# 13-2. Annuities Due. Chapter 13. MH Ryerson

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2 13-2 Annuities Due Chapter 13

3 13-3 Learning Objectives After completing this chapter, you will be able to: > Calculate the future value and present value of annuities due. > Calculate the payment size, number of payments, and interest rate in annuities due.

4 13-4 Classification of Annuities Ordinary annuity - regular deposits/payments made at the end of the period Annuity due - regular deposits/payments made at the beginning of the period Jan. 31 Monthly Jan. 1 Jun. 30 Quarterly Apr. 1 Dec. 31 Semiannually Jul. 1 Dec. 31 Annually Jan. 1

5 13-5 Distinguishing Characteristics of Annuity Categories TABLE 13.1 Distinguishing Characteristics of Annuity Categories Annuity Category Is the payment at the end or at the beginning of each payment interval? Compare the payment interval to the compounding interval. Ordinary simple annuity End Equal Ordinary general annuity End Not equal Simple annuity due Beginning Equal General annuity due Beginning Not equal

6 Time Diagram for an n-payment Annuity Due n-2 n-1 n Interval Number PMT PMT PMT PMT PMT PMT PMT PMT PV FV

7 13-7 Future Value of a Simple Annuity Due FV(due) = PMT[(1+ i) n - 1] * ( 1 + i ) i = FV * (1 + i)

8 How much will Elyse accumulate in her RRSP by age 60 if she makes semi-annual contributions of \$1000 starting on her 30th birthday? Assume that the RRSP earns 8% compounded semi-annually and that no contribution is made on her 60th birthday. Financial Calculator sol n: 8/2 = 30*2= /- comp I/Y n PV PMT FV Set your financial calculator to the BGN mode FV(due) = \$

9 How much will Elyse accumulate in her RRSP by age 60 if she makes semi-annual contributions of \$1000 starting on her 30th birthday? Assume that the RRSP earns 8% compounded semi-annually and that no contribution is made on her 60th birthday. Now let s use the formula to answer the same question FV(due) = PMT[(1+ i) n - 1] i * ( 1 + i ) FV (due) =\$ Algebraic Solution:.08 / 2 = sto1 + 1 = sto 2 y x 60 = - 1 = 13-9 / rcl 1 * 500 * rcl 2 =

10 13-10 Present Value of an Annuity Due PV(due) = PMT[1 - (1+ i) -n ] i * ( 1 + i ) = PV * (1 + i)

11 A lottery offers the winner a choice between a \$ cash prize, or quarterly payments of \$7000 beginning immediately and continuing for 20 years. Which alternative should the winner pick if money is worth 8% compounded quarterly? Financial Calculator sol n: 8/4 = 20*4= /- comp I/Y n FV PMT PV Set your financial calculator to the BGN mode PV(due) = \$ The cash prize should be taken - it is worth \$16224 more in current dollars 13-11

12 A lottery offers the winner a choice between a \$ cash prize, or quarterly payments of \$7000 beginning immediately and continuing for 20 years. Which alternative should the winner pick if money is worth 8% compounded quarterly? PV (due) = PV (1+ i) Now let s use the formula to answer the same question ( 1 i) 1 + = PMT i n ( 1+ i).08 / 4 = sto1 + 1 = sto 2 y x 80 +/- = - 1 = +/ / rcl 1 * 7000 * rcl 2 = PV(due) = \$ The cash prize should be taken

13 Mark has already accumulated \$ in his RRSP. His goal is to build it to \$ with equal contributions every 6 months for the next 7 years. If he earns 8.5% cmpd sa, and starts today, find the size of his contributions? Financial Calculator sol n: 8.5/2 = 7*2= comp I/Y n PV FV PMT Set your financial calculator to the BGN mode Payment needed = \$

14 A car sells for \$ The manufacturer offers an interest rate of 1.8% compounded monthly on a three year lease. If the residual value is \$14 500, find the lease payments assuming \$2500 down payment. Purchase price Down = payment Present value of + the lease payments Present value of the residual value \$ \$2500 = Present value of the lease payments Present value of + the \$14 500

15 A car sells for \$ The manufacturer offers an interest rate of 1.8% compounded monthly on a three year lease. If the residual value is \$14 500, find the lease payments assuming \$2500 down payment. 1.8/12 = 3*12 = /- I/Y n PV Set your financial calculator to the BGN mode FV Lease Payment = \$ comp PMT 13-15

16 13-16 Calculating the Number of Payments using the algebraic method n = ln i FV (due) 1 + PMT ln 1 ( 1+ i) ( + i) n = ln i PV (due) 1 PMT ln 1 ( 1+ i) ( + i)

17 13-17 How long will it take to accumulate \$1 million in an RRSP if the first quarterly payment of \$2000 is made today? Assume the RRSP earns 8% cmpd quarterly. 8/4 = / comp I/Y PMT PV FV n Set your financial calculator to the BGN mode payments = 120*3 = 360 months = 30 years + a bit

18 13-18 How long will it take to accumulate \$1 million in an RRSP if the first quarterly payment of \$2000 is made today? Assume the RRSP earns 8% cmpd quarterly. Algebraic Sol n n = ln i PV (due) 1 PMT ln 1 ( 1+ i) ( + i) n = ln i FV (due) 1 + PMT ln 1 ( 1+ i) ( + i) Which do we use? Does this involve FV or PV?

19 13-19 How long will it take to accumulate \$1 million in an RRSP if the first quarterly payment of \$2000 is made today? Assume the RRSP earns 8% cmpd quarterly..08 / 4 = sto1 + 1 = sto2 lnx sto3 rcl1 * / 2000 / rcl2 = + 1 = lnx / rcl3 = n = Algebraic Sol n ln i FV (due) 1 + PMT ln 1 ( 1+ i) ( + i) payments = 120*3 = 360 months = 30 years + a bit

20 A \$ life insurance policy requires an annual premium of \$420 or a monthly premium of \$37. In either case, the premium is due at the beginning of the period of coverage. What is the effective rate of interest charged to those who pay monthly? n Set your financial calculator to the BGN mode 37 +/- PMT comp PV FV I/Y f = (1+i) m - 1 = ( ) 12-1 = = 13.04%

21 13-21 THE END

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