notonal Lbor DF) + Proft = (Notonal F = DF) f dealer s party recevng fxed; and = (Notonal Lbor Days DF) = (Notonal F Days DF ) + Proft f dealer s party recevng Lbor = 4

5 IRS formula when notonal amount changes over tme When IRS notonal decreases over tme Amortzng swap When IRS notonal ncreases over tme Accretng swap IRS rate F may be derved from followng equaton: = Days Days (Notonal Lbor DF) = (Notonal F = DF) Notonal = swap notonal at begnnng of perod Swap rate for amortzng/accretng notonal, as compared to swap rate for bullet notonal s summarzed n ths table: Lbor spot/forward curve Upward Inverted slopng Amortzng notonal Lower Hgher Accretng notonal Hgher Lower Forward-startng swap Prced usng precedng equaton by settng Notonal = 0 untl start date. If curve s upward slopng: forward-start IRS rate > spot-start IRS rate; and vce-versa f curve s nverted. Frequences and day-counts Each leg of IRS may have dfferent frequency and/or day-count. Example: Swap 3m Lbor-based 3-year bullet loan nto sem-annual fxed wth 30/ day-count. In ths case, IRS rate F s the soluton to followng equaton: Notonal Lbor Days DF = Notonal F 0.5 DF = 2 = 5

6 Lbor-n-arrears (LIA) swap Works exactly lke a regular swap n all respects, except that fxng of Lbor for a perod takes place 2 busness days before the end of that perod, nstead of 2 busness days before the begnnng. Regular Swap: LIA swap: Fxng Begnnng Settlement End Begnnng Fxng Settlement End 2 bus. days Interest Perod Interest Perod 2 bus. days Regular IRS formula apples for LIA swap where Lbor s Lbor rate that fxes 2 busness days before end of nterest perod (as opposed to begnnng n regular IRS). If curve s upward slopng rate on LIA swap fxed leg. Accurate prcng needs a small convexty adjustment, especally for long-dated swaps and n volatle rate envronments. IRS Replcaton Any IRS can be vewed as 2 bond postons: one long and one short. To receve fxed / pay floatng under IRS s _ to: Issung Lbor-based lablty and Investng proceeds n fxed-rate bond; and vce-versa -month Lbor Coupons on fxed rate bond Dealer -month Lbor Fxed Bass swap Involves exchange of floatng leg aganst another floatng leg but n another currency, and ncludes prncpal exchange at maturty. Example: suppose /$ spot =.50 US bank wants to borrow 00MM for 5 yrs but has no access to fundng market. UK bank wants to borrow $50MM for 5 yrs but has no access to $ fundng market. US bank ssues $50MM 5-yr FRN whch pays m $ Lbor, and converts proceeds nto 00MM n spot FX market UK bank ssues 00MM 5-yr FRN whch pays m Lbor and converts proceeds nto $50MM n spot FX market

7 In fact 2 banks can execute spot FX transactons wth each other 2 banks now enter nto swap descrbed n ths dagram: 00 MM at maturty $50 MM bond at $ Lbor US Bank Lbor on 00 MM sem-annually $ Lbor on $50 MM sem-annually UK Bank 00 MM bond at Lbor $50 MM at maturty Ths s a Lbor bass swap that, unlke IRS, ncludes an exchange of prncpal at maturty A postve or negatve bass pont adjustment to ether leg may be necessary, dependng on demand/supply, relatve credtworthness of US Bank v. UK Bank, and other factors. Cross-currency Swap Company ssues $50MM 3-yr 5.5% bond (S.A.) but needs to fund a UK expanson. Wth /$ spot at.50, Company converts $ proceeds nto, and seeks to hedge future $ coupon and prncpal. Company would lke: Bank to make all payments requred (n $) under the bond; n return Company makes to the bank, usually on same dates, payments n whose profle s dentcal to that of a fxed-rate sem-annual bond: 00 MM at maturty $50 MM bond at $ 5.5% Your Company Fxed on 00 MM sem-annually $ 5.5% on $50 MM sem-annually Bank $50 MM at maturty The GBP fxed rate under the CCS, F, s derved from the followng equaton: Notonal $ F $ DCF $ $ ( DF ) + Notonal $ $ DF = = FXSpot Notonal F DCF j DF j + Notonal DF j= 7

8 Where: Notonal $ and Notonal = notonals n $ and respectvely = $50MM and 00MM. DF $ and DF = dscount factors for $ and curves respectvely. DCF $ and DCF = Day-count factor, accordng to approprate conventon, for $ and respectvely. F $ = US $ fxed rate = 5.5%. FX Spot = /$ FX spot rate =.50. Bass Pont Converson Cannot smply add or subtract equal numbers of bps to each leg ($ and ) after prcng CCS. For example, addng 25bps to $ fxed rate does not necessarly result n dentcal ncrease of 25 bps to leg. PVs obtaned by dscountng off 2 dfferent curves must be equal. Equvalent number of bass ponts n can be derved from ths equaton: Notonal $ 25bps DCF $ $ ( DF ) + Notonal $ $ DF = j= = FXSpot Notonal Equv DCF DF + Notonal DF Constructng/hedgng CCS Typcally ths s done usng 3 swaps: 2 IRS and Lbor bass swap: $50 MM at maturty $Fxed on $50 MM $L on $50 MM IRS $ Bond at 5.50% Company $Fxed on $50 MM Fxed on 00 MM Bank 00 MM at maturty L+ x bps on 00 MM $L on $50 MM $50 MM at maturty Lbor Bass Swap 00 MM at maturty L on 00 MM Fxed on 00 MM IRS2 If we are constructng a fxed/floatng CCS, one of the IRSs would become unnecessary. Frequences and day-count for the 2 legs do not have to match: could have $ leg based on quarterly, 30/, and leg on m Lbor, act/. 8

9 $ Lbor Curve $ DFs $ Pmts PV of $ Pmts Days Lbor Curve PV of DFs Pmts Pmts 3.94% % % % % % % % % % % % Totals n $ Fxed 5.50% PV Dff 0.00 Fxed 4.5% 50.2 n $ $ Notonal 50 FX Spot.5 Notonal 00 IRS Swap Revaluaton Fxed rate recever has MTM gan when rates : fxed payments are dscounted at dscount rates recevng fxed under a swap s one way of takng a vew on nterest rates. Fxed rate recever poston = ownng a fxed rate bond funded at Lbor gan on asset when rates whle lablty s value remans almost the same. DV0 of IRS DV0 of fxed rate bond whose prncpal, coupon, maturty and daycount/frequency match those of IRS fxed leg. IRS also has convexty changes n ts value are not lnear relatve to changes n rates 9

Multiple discount and forward curves

Multiple discount and forward curves Multple dscount and forward curves TopQuants presentaton 21 ovember 2012 Ton Broekhuzen, Head Market Rsk and Basel coordnator, IBC Ths presentaton reflects personal vews and not necessarly the vews of

More information

Time Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6 - The Time Value of Money. The Time Value of Money

Time Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6 - The Time Value of Money. The Time Value of Money Ch. 6 - The Tme Value of Money Tme Value of Money The Interest Rate Smple Interest Compound Interest Amortzng a Loan FIN21- Ahmed Y, Dasht TIME VALUE OF MONEY OR DISCOUNTED CASH FLOW ANALYSIS Very Important

More information

An Alternative Way to Measure Private Equity Performance

An Alternative Way to Measure Private Equity Performance An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate

More information

Interest Rate Futures

Interest Rate Futures Interest Rate Futures Chapter 6 6.1 Day Count Conventons n the U.S. (Page 129) Treasury Bonds: Corporate Bonds: Money Market Instruments: Actual/Actual (n perod) 30/360 Actual/360 The day count conventon

More information

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt. Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces

More information

Simple Interest Loans (Section 5.1) :

Simple Interest Loans (Section 5.1) : Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part

More information

Section 5.4 Annuities, Present Value, and Amortization

Section 5.4 Annuities, Present Value, and Amortization Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today

More information

10.2 Future Value and Present Value of an Ordinary Simple Annuity

10.2 Future Value and Present Value of an Ordinary Simple Annuity 348 Chapter 10 Annutes 10.2 Future Value and Present Value of an Ordnary Smple Annuty In compound nterest, 'n' s the number of compoundng perods durng the term. In an ordnary smple annuty, payments are

More information

Bond futures. Bond futures contracts are futures contracts that allow investor to buy in the

Bond futures. Bond futures contracts are futures contracts that allow investor to buy in the Bond futures INRODUCION Bond futures contracts are futures contracts that allow nvestor to buy n the future a theoretcal government notonal bond at a gven prce at a specfc date n a gven quantty. Compared

More information

Chapter 4 Financial Markets

Chapter 4 Financial Markets Chapter 4 Fnancal Markets ECON2123 (Sprng 2012) 14 & 15.3.2012 (Tutoral 5) The demand for money Assumptons: There are only two assets n the fnancal market: money and bonds Prce s fxed and s gven, that

More information

Compound Interest: Further Topics and Applications. Chapter 9

Compound Interest: Further Topics and Applications. Chapter 9 9-2 Compound Interest: Further Topcs and Applcatons Chapter 9 9-3 Learnng Objectves After letng ths chapter, you wll be able to:? Calculate the nterest rate and term n ound nterest applcatons? Gven a nomnal

More information

7.5. Present Value of an Annuity. Investigate

7.5. Present Value of an Annuity. Investigate 7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on

More information

FINANCIAL MATHEMATICS. A Practical Guide for Actuaries. and other Business Professionals

FINANCIAL MATHEMATICS. A Practical Guide for Actuaries. and other Business Professionals FINANCIAL MATHEMATICS A Practcal Gude for Actuares and other Busness Professonals Second Edton CHRIS RUCKMAN, FSA, MAAA JOE FRANCIS, FSA, MAAA, CFA Study Notes Prepared by Kevn Shand, FSA, FCIA Assstant

More information

Joe Pimbley, unpublished, 2005. Yield Curve Calculations

Joe Pimbley, unpublished, 2005. Yield Curve Calculations Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward

More information

Hedging Interest-Rate Risk with Duration

Hedging Interest-Rate Risk with Duration FIXED-INCOME SECURITIES Chapter 5 Hedgng Interest-Rate Rsk wth Duraton Outlne Prcng and Hedgng Prcng certan cash-flows Interest rate rsk Hedgng prncples Duraton-Based Hedgng Technques Defnton of duraton

More information

Level Annuities with Payments Less Frequent than Each Interest Period

Level Annuities with Payments Less Frequent than Each Interest Period Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Symoblc approach

More information

Section 5.3 Annuities, Future Value, and Sinking Funds

Section 5.3 Annuities, Future Value, and Sinking Funds Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme

More information

Using Series to Analyze Financial Situations: Present Value

Using Series to Analyze Financial Situations: Present Value 2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated

More information

Lecture 3: Force of Interest, Real Interest Rate, Annuity

Lecture 3: Force of Interest, Real Interest Rate, Annuity Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annuty-mmedate, and ts present value Study annuty-due, and

More information

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression. Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook

More information

10. (# 45, May 2001). At time t = 0, 1 is deposited into each of Fund X and Fund Y. Fund X accumulates at a force of interest

10. (# 45, May 2001). At time t = 0, 1 is deposited into each of Fund X and Fund Y. Fund X accumulates at a force of interest 1 Exam FM questons 1. (# 12, May 2001). Bruce and Robbe each open up new bank accounts at tme 0. Bruce deposts 100 nto hs bank account, and Robbe deposts 50 nto hs. Each account earns an annual e ectve

More information

EXAMPLE PROBLEMS SOLVED USING THE SHARP EL-733A CALCULATOR

EXAMPLE PROBLEMS SOLVED USING THE SHARP EL-733A CALCULATOR EXAMPLE PROBLEMS SOLVED USING THE SHARP EL-733A CALCULATOR 8S CHAPTER 8 EXAMPLES EXAMPLE 8.4A THE INVESTMENT NEEDED TO REACH A PARTICULAR FUTURE VALUE What amount must you nvest now at 4% compoune monthly

More information

Texas Instruments 30X IIS Calculator

Texas Instruments 30X IIS Calculator Texas Instruments 30X IIS Calculator Keystrokes for the TI-30X IIS are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the

More information

In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount owed is. P (1 + i) A

In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount owed is. P (1 + i) A Amortzed loans: Suppose you borrow P dollars, e.g., P = 100, 000 for a house wth a 30 year mortgage wth an nterest rate of 8.25% (compounded monthly). In ths type of loan you make equal payments of A dollars

More information

Finite Math Chapter 10: Study Guide and Solution to Problems

Finite Math Chapter 10: Study Guide and Solution to Problems Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount

More information

0.02t if 0 t 3 δ t = 0.045 if 3 < t

0.02t if 0 t 3 δ t = 0.045 if 3 < t 1 Exam FM questons 1. (# 12, May 2001). Bruce and Robbe each open up new bank accounts at tme 0. Bruce deposts 100 nto hs bank account, and Robbe deposts 50 nto hs. Each account earns an annual effectve

More information

Capital asset pricing model, arbitrage pricing theory and portfolio management

Capital asset pricing model, arbitrage pricing theory and portfolio management Captal asset prcng model, arbtrage prcng theory and portfolo management Vnod Kothar The captal asset prcng model (CAPM) s great n terms of ts understandng of rsk decomposton of rsk nto securty-specfc rsk

More information

Time Value of Money Module

Time Value of Money Module Tme Value of Money Module O BJECTIVES After readng ths Module, you wll be able to: Understand smple nterest and compound nterest. 2 Compute and use the future value of a sngle sum. 3 Compute and use the

More information

IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS

IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS Chrs Deeley* Last revsed: September 22, 200 * Chrs Deeley s a Senor Lecturer n the School of Accountng, Charles Sturt Unversty,

More information

8.4. Annuities: Future Value. INVESTIGATE the Math. 504 8.4 Annuities: Future Value

8.4. Annuities: Future Value. INVESTIGATE the Math. 504 8.4 Annuities: Future Value 8. Annutes: Future Value YOU WILL NEED graphng calculator spreadsheet software GOAL Determne the future value of an annuty earnng compound nterest. INVESTIGATE the Math Chrstne decdes to nvest $000 at

More information

Problem Set 3. a) We are asked how people will react, if the interest rate i on bonds is negative.

Problem Set 3. a) We are asked how people will react, if the interest rate i on bonds is negative. Queston roblem Set 3 a) We are asked how people wll react, f the nterest rate on bonds s negatve. When

More information

Nasdaq Iceland Bond Indices 01 April 2015

Nasdaq Iceland Bond Indices 01 April 2015 Nasdaq Iceland Bond Indces 01 Aprl 2015 -Fxed duraton Indces Introducton Nasdaq Iceland (the Exchange) began calculatng ts current bond ndces n the begnnng of 2005. They were a response to recent changes

More information

A Master Time Value of Money Formula. Floyd Vest

A Master Time Value of Money Formula. Floyd Vest A Master Tme Value of Money Formula Floyd Vest For Fnancal Functons on a calculator or computer, Master Tme Value of Money (TVM) Formulas are usually used for the Compound Interest Formula and for Annutes.

More information

Texas Instruments 30Xa Calculator

Texas Instruments 30Xa Calculator Teas Instruments 30Xa Calculator Keystrokes for the TI-30Xa are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the tet, check

More information

1. Math 210 Finite Mathematics

1. Math 210 Finite Mathematics 1. ath 210 Fnte athematcs Chapter 5.2 and 5.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210

More information

3. Present value of Annuity Problems

3. Present value of Annuity Problems Mathematcs of Fnance The formulae 1. A = P(1 +.n) smple nterest 2. A = P(1 + ) n compound nterest formula 3. A = P(1-.n) deprecaton straght lne 4. A = P(1 ) n compound decrease dmshng balance 5. P = -

More information

Section 2.3 Present Value of an Annuity; Amortization

Section 2.3 Present Value of an Annuity; Amortization Secton 2.3 Present Value of an Annuty; Amortzaton Prncpal Intal Value PV s the present value or present sum of the payments. PMT s the perodc payments. Gven r = 6% semannually, n order to wthdraw $1,000.00

More information

Thursday, December 10, 2009 Noon - 1:50 pm Faraday 143

Thursday, December 10, 2009 Noon - 1:50 pm Faraday 143 1. ath 210 Fnte athematcs Chapter 5.2 and 4.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210

More information

A) 3.1 B) 3.3 C) 3.5 D) 3.7 E) 3.9 Solution.

A) 3.1 B) 3.3 C) 3.5 D) 3.7 E) 3.9 Solution. ACTS 408 Instructor: Natala A. Humphreys SOLUTION TO HOMEWOR 4 Secton 7: Annutes whose payments follow a geometrc progresson. Secton 8: Annutes whose payments follow an arthmetc progresson. Problem Suppose

More information

Reporting Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (including SME Corporate), Sovereign and Bank Instruction Guide

Reporting Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (including SME Corporate), Sovereign and Bank Instruction Guide Reportng Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (ncludng SME Corporate), Soveregn and Bank Instructon Gude Ths nstructon gude s desgned to assst n the completon of the FIRB

More information

Solutions to First Midterm

Solutions to First Midterm rofessor Chrstano Economcs 3, Wnter 2004 Solutons to Frst Mdterm. Multple Choce. 2. (a) v. (b). (c) v. (d) v. (e). (f). (g) v. (a) The goods market s n equlbrum when total demand equals total producton,.e.

More information

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy 4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

More information

Stress test for measuring insurance risks in non-life insurance

Stress test for measuring insurance risks in non-life insurance PROMEMORIA Datum June 01 Fnansnspektonen Författare Bengt von Bahr, Younes Elonq and Erk Elvers Stress test for measurng nsurance rsks n non-lfe nsurance Summary Ths memo descrbes stress testng of nsurance

More information

Underwriting Risk. Glenn Meyers. Insurance Services Office, Inc.

Underwriting Risk. Glenn Meyers. Insurance Services Office, Inc. Underwrtng Rsk By Glenn Meyers Insurance Servces Offce, Inc. Abstract In a compettve nsurance market, nsurers have lmted nfluence on the premum charged for an nsurance contract. hey must decde whether

More information

Nordea G10 Alpha Carry Index

Nordea G10 Alpha Carry Index Nordea G10 Alpha Carry Index Index Rules v1.1 Verson as of 10/10/2013 1 (6) Page 1 Index Descrpton The G10 Alpha Carry Index, the Index, follows the development of a rule based strategy whch nvests and

More information

In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount in the account, the balance, is

In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount in the account, the balance, is Payout annutes: Start wth P dollars, e.g., P = 100, 000. Over a 30 year perod you receve equal payments of A dollars at the end of each month. The amount of money left n the account, the balance, earns

More information

Recurrence. 1 Definitions and main statements

Recurrence. 1 Definitions and main statements Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

More information

Interest Rate Fundamentals

Interest Rate Fundamentals Lecture Part II Interest Rate Fundamentals Topcs n Quanttatve Fnance: Inflaton Dervatves Instructor: Iraj Kan Fundamentals of Interest Rates In part II of ths lecture we wll consder fundamental concepts

More information

Chapter 15 Debt and Taxes

Chapter 15 Debt and Taxes hapter 15 Debt and Taxes 15-1. Pelamed Pharmaceutcals has EBIT of $325 mllon n 2006. In addton, Pelamed has nterest expenses of $125 mllon and a corporate tax rate of 40%. a. What s Pelamed s 2006 net

More information

Intra-year Cash Flow Patterns: A Simple Solution for an Unnecessary Appraisal Error

Intra-year Cash Flow Patterns: A Simple Solution for an Unnecessary Appraisal Error Intra-year Cash Flow Patterns: A Smple Soluton for an Unnecessary Apprasal Error By C. Donald Wggns (Professor of Accountng and Fnance, the Unversty of North Florda), B. Perry Woodsde (Assocate Professor

More information

Overview of Lecture 4

Overview of Lecture 4 Overview of Lecture 4 Examples of Quoted vs. True Interest Rates Banks Auto Loan Forward rates, spot rates and bond prices How do things change when interest rates vary over different periods? Present

More information

An Overview of Financial Mathematics

An Overview of Financial Mathematics An Overvew of Fnancal Mathematcs Wllam Benedct McCartney July 2012 Abstract Ths document s meant to be a quck ntroducton to nterest theory. It s wrtten specfcally for actuaral students preparng to take

More information

Chapter 15: Debt and Taxes

Chapter 15: Debt and Taxes Chapter 15: Debt and Taxes-1 Chapter 15: Debt and Taxes I. Basc Ideas 1. Corporate Taxes => nterest expense s tax deductble => as debt ncreases, corporate taxes fall => ncentve to fund the frm wth debt

More information

YIELD CURVE FITTING 2.0 Constructing Bond and Money Market Yield Curves using Cubic B-Spline and Natural Cubic Spline Methodology.

YIELD CURVE FITTING 2.0 Constructing Bond and Money Market Yield Curves using Cubic B-Spline and Natural Cubic Spline Methodology. YIELD CURVE FITTING 2.0 Constructng Bond and Money Market Yeld Curves usng Cubc B-Splne and Natural Cubc Splne Methodology Users Manual YIELD CURVE FITTING 2.0 Users Manual Authors: Zhuosh Lu, Moorad Choudhry

More information

Mathematics of Finance

Mathematics of Finance Mathematcs of Fnance 5 C H A P T E R CHAPTER OUTLINE 5.1 Smple Interest and Dscount 5.2 Compound Interest 5.3 Annutes, Future Value, and Snkng Funds 5.4 Annutes, Present Value, and Amortzaton CASE STUDY

More information

Properties of American Derivative Securities

Properties of American Derivative Securities Capter 6 Propertes of Amercan Dervatve Securtes 6.1 Te propertes Defnton 6.1 An Amercan dervatve securty s a sequence of non-negatve random varables fg k g n k= suc tat eac G k s F k -measurable. Te owner

More information

Financial Mathemetics

Financial Mathemetics Fnancal Mathemetcs 15 Mathematcs Grade 12 Teacher Gude Fnancal Maths Seres Overvew In ths seres we am to show how Mathematcs can be used to support personal fnancal decsons. In ths seres we jon Tebogo,

More information

Mathematics of Finance

Mathematics of Finance 5 Mathematcs of Fnance 5.1 Smple and Compound Interest 5.2 Future Value of an Annuty 5.3 Present Value of an Annuty;Amortzaton Chapter 5 Revew Extended Applcaton:Tme, Money, and Polynomals Buyng a car

More information

CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol

CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL

More information

DEFINING %COMPLETE IN MICROSOFT PROJECT

DEFINING %COMPLETE IN MICROSOFT PROJECT CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMI-SP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,

More information

FINANCIAL MATHEMATICS

FINANCIAL MATHEMATICS 3 LESSON FINANCIAL MATHEMATICS Annutes What s an annuty? The term annuty s used n fnancal mathematcs to refer to any termnatng sequence of regular fxed payments over a specfed perod of tme. Loans are usually

More information

Calculation of Sampling Weights

Calculation of Sampling Weights Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample

More information

SIMPLE LINEAR CORRELATION

SIMPLE LINEAR CORRELATION SIMPLE LINEAR CORRELATION Smple lnear correlaton s a measure of the degree to whch two varables vary together, or a measure of the ntensty of the assocaton between two varables. Correlaton often s abused.

More information

ADVA FINAN QUAN ADVANCED FINANCE AND QUANTITATIVE INTERVIEWS VAULT GUIDE TO. Customized for: Jason (jason.barquero@cgu.edu) 2002 Vault Inc.

ADVA FINAN QUAN ADVANCED FINANCE AND QUANTITATIVE INTERVIEWS VAULT GUIDE TO. Customized for: Jason (jason.barquero@cgu.edu) 2002 Vault Inc. ADVA FINAN QUAN 00 Vault Inc. VAULT GUIDE TO ADVANCED FINANCE AND QUANTITATIVE INTERVIEWS Copyrght 00 by Vault Inc. All rghts reserved. All nformaton n ths book s subject to change wthout notce. Vault

More information

The OC Curve of Attribute Acceptance Plans

The OC Curve of Attribute Acceptance Plans The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

More information

Documentation about calculation methods used for the electricity supply price index (SPIN 35.1),

Documentation about calculation methods used for the electricity supply price index (SPIN 35.1), STATISTICS SWEDEN Documentaton (6) ES/PR-S 0-- artn Kullendorff arcus rdén Documentaton about calculaton methods used for the electrct suppl prce ndex (SPIN 35.), home sales (HPI) The ndex fgure for electrct

More information

9.1 The Cumulative Sum Control Chart

9.1 The Cumulative Sum Control Chart Learnng Objectves 9.1 The Cumulatve Sum Control Chart 9.1.1 Basc Prncples: Cusum Control Chart for Montorng the Process Mean If s the target for the process mean, then the cumulatve sum control chart s

More information

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange

More information

ANALYSIS OF FINANCIAL FLOWS

ANALYSIS OF FINANCIAL FLOWS ANALYSIS OF FINANCIAL FLOWS AND INVESTMENTS II 4 Annutes Only rarely wll one encounter an nvestment or loan where the underlyng fnancal arrangement s as smple as the lump sum, sngle cash flow problems

More information

Graph Theory and Cayley s Formula

Graph Theory and Cayley s Formula Graph Theory and Cayley s Formula Chad Casarotto August 10, 2006 Contents 1 Introducton 1 2 Bascs and Defntons 1 Cayley s Formula 4 4 Prüfer Encodng A Forest of Trees 7 1 Introducton In ths paper, I wll

More information

Calculating the Trend Data

Calculating the Trend Data LASER INTERFEROMETER GRAVITATIONAL WAVE OBSERVATORY - LIGO - CALIFORNIA INSTITUTE OF TECHNOLOGY MASSACHUSETTS INSTITUTE OF TECHNOLOGY Techncal Note LIGO-T990110-B - D 3/29/07 Calculatng the Trend Data

More information

Section 2.2 Future Value of an Annuity

Section 2.2 Future Value of an Annuity Secton 2.2 Future Value of an Annuty Annuty s any sequence of equal perodc payments. Depost s equal payment each nterval There are two basc types of annutes. An annuty due requres that the frst payment

More information

Number of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000

Number of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000 Problem Set 5 Solutons 1 MIT s consderng buldng a new car park near Kendall Square. o unversty funds are avalable (overhead rates are under pressure and the new faclty would have to pay for tself from

More information

Morningstar After-Tax Return Methodology

Morningstar After-Tax Return Methodology Mornngstar After-Tax Return Methodology Mornngstar Research Report March 1, 2013 2013 Mornngstar, Inc. All rghts reserved. The nformaton n ths document s the property of Mornngstar, Inc. Reproducton or

More information

Mathematics of Finance

Mathematics of Finance CHAPTER 5 Mathematcs of Fnance 5.1 Smple and Compound Interest 5.2 Future Value of an Annuty 5.3 Present Value of an Annuty; Amortzaton Revew Exercses Extended Applcaton: Tme, Money, and Polynomals Buyng

More information

Return decomposing of absolute-performance multi-asset class portfolios. Working Paper - Nummer: 16

Return decomposing of absolute-performance multi-asset class portfolios. Working Paper - Nummer: 16 Return decomposng of absolute-performance mult-asset class portfolos Workng Paper - Nummer: 16 2007 by Dr. Stefan J. Illmer und Wolfgang Marty; n: Fnancal Markets and Portfolo Management; March 2007; Volume

More information

Electric circuit components. Direct Current (DC) circuits

Electric circuit components. Direct Current (DC) circuits Electrc crcut components Capactor stores charge and potental energy, measured n Farads (F) Battery generates a constant electrcal potental dfference ( ) across t. Measured n olts (). Resstor ressts flow

More information

A Critical Note on MCEV Calculations Used in the Life Insurance Industry

A Critical Note on MCEV Calculations Used in the Life Insurance Industry A Crtcal Note on MCEV Calculatons Used n the Lfe Insurance Industry Faban Suarez 1 and Steven Vanduffel 2 Abstract. Snce the begnnng of the development of the socalled embedded value methodology, actuares

More information

The Development of Web Log Mining Based on Improve-K-Means Clustering Analysis

The Development of Web Log Mining Based on Improve-K-Means Clustering Analysis The Development of Web Log Mnng Based on Improve-K-Means Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.

More information

Basel Committee on Banking Supervision

Basel Committee on Banking Supervision Basel Commttee on Banng Supervson The standardsed approach for measurng counterparty credt rs exposures March 014 (rev. Aprl 014) Ths publcaton s avalable on the BIS webste (www.bs.org). Ban for Internatonal

More information

Stock Profit Patterns

Stock Profit Patterns Stock Proft Patterns Suppose a share of Farsta Shppng stock n January 004 s prce n the market to 56. Assume that a September call opton at exercse prce 50 costs 8. A September put opton at exercse prce

More information

State function: eigenfunctions of hermitian operators-> normalization, orthogonality completeness

State function: eigenfunctions of hermitian operators-> normalization, orthogonality completeness Schroednger equaton Basc postulates of quantum mechancs. Operators: Hermtan operators, commutators State functon: egenfunctons of hermtan operators-> normalzaton, orthogonalty completeness egenvalues and

More information

Risk-based Fatigue Estimate of Deep Water Risers -- Course Project for EM388F: Fracture Mechanics, Spring 2008

Risk-based Fatigue Estimate of Deep Water Risers -- Course Project for EM388F: Fracture Mechanics, Spring 2008 Rsk-based Fatgue Estmate of Deep Water Rsers -- Course Project for EM388F: Fracture Mechancs, Sprng 2008 Chen Sh Department of Cvl, Archtectural, and Envronmental Engneerng The Unversty of Texas at Austn

More information

CHAPTER 18 INFLATION, UNEMPLOYMENT AND AGGREGATE SUPPLY. Themes of the chapter. Nominal rigidities, expectational errors and employment fluctuations

CHAPTER 18 INFLATION, UNEMPLOYMENT AND AGGREGATE SUPPLY. Themes of the chapter. Nominal rigidities, expectational errors and employment fluctuations CHAPTER 18 INFLATION, UNEMPLOYMENT AND AGGREGATE SUPPLY Themes of the chapter Nomnal rgdtes, expectatonal errors and employment fluctuatons The short-run trade-off between nflaton and unemployment The

More information

Introduction: Analysis of Electronic Circuits

Introduction: Analysis of Electronic Circuits /30/008 ntroducton / ntroducton: Analyss of Electronc Crcuts Readng Assgnment: KVL and KCL text from EECS Just lke EECS, the majorty of problems (hw and exam) n EECS 3 wll be crcut analyss problems. Thus,

More information

ErrorPropagation.nb 1. Error Propagation

ErrorPropagation.nb 1. Error Propagation ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then

More information

Portfolio Risk Decomposition (and Risk Budgeting)

Portfolio Risk Decomposition (and Risk Budgeting) ortfolo Rsk Decomposton (and Rsk Budgetng) Jason MacQueen R-Squared Rsk Management Introducton to Rsk Decomposton Actve managers take rsk n the expectaton of achevng outperformance of ther benchmark Mandates

More information

Abstract # 015-0399 Working Capital Exposure: A Methodology to Control Economic Performance in Production Environment Projects

Abstract # 015-0399 Working Capital Exposure: A Methodology to Control Economic Performance in Production Environment Projects Abstract # 015-0399 Workng Captal Exposure: A Methodology to Control Economc Performance n Producton Envronment Projects Dego F. Manotas. School of Industral Engneerng and Statstcs, Unversdad del Valle.

More information

FIN 472 Fixed-Income Securities Forward Rates

FIN 472 Fixed-Income Securities Forward Rates FIN 472 Fixed-Income Securities Forward Rates Professor Robert B.H. Hauswald Kogod School of Business, AU Interest-Rate Forwards Review of yield curve analysis Forwards yet another use of yield curve forward

More information

SUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW.

SUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW. SUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW. Lucía Isabel García Cebrán Departamento de Economía y Dreccón de Empresas Unversdad de Zaragoza Gran Vía, 2 50.005 Zaragoza (Span) Phone: 976-76-10-00

More information

Multiple stage amplifiers

Multiple stage amplifiers Multple stage amplfers Ams: Examne a few common 2-transstor amplfers: -- Dfferental amplfers -- Cascode amplfers -- Darlngton pars -- current mrrors Introduce formal methods for exactly analysng multple

More information

S&P/CITIC CHINA BOND INDICES

S&P/CITIC CHINA BOND INDICES October 2006 S&P/CITIC CHINA BOND INDICES INDEX METHODOLOGY Table of Contents Introducton 3 Hghlghts 3 Partnershp 4 Index Famly 4 Elgblty Crtera 5 S&P/CITIC Government Bond Index 5 S&P/CITIC Corporate

More information

Staff Paper. Farm Savings Accounts: Examining Income Variability, Eligibility, and Benefits. Brent Gloy, Eddy LaDue, and Charles Cuykendall

Staff Paper. Farm Savings Accounts: Examining Income Variability, Eligibility, and Benefits. Brent Gloy, Eddy LaDue, and Charles Cuykendall SP 2005-02 August 2005 Staff Paper Department of Appled Economcs and Management Cornell Unversty, Ithaca, New York 14853-7801 USA Farm Savngs Accounts: Examnng Income Varablty, Elgblty, and Benefts Brent

More information

We first solve for the present value of the cost per two barrels: (1.065) 2 = 41.033 (1.07) 3 = 55.341. x = 20.9519

We first solve for the present value of the cost per two barrels: (1.065) 2 = 41.033 (1.07) 3 = 55.341. x = 20.9519 Chapter 8 Swaps Question 8.1. We first solve for the present value of the cost per two barrels: $22 1.06 + $23 (1.065) 2 = 41.033. We then obtain the swap price per barrel by solving: which was to be shown.

More information

Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6)

Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6) Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called

More information

1. Measuring association using correlation and regression

1. Measuring association using correlation and regression How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a

More information

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001. Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.

More information

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..

More information

Support Vector Machines

Support Vector Machines Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.

More information

IN THE UNITED STATES THIS REPORT IS AVAILABLE ONLY TO PERSONS WHO HAVE RECEIVED THE PROPER OPTION RISK DISCLOSURE DOCUMENTS.

IN THE UNITED STATES THIS REPORT IS AVAILABLE ONLY TO PERSONS WHO HAVE RECEIVED THE PROPER OPTION RISK DISCLOSURE DOCUMENTS. http://mm.pmorgan.com European Equty Dervatves Strategy 4 May 005 N THE UNTED STATES THS REPORT S AVALABLE ONLY TO PERSONS WHO HAVE RECEVED THE PROPER OPTON RS DSCLOSURE DOCUMENTS. Correlaton Vehcles Technques

More information