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1、Chapter 2,A review of financial mathematics,Overview,In this lecture we will Review the basic mathematics used in valuing assets Examine the mathematics for calculating and comparing: Financial arrangements consisting of different interest rates The present value of different cash flow structures Ac
2、cumulated future cash positions for different cash flow structures,Aim of financial mathematics,The aim of financial mathematics is to convert single or multiple cash flows that will be received at different points in time to one numberThis number represents the value of all of an assets cash flows
3、at a given point in timeThis is typically used to: Make a rational choice between different assets Determine the maximum amount an investor is willing to pay for an asset i.e. the intrinsic value of the asset,Time value of money,FIGURE 2.1,Time value of money (cont),The intrinsic value of Asset 2 is
4、 greater because of the time value of moneyIf you intended to consume in year 5, rather than year 3, you would still prefer Asset 2 because you could reinvest the $100 received in year 3 and accumulate more than $100 by year 5If you intended to consume in year 2, you would still prefer Asset 2 since
5、 you will have to borrow in year 2 you will be better off repaying the loan in year 3 rather than year 5,Interest rate arrangements,The time value of money is often measured by using an interest rate, which compensates those who defer consumption until later and imposes a charge on those who wish to
6、 consume more now than their income allowsWe will use the symbol r to denote the interest rate in the following calculations,Simple interest,Interest is earned on the initial amount invested or principalIf you invest an initial amount (PV, the present value) you would accumulate an amount (FV, the f
7、uture value) equal to the principal plus interest:The interest is the product of PV and the simple interest rate, r,Thus:where:FV = the accumulated (future) valuePV = the initial amount invested or borrowedr = the simple interest rate over the entire periodIf simple interest is applied to periods of
8、 less than 1 year, the interest rate is:where dtm = days to maturity of the loan,Simple interest (cont),(2.1),Example 2.1A credit union pays 5% p.a. simple interest. If $1000 is invested today, how much will the account accrue in 4 years?_FV = PV(1 + r)r = 4-year interest rate = 0.05 x 4 = 0.20Hence
9、: FV= 1000 (1 + 0.20) = $1200,Simple interest (cont),Compound interest,The amount of interest accrued each period is added to the principal, and this new balance is used to calculate the interest amount for the next periodThus, interest is paid on interest that has accrued in previous periods,The fo
10、rmula for the accrued value under compound interest is:where:FV = the accumulated amount in period nPV = the initial amount investedr = the interest rate per periodn = the number of periods,Compound interest (cont),(2.2),Example 2.2A credit union pays 5% p.a. interest compounded annually. If $1000 i
11、s invested today, how much will the account accrue in 4 years?_FV = PV(1 + r)nr = 4-year interest rate = 0.05 x 4 = 0.20Hence: FV= 1000 (1 + 0.05)4 = $1216,Compound interest (cont),Comparing different financing arrangements,Compound interest rates are quoted as:Nominal interest rate Quoted annual in
12、terest rate that is adjusted to match the frequency of payments or compounding by taking a proportion of the quoted nominal rate to obtain the actual interest rate per period. E.g. 10% p.a. compounded semi-annually = 5% per half-year Effective interest rate Accounts for the true amount of interest t
13、hat is earned on both reinvested interest and principal earned over a year.,The effective rate is an annual rate that takes into account the effect of compounding:where:rnom = the nominal ratem = the number of compounding periods underlying the nominal rateThe effective rate will be greater than the
14、 nominal rate for compounding periods of less than 1 year,Effective interest rate,(2.3),Example 2.3What is the effective interest rate on the outstanding balance on a credit card if the nominal rate quoted on the card is 15.75% per annum, compounding daily?_,Comparing interest rates,Example 2.4Is a
15、loan that charges 14% annual interest with monthly compounding cheaper than a loan with a 14.75% interest rate with annual compounding? Would you prefer a loan with a 14.5% interest rate with semi-annual compounding?_,Comparing interest rates (cont),We can compare simple interest with compound inter
16、est by converting the compound interest into an equivalent simple interest ratewhere:m = the compounding intervalt = the number of years over which the amount is investedcompounding interest ratem = interest rate with compounding frequency m,(2.4),Comparing interest rates (cont),Example 2.6A credit union pays interest at 5% p.a. compounded annually. If $1000 is invested for five years, what is the equivalent simple interest rate that the amount will earn?_,
