Present-Value-Of-An-Ordinary-simple-Annuity.ppt

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,Chapter 10 Ordinary Annuities:Future Value And Present Value,10.3 Present Value Of An Ordinary simple Annuity,1,Present value of an annuity,:It is the single amount,at the beginning of the annuity,that is economically equivalent to the annuity.,2,Present Value Using The Algebraic Method,0,1,2,3,4,n-1,n,R,R,R,R,R,R,P,1,P,4,P,3,P,n-1,P,n,P,2,3,The formula used to evaluated the present value for an ordinary annuity is:,A,n,represents the present value of an ordinary annuity,R represents the value of the payments,P represents the interest rate per payment interval,n represents the number of payments in the term,(10-4),4,The factor is called the,discount,factor for annuities,or the,present-value,factor for annuities,.,5,Present value using the financial calculator functions,The correspondence between the key labels and the algebraic variables in formula(10-4)is:,=n,the number of payments in the term,=p,the interest rate per payment interval,=R,the size of the annuity payment,=A,n,the present value of the annuity.,n,1/y,PMT,PV,6,n,+/-,PMT,1/y,FV,PV,results,$500,$500,$500,$500,$500,+($1200,),P,1,P,2,P,4,P,3,P,5,P,0,1,2,3,4,5,A,n,n,R,p,S,CPT,7,Applications Of The Present Value Of An Annuity,10,The Marked Value Of An Annuity,Valuation principal,:The fair market value of an investment is the sum of the present value of the expected cash flows.The discount rate used in the present-value calculations should be the marked-based rate of return required for this type of investment.,11,Fair market,value of an,annuity,Present value of,the annuity,payments,(discounted at the,market rate of,return),=,12,Example,A certain investment pays back$50 at the end of every 6 months for 17 years.At the end of the 17 years,the investment pays back$1000 in addition to the regular$50 payment.What is the present value of all of the payments if money can earn 8.5%compounded semiannually?,13,for the annuity,R=$50 term=17 years,payment interval=6 months,j=8.5%compounded semiannually,p=i=j/m=8.5%/2=4.25%(per 6 months),n=,Solution:,14,For the additional$1000 at the end of 17 years,S=$1000 n=34 i=4.25%(per 6 months),the combined present value,=S+A,n,=$1133.61,The present value of all of the payments is$1133.61.,15,Financial calculator method,The present value of all of the payments is$1133.61.,1000,50,4.25,34,-1133.61,FV,PMT,1/Y,n,PV,CPT,16,Example,A two-level retirement annuity will pay$2000 per month for the first 10 years and$3000 per month for the next 15 years.What is the present value,1 month before the first payment,of the entire stream of annuity payments?Money can earn 7.5%compounded monthly.,17,0,1,2,120,300,0,10,25,month,year,$2000,$2000,$2000,$3000,$3000,A,n1,Annuity 2,Annuity 1,P*,A,n2,Sum of A,n2,and P*is the present value of payments,$3000,j=7.5%,compounded,monthly,18,Step 1:,calculate the present value of annuity 2 at 120th month.,Term=15 years R=$3000,j=7.5%compounded monthly,payment interval=1 month,n=,(per month),19,0,3000,180,0.625,FV,PMT,n,1/y,PV,-323,620.28,CPT,20,Step 2:,calculate the present value of A,n1,at the beginning of the two-level retirement annuity,S=A,n1,=$323,620.28 term=10 years,i=j/m=0.625%(per month),21,Step 3:,calculate the present value of annuity 1 at the beginning of the two-level retirement annuity,R=$2000 term=10 years,p=i=0.625%(per month),payment interval=1 month n=120,=$168,489.49,22,Step 4:,work out the sum of A,n2,and P*,The combined present value=A,n2,+P*,=$321,714.10,The combined present value is$321,714.10.,153,224.61,2000,120,-321,714.10,FV,PMT,n,PV,1/Y,0.625,CPT,23,Example,Crazy Eds Furniture Mart held a promotion for purchases exceeding$1000.Customers can pay six equal monthly payments with no interest charged.On such installment sales,the customer signs a conditional sale contract.Crazy Eds sold the conditional sale contracts to a finance company.The finance company purchased the contracts at a discounted price so that the six payments would provide a rate of return on the purchase price of 15%compounded monthly.What did it pay for a$1200 contract(six payments of$200)?,24,Solution:,n=6 ,R=$200,j=15%compounded monthly,compounding period=payment interval,p=i=1.25%per month,The discounted price,The present value of,the annuity,=,25,Financial calculator solution,The finance company paid$1149.20 for the contract.,PMT,FV,PV,1/y,n,-1149.20,200,0,1.25,6,CPT,26,Loan Balance And Selling Price,Loan Balance,Original loan,Present value of,all the payments,(discounted at the,contractual rate,of interest on,the loan),=,27,The balance owed on a loan at any point,principal balance,Present value of the remaining,payments,(discounted at the contractual,rate of interest on the loan),=,28,Selling Price(Or Proceeds From The Sale)Of A Loan Contract,selling price of a loan contract,present value of the remaining,payments(discounted at the,negotiated rate of return,),=,29,Example 10.3F,The monthly payment on a 5-year loan with interest at 9%compounded monthly is$249.10.,A,what was the original principal amount of the loan?,B,what is the balance owed just after the 17th payment?,30,In this case,j=9%compounded monthly,i=9%/12=0.75%(per month),payment interval=1 month,p=i=1%(per month),Solution:,31,A,what was the original principal amount of the loan?,R=$249.10 p=1%term=5 years,n=5*12=60,substituting them to formula(10-4),get,32,PMT,FV,n,1/y,PV,-11,999.99,249.10,0,60,0.75,The original principal of the loan is$11,999.99,CPT,33,B,what is the balance owed just after the 17th payment?,There are 60-17=43 payments remaining.,The balance owed=present value of the,remaining payments,=,=,=$9126.76,34,n,PV,43,-9126.76,The balance after 17th payment is$9126.76.,CPT,35,Example,What will be the proceeds from the sale of the loan in Example 10.3F just after the seventeenth payment,if the purchasers required monthly compounded rate of return is 10.5%?,36,Solution:,j=10.5%compounded monthly,p=i=j/m=10.5%/12=0.875%,R=$249.10 n=43,PMT,FV,n,1/y,PV,-8894.86,249.10,0,43,0.875,The proceeds will be$8894.86.,CPT,37,The Economic Value Of An Annuity,The economic value of a payment stream on a particular date(focal date)refers to a single amount that is an economic substitute for the payment stream.,The economic value of an annuity at the beginning of the annuity is just its present value(using the time value of money as the discount rate).,38,Current,economic,value of,an annuity,Present value of the,annuity payments,(discounted at a rate of,return that money can earn,on low-risk investments),=,39,Example,An eligible individual may elect to start collecting the Canada Pension Plan(CPP)monthly retirement benefit at any time between the ages of 60 and 65.The payments are then reduced(from the amount that would otherwise be paid after age 65)by 0.5%for each month before age 65.For example,if the retiree starts 5 years early,at age 60,the,40,monthly payment will be decreased by(60months)*0.5%=30%.The reduction is permanent,extending to payments after age 65 as well.,The average life expectancy of a woman aged 60 is another 22 years.If a woman aged 60 lives just the expected 22 years,compare the economic values at age 60 of the two alternatives:,1,collect a 100%pension from age 65.,2,collect a 70%pension from age 60.,Assume that money is worth 6%compounded monthly.,41,Solution:,j=6%compounded monthly,payment interval=1 month p=i=0.5%,Annuity 1:,if a woman aged,60 lives just the,expected 22 years,she get 100%,pension from age,65.,Annuity 2:,if a woman aged,60 lives just the,expected 22 years,she get 70%,pension from 60,years old.,42,age,60,65,82,R,R,R,R,R,R,A,n1,PV,age,60,82,0.7R,0.7R,0.7R,0.7R,0.7R,0.7R,A,n2,0.7R,Annuity 2,Annuity 1,43,For annuity 1,82-65=17 years n=17*12=204,p=i=0.5%R=$1000,44,PMT,FV,PV,n,-127,679,1000,0,0.5,204,PMT,+/-,PV,n,-94,671,0,60,FV,1/y,0.5,The present value of annuity,1 is$94,671.,1/y,CPT,CPT,45,n=(82-60)*12=264,For annuity 2,PMT,FV,PV,n,-102,478,700,0,264,1/y,0.5,The present value,of annuity 2,is$102,478.,A,n2,PV,CPT,46,