The Theory of Interest - banach.millersville.edu

1 The Theory of InterestAn Undergraduate Introduction to Financial MathematicsJ. Robert Buchanan2014J. Robert BuchananThe Theory of InterestSimple Interest (1 of 2)DefinitionInterestis money paid by a bank or other financial institution toan investor or depositor in exchange for the use of thedepositor s of Interest is (usually) a fraction (called theinterestrate) of the initial amount deposited called : a bank whose Interest rate for depositors is the sameas its Interest rate for borrowers is called anideal Robert BuchananThe Theory of InterestSimple Interest (1 of 2)DefinitionInterestis money paid by a bank or other financial institution toan investor or depositor in exchange for the use of thedepositor s of Interest is (usually) a fraction (called theinterestrate) of the initial amount deposited called : a bank whose Interest rate for depositors is the sameas its Interest rate for borrowers is called anideal Robert BuchananThe Theory of InterestSimple Interest (2 of 2)Notation:r: Interest rate per unit timeP: principal amountA: amount due (account balance)t.

2 TimeThese quantities are related through the equation:A=P(1+rt).J. Robert BuchananThe Theory of InterestCompound Interest (1 of 2)Once credited to the investor, the Interest may be kept by theinvestor, and may earn Interest Interest is credited once per year, then aftertyears theamount due isA=P(1+r) Robert BuchananThe Theory of InterestCompound Interest (2 of 2)If a portion of the Interest is credited after a fraction of a year,then the Interest is said to there arencompounding periodsper year, then intyearsthe amount due isA=P(1+rn) Robert BuchananThe Theory of InterestExamples (1 of 2)ExampleSuppose an account earns annually compoundedmonthly. If the principal amount is $3104 what is the amountdue after three and one-half years?

3 Solution:A=P(1+rn)tn=3104(1+ )( )(12) Robert BuchananThe Theory of InterestExamples (1 of 2)ExampleSuppose an account earns annually compoundedmonthly. If the principal amount is $3104 what is the amountdue after three and one-half years?Solution:A=P(1+rn)tn=3104(1+ )( )(12) Robert BuchananThe Theory of InterestExamples (2 of 2)ExampleSuppose an account earns annual simple Interest . If theprincipal amount is $3104 what is the amount due after threeand one-half years?Solution:A=P(1+rt)=3104(1+ ( )) Robert BuchananThe Theory of InterestExamples (2 of 2)ExampleSuppose an account earns annual simple Interest . If theprincipal amount is $3104 what is the amount due after threeand one-half years?

4 Solution:A=P(1+rt)=3104(1+ ( )) Robert BuchananThe Theory of InterestEffective Interest RateDefinitionThe annual Interest rate equivalent to a given compoundinterest rate is called theeffective Interest (1+rn)n 1 Remark: the effective Interest rate is also known as theeffective yieldor simply as Robert BuchananThe Theory of InterestEffective Interest RateDefinitionThe annual Interest rate equivalent to a given compoundinterest rate is called theeffective Interest (1+rn)n 1 Remark: the effective Interest rate is also known as theeffective yieldor simply as Robert BuchananThe Theory of InterestExampleSuppose an account earns annually compoundedmonthly. What is the effective Interest rate?

5 Re=(1+rn)n 1=(1+ )12 1 Robert BuchananThe Theory of InterestExampleSuppose an account earns annually compoundedmonthly. What is the effective Interest rate?re=(1+rn)n 1=(1+ )12 1 Robert BuchananThe Theory of InterestContinuous CompoundingWhat happens as we increase the frequency of compounding?A=limn P(1+rn)n t=P er tEvaluate the limit using l H pital s amount due forcontinuously compounded interestisA=P er tJ. Robert BuchananThe Theory of InterestContinuous CompoundingWhat happens as we increase the frequency of compounding?A=limn P(1+rn)n t=P er tEvaluate the limit using l H pital s amount due forcontinuously compounded interestisA=P er tJ. Robert BuchananThe Theory of InterestContinuous CompoundingWhat happens as we increase the frequency of compounding?

6 A=limn P(1+rn)n t=P er tEvaluate the limit using l H pital s amount due forcontinuously compounded interestisA=P er tJ. Robert BuchananThe Theory of InterestExample (1 of 2)Suppose $3585 is deposited in an account which pays interestat an annual rate of compounded the amount due after two and one half the equivalent annual effective simple Interest Robert BuchananThe Theory of InterestExample (2 of 2)1 Amount due:A=Pert= ( ) rate: since limn (1+rn)n 1=er 1 thenre=er 1= 1 Robert BuchananThe Theory of InterestExample (2 of 2)1 Amount due:A=Pert= ( ) rate: since limn (1+rn)n 1=er 1 thenre=er 1= 1 Robert BuchananThe Theory of InterestPresent ValueHow do we rationally compare amounts of money paid atdifferent times in an Interest -earning environment?

7 DefinitionThepresent valueofA, an amount duetyears from nowsubject to an Interest rateris the principal amountPwhichmust to invested now so thattyears from now the accumulatedprincipal and Interest (1+rn) n t(discrete compounding)P=A e r t(continuous compounding)J. Robert BuchananThe Theory of InterestPresent ValueHow do we rationally compare amounts of money paid atdifferent times in an Interest -earning environment?DefinitionThepresent valueofA, an amount duetyears from nowsubject to an Interest rateris the principal amountPwhichmust to invested now so thattyears from now the accumulatedprincipal and Interest (1+rn) n t(discrete compounding)P=A e r t(continuous compounding)J. Robert BuchananThe Theory of InterestPresent ValueHow do we rationally compare amounts of money paid atdifferent times in an Interest -earning environment?

8 DefinitionThepresent valueofA, an amount duetyears from nowsubject to an Interest rateris the principal amountPwhichmust to invested now so thattyears from now the accumulatedprincipal and Interest (1+rn) n t(discrete compounding)P=A e r t(continuous compounding)J. Robert BuchananThe Theory of InterestPresent ValueHow do we rationally compare amounts of money paid atdifferent times in an Interest -earning environment?DefinitionThepresent valueofA, an amount duetyears from nowsubject to an Interest rateris the principal amountPwhichmust to invested now so thattyears from now the accumulatedprincipal and Interest (1+rn) n t(discrete compounding)P=A e r t(continuous compounding)J. Robert BuchananThe Theory of InterestExample (1 of 2)Suppose an investor will receive payments at the end of thenext six years in the amounts shown in the the Interest rate is compounded monthly, what is thetotal present value of the investments?

9 J. Robert BuchananThe Theory of InterestExample (2 of 2)Solution:P=6 t=1(At(1+ ) 12t)=6 t=1At( )t Robert BuchananThe Theory of InterestExample: LotteryA lottery has a grand prize of $10M which is paid in tenpayments of $1M annually with the first payment madeimmediately. If the prevailing annual Interest rate is monthly, find the present value of the lottery sgrand Robert BuchananThe Theory of InterestEquivalence of Cash Flow StreamsThe cash flow streamsx={x0,x1,..,xn}andy={y0,y1,..,yn} areequivalentfor an ideal bank if and onlyif the present values of the two streams are Robert BuchananThe Theory of InterestExample: Harvesting a CropSuppose you can stock a pond with fish that you can later sellfor food.

10 Stocking the pond requires an initial outlay of capital,but once stocked the fish and pond are self-sustaining. You canchoose when the harvest the fish, but the longer you wait toharvest, the larger the fish will be. The annually compoundedinterest rate is 5%. If you harvest after one year the cash flowstream is{ 100,200}. If you harvest after two years the cashflow stream is{ 100,0,250}. When should you harvest?J. Robert BuchananThe Theory of InterestGeometric SeriesTheoremIf a6=1then S=1+a+a2+ +an=1 an+11 +a+a2+ +anthenaS=a+a2+ +an+an+1andS aS= (1+a+ +an) (a+a2+ +an+1)S(1 a) =1 an+1S=1 an+11 aJ. Robert BuchananThe Theory of InterestGeometric SeriesTheoremIf a6=1then S=1+a+a2+ +an=1 an+11 +a+a2+ +anthenaS=a+a2+ +an+an+1andS aS= (1+a+ +an) (a+a2+ +an+1)S(1 a) =1 an+1S=1 an+11 aJ.