1、Introduction to Financial Mathematics,李社环上海财经大学金融学院保险系 2010年9月,Lecture 1 An introduction,Aims What? Why? How? References An introduction to interestAn introduction to financial market,Structure and CONTENTS,相关学习内容,教材 CT1,练习,参考文献,练习册,社会实践考察,教材中的questions,Structure and CONTENTS,Theory of Interest in C
2、T1,Part I:Basic conceptsGeneral formulaeGeneral principles,direct application,more complex application,Ch. 17,Ch. 810,Ch. 1113,Part II:,Part III:,generalization,Part IV:,Ch. 1415,What? (Part I),cash-flow model and financial transactions Value of money , Value of a cash-flow interest rates and return
3、 formulae of value determination and equation of value,What? (Part II),How to calculate a schedule of repayments under a loan How to appraise an investment project How to calculate an ordinary share and a property or other financial assets,What? (Part III&IV),How to calculate the delivery price and
4、the value of a forward contract? How to price the value of a future contract? How to measure the impact of taxation and inflation on yields of investment? How to measure and control risk of interest rates the term structure of interest rates simple stochastic interest rate models,Application,In insu
5、rance industry Annuities Pricing of insurance products Asset liability management & Insolvency management ,In banking industry Loan schedule Asset liability management Interest risk management: measurement of risk and immunity Determination of yields ,In security exchange industry Determination of y
6、ields Appraisal of financial investment Appraisal of fund management pricing of securities Financial risk management: pricing of risk transferring, like pricing of derivatives ,In others industries Appraisal of business investment Enterprise risk management ,Why ?,CT3,CT4,CT7,CT8,ST1,ST4,ST5,CT1,微积分
7、,证券市场,货币银行,How?,Three Steps Write down cashflows related Make up a equation Solve the equation,References,S.G. Kellison, The Theory of Interest, RICHARD D. IRWIN, INC. 1991The Theory of Interest, The Actuary Education Company, (精算中心)Financial Mathematics? (精算中心),孟生旺 袁卫，利息理论及其应用，中国人民大学出版社 中国证券业协会，固定收
8、益证券估值与分析，中国人民大学证券研究所，证券市场分析，中国人民大学出版社,http:/www.actuarial.unsw.edu.au/ http:/www.actuaries.org.uk/Display_Page.cgi?url=/students/specimen_papers.xml,An introduction to Theory of interest,two versions of Theory of Interest What is interest? How many kinds of interest rates are there?,What is Interest
9、?,A reward paid by one person or organization for use of an asset, referred to as capital, belonging ti another person or organisation (the lender)A return for lending capital（CH.2, P1）： Interest may be regarded as a reward paid by one person or organisation (the borrower) for the use of an asset, r
10、eferred to as capital, belonging to another person or organisation (the lender). Time value,What is Interest rate?,Interest rate is the amount of interest a unit initial capital will earn at the end of the time period. Price of money time value of unit money,An Example,You lent 50$ for one year at t
11、he beginning of this year and will receive 60$ at the end of the year. What interest rate will you get? What is the interest rate will you get if the contract of loan ended half year later after issued and others are not changed ?,How many kinds of interest rates are there?,interest rates of banking
12、 rates of loan yields of investment coupon rates rent rates insurance premium ,More interest rates,nominal Effective single complex real ,An introduction to financial market,Financial market Capital market Stock market bond market money market Insurance market Foreign exchange market Gold market ,Fi
13、nancial goods,Securities Broadly Narrowly ：bonds and equitiesBonds and Bills ( lendersborrowers, or creditors - debtors) Government bills & bonds corporate bonds & bills （debentures, unsecured loan stocks ,Eurobonds） Financial bonds & billsCertificates of deposit,equities (managers - owners) Common
14、Stock or ordinary shares Precedence Stock (or shares),Important items of a bond,Name of the issuerFace valueCoupon rateIssuing dateMaturity dateCoupon payment,Financial transaction,Issuing styles at par-face value = sale price at premium-face value sale price Remark: interest is determined by the fa
15、ce value,Examples,A zero-coupon bond Stock issuing in China,What we have known in this lecture?,Contents we will study Techniques for the study An introduction to interest and interest rates An introduction to financial markets,Financial Mathematics,李社环上海财经大学金融学院保险系（E-mail: ）2010年9月,Lecture 1 Gener
16、alized Cashflow Models,What is a cash flow? Main factors for determining a cash flow Types of cash flow special cash flows,What is a cash flow?,Definition A cashflow model is a mathematical projection of the payments arising from a financial transaction：CFtiGeometrical expression,main factorsAmount
17、(of each payment) Cfti Timing (or interval of payments) ti Number (of payments) nPositive or negative?Certain or uncertain?,A cashflow of a zero-coupon bill,-98$,A cashflow of a fixed-interest security with coupon of 5%,a lump sum of payment of 50000 at beginninga series of regular level interest pa
18、yments of 2500a lump sum of redemption of 50000 at maturity,Cashflows in real term,How to measure Price Inflation?,Price indexfixed base index . for instant, The price index based on 1990:Chain Price Index,How to measure Price Inflation?,Price indexYear-Based Price Index,How to measure Price Inflati
19、on?,Rate of inflation A rate of inflation per annum, st,3. Relation,An Example,An inflation index takes the value as the following table. Find the annual inflation rates for these years.,Answer,Prove that r= i/(1+)where,r: real rate of interest or real payment,i: nominal rate of interest, or nominal
20、 payment : inflation rate,Cashflows in real term,Question 1.5. An investor purchases a three-year indexed security on 1.1.2001. In return the investor receives payments at the end of each year plus a final redemption amount, all of which were increased in line with the index given in the following t
21、able. The payments would have been $600 each year and $11,000 on redemption if there had been no inflation. Calculate the payments actually received by the investor.,1.1.2001,1.1.2003,600,1.1.2002,1.1.2003,600,11600,If no inflation,If inflation,600*1.05=630,600*1.08=648,11600*1.13=13108,uncertain ca
22、shflows,Ex.1. A speculator spent $300 and bought a lottery on June 1,1999, got $20 immediately , and furthermore wan $10,000 on March 1, 2000. Write down the cashflow for the speculator.,Ex.2. There is a kind of lottery with face value of $1 and a term of 1 year. The lottery is issued at the beginni
23、ng of every year and discloses the winners who could get a prize of $10,000 each time with a probability of millionth at the end of each month during the year. Write down the cashflow for a speculator with a single lottery note.,An annuity,Definition and featuresAn annuity provides a series of regul
24、ar payments in return for a single premium (i.e. a lump sum paid at the outset) Regular: timing interval ti-ti-1=cont.,Which of the followings are annuities?Cashlflow of a fixed-interest security dividend cashlflow of a stockpayments of a stock payments in your daily life Payments of a pension schem
25、e,Types of annuities,In generalLevel annuities: Cfi=CFjVariable annuities: Cfi mmay not equal CFjInsurance products a temporary life annuity a guaranteed annuityA “whole life” annuity,An Example: Question 1.7 (Ch.1 P10) Certain or not,An Example,An individual purchases the following three annuities
26、at time 0:A whole life annuity paying $800 at the end of each year.A life annuity with a 10 year guarantee period paying $900 at the end of each year.A temporary life annuity paying $1000 at the end of each year with a maximum term of 8 years.,What would the total annuity payments made at times 5, 9
27、 and 12 be if the individual died at time 14.5?What would the total annuity payments made at times 6, 10 and 13 be if the individual died at time 7.5?,What we have known in this lecture?,Definition and factors of a cash flow? Types of cash flow Some specified cash flowsAnnuities,Exercises,Finish all
28、 exercises of Ch. 1,Financial Mathematics,李社环上海财经大学金融学院保险系（E-mail: ）2010年9月,Lecture 2 The time value of money,What well knowconcepts simple interest compound interest Effective rate of interest Discount interest rate present value Accumulation value Discount factor time value of money,ProcessDiscou
29、ntingAccumulating,Pay attention!,simple interest & compound interest interest rate & discount rate present value & Accumulation value,Well answer the following questions,What is the value of a cash flow?For two cash flows, how can we know one is more valued than the other?,$1 in this year = $1 in ne
30、xt year?,A DOLLAR TODAY IS WORTH MORE THAN A DOLLAR IN THE FUTURE,Factors affecting money value,Gain of investment Price inflation Change of exchange rates Other risks, like,DefaultLiquidate and so on.,Time value,How much one dollar today can earn during the year，in a economic activity? interest int
31、erest rate,Composing of interest rate,Interest rate = risk-free rate + risk premium,How much risk premium ?,In 2000, in the Fortune 1000 companies, Aaa rated corporations paid an average spread of 1.6% over risk-free Bonds and Baa rated corporations paid an average spread of 2.3% . Given the total v
32、olume of corporate bonds is USD 7.8 trillion, this provides an estimated range of USD 125 billion to USD 180 billion for risk premium to investors in US corporate bonds. (Swiss Re, The picture of ART, sigma,No.1/2003),Various costs of risk taking for Fortune 1000 corporations in 2000,Esp. (if the on
33、ly risk is price inflation):nominal rate = real rate + inflation rate,2 forms of interest rates in practice,Simple interest： simple interest can not earn further interest. i interest rate; Ccapital; nterm-C （Ci） （Ci） （Ci） C+nCi 0 1 2 n-1 n,（Ci）means interest of Ci paid nominally not actually,Simple
34、Interest Rate,An investment of $1 under an annual simple interest rate of i will have value $(1+ i t) at time t,compound interest: interest can also earn interest.-C Ci Ci Ci Ci+C 0 1 2 n-1 n,compound interest,Question and thinking,An investor initials a time deposit of saving that pays interest rat
35、e of 2.80% pa, with a term of 3 years and amount of principle ￥10000, in the Commercial Bank of China. (a) How much will he receive on the redemption date? (b) Is simple or compound interest used?,On June 10, 1998, our government issued a kind of credit-booked Treasure bonds (凭证式国债) with a term of 3
36、 years and a coupon of 7.11% pa. Investors got interest annually from those appointed banks by their credit books and got the capital on the redemption date. Did they get simple or compound interest ?,Value on redemption,Symbols Anthe amount which will be received by the investor if the account is c
37、losed after n years A0 = C the initial principle i a interest rate What is An?,Simple interest,The future value of $1 today is linear to the length of time under simple interest rate,Under compound interest rate,An investment of $1 at a yearly compound interest rate of i will have value $(1+ i )t at
38、 time t (in years),Comparison of simple and compound rate,Note, nCi C(1+i)n 1 = n Ci If n 1 =,Accumulation values,C(1+ni): the accumulation value at n of initial capital C under simple interest rate. C(1+i)n: the accumulation value at n of initial capital C under Compound interest rate. Or (1+i)n is
39、 the accumulation value at n of a unit initial capital under Compound interest rate.,What does the accumulation value mean?,With a simple interest i=6% pa, which of the followings would prefer?$100 at the beginning of this year;$106 at the end of this year.With a Compound interest i, which of the fo
40、llowings would prefer?$C at 0;$C (1+i)n at n.,Remarks,$C at 0 is equivalent to $ C (1+i)n at n; Or $C/ (1+i)n at 0 is equivalent to $ C at n; or$1/ (1+i)n at 0 is equivalent to $ 1 at n.,$1/ (1+i)n,$ 1,n,0,Present values,1/(1+ni): the Present value at time 0 of a unit of capital at time n by simple
41、interest. 1/(1+i)n: the Present value at time 0 of a unit of capital at time n by compound interest. v = 1/(1+i) is called a discount factor.,practice,Question 2.5 _P11,Present value & accumulation value of a cashflow (ch.4),What is the present value of CFti i=1 n?What is the accumulation value of C
42、Fti i=1 n?,identities,Ci(1+i)n-1+ Ci(1+i)n-2 +Ci(1+i)+Ci+C C(1+i)n 或者,Examples,Example 1. (P6)Example 2. An investor makes an initial investment of $3000 and is credited with $158 interest at the end of the second year. What amount of interest the investor will get for the initial $3000 after half p
43、ast 3 years if the same compound interest is used?,Discount rate,A rate of commercial discount is a way of expressing the amount of the repayment requires for a loan usually in a period t 1 year. commercial discount d -( 1-nd ) 1 payment 0 n time,for a rate of compound discount or effective discount
44、 d (1-d)n 1 0 n,Examples,An one-month treasure bill is issued at a rate of commercial discount of 18% pa. If the repayment is $20,000, how much was paid to the government initially?,Relation of i and d,Examples,An Investor buys one year Treasury Bill discounted at a rate of discount of 7% p.a. Find
45、the effective rate of interest per annum for the transaction.,If the above question is changed as the following, what is the result? An Investor buys a 91-day Treasury Bill discounted at a simple rate of discount of 7% p.a. Find the effective rate of interest per annum for the transaction.,Summary,E
46、xample,Carl puts $10,000 into a bank account that pays an annual interest rate of 4% for ten years. If a withdrawal is made during the first 5 years, a penalty of 5% of the withdrawal is made. Carl withdraws $K at the end of years 4, 5, 6 and 7.The balance at the end of 10 years is $10,000. What is the value of K?,Exercises,Assignment X1: X1.1; X1.3; X1.4 Finish all exercises of Ch. 2,Introduction to Financial Mathematics,李社环上海财经大学金融学院保险系（E-mail: ）2010年10月,Lecture 3 interest rates,
