期货理论与实务英文版课件：Chapter 2 Risk Free Asset

1、chapter 2Risk-Free Assets2.1 Time Value of Money2.1.1 Simple Interest2.1.2 Periodic Compounding2.1.3 Streams of Payments2.1.4 Continuous Compounding2.2 Money Market2.2.1 Zero-Coupon Bonds2.2.2 Coupon Bondsmain content2.1 Time Value of MoneyThe way in which money changes its value in time is a comple

2、x issue of fundamental importance in finance. What is the future value of an amount invested or borrowed today?What is the present value of an amount to be paid or received at a certain time in the future?2.1.1 Simple InterestSuppose that an amount is paid into a bank account, where it is to earn in

3、terest. The future value of this investment consists of the initial deposit, called the principal and denoted by P, plus all the interest earned since the money was deposited in the account.The value of the investment at time t, denoted by V (t), is given by V (t) = (1+tr)PIf the principal P is inve

4、sted at time s, rather than at time 0,then the value at time t s will be V (t) = (1+(t s)r)P.Simple InterestReturn:Some ConceptsFuture Value, Growth FactorDiscounted Value(present), Discount FactorIn practice simple interest is used only for short-term investments and for certain types of loans and

5、deposits. The interest already earned can be reinvested to attract even more interest.2.1.2 Periodic CompoundingIn general, if m interest payments are made per annum, the time between two consecutive payments measured in years will be ,the first interest payment being due at time . Each interest pay

6、ment will increase the principal by a factor of 1 + .Given that the interest rate r remains unchanged, after t years the future value of an initial principal P will become V (t) = Pbecause there will be tm interest payments during this period. In this formula t must be a whole multiple of the period

7、 .The number is the growth factor .ExerciseTo show that V (t) increases as the compounding frequency m increases,the others remaining unchanged.we need to verify that if m 0 compounded m times a year can be written as:In the limit as m, we obtainThis is known as continuous compounding.The correspond

8、ing growth factor is .2.1.4 Continuous Compounding2.1.5 How to Compare Compounding MethodsFrequent compounding will produce a higher future value than less frequent compounding if the interest rates and the initial principal are the same.For a given compounding method with interest rate r the effect

9、ive rate re is one that gives the same growth factor over a one year period under annual compounding.In particular, in the case of periodic compounding with frequency m and rate r the effective rate satisfiesIn the case of continuous compounding with rate r2.2 Money MarketThe money market consists o

10、f risk-free (more precisely, default-free) securities. An example is a bond, which is a financial security promising the holder a sequence of guaranteed future payments. Risk-free means here that these payments will be delivered with certainty.There are many kinds of bonds like treasury bills and no

11、tes, treasury, mortgage and debenture bonds, commercial papers, and others with various particular arrangements concerning the issuing institution, duration, number of payments, embedded rights and guarantees.2.2.1 Zero-Coupon BondsThe simplest case of a bond is a zero-coupon bond, which involves ju

12、st a single payment. The issuing institution (for example, a government, a bank or a company) promises to exchange the bond for a certain amount of money F, called the face value, on a given day T, called the maturity date.Typically, a bond can be sold at any time prior to maturity at the market pri

13、ce. This price at time t is denoted B(t, T). In particular, B(0, T) is the current, time 0 price of the bond, and B(T,T) = 1 is equal to the face value.So V (t) = B(t, T), V (T) = 1.Using periodic compounding with frequency m, we need to solve the equationIn the case of continuous compounding the eq

14、uation for the implied rate satisfies2.2.2 Coupon BondsBonds promising a sequence of payments are called coupon bonds. These payments consist of the face value due at maturity, and coupons paid regularly, typically annually, semi-annually, or quarterly, the last coupon due at maturity. The assumptio

15、n of constant interest rates allows us to compute the price of a coupon bond by discounting all the future payments.Question: what is the price of the bond at time s?2.2.2 Coupon BondsQuestion 1: what is the price, when ir?Question 2: what is the r, when giving with a streams of payments?2.2.2 Coupon BondsWhen the interest rate is constant, the function A(t) does not depend on the way the money market account is run, that is, it neither depends on the types of bonds selecte