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BUS 249 – FALL 2018 - Chapter 13: Annuities Due, Deferred Annuities, and Perpetuities Objectives: After completing chapter thirteen, the student will be able to: · Compute the future value, present value, periodic payment, term, and interest rate for simple annuities due. · Compute the future value, present value, periodic payment, term, and interest rate for general annuities due. · Compute the future value, present value, periodic payment, term, and interest rate for ordinary deferred annuities. · Compute the future value, present value, periodic payment, term, and interest rate for deferred annuities due. · Compute the present value, periodic payment, and interest rate for ordinary perpetuities, perpetuities due, and deferred perpetuities Annuities Due Simple vs. Ordinary Simple Annuities Due “Due” means that the payments are made at the ___beginning__________ of the payment interval We can identify an annuity due by wordings such as: -payable in advance -payments (or deposits) made at the beginning of each (or every)….. -first payment is due…now……….. Applications: Leases and rental (1st payment is due immediately) “Simple” means that the payment interval and interest conversion interval are the sameeee___________________________________ If they are not equal = Find the FV of $100 payments at 6% compounded annually made at beginning of the year for four years Formula for FV of Simple Annuity Due Applying the FV Formula for the Previous Example $100 payments at 6% compounded annually made at beginning of the year for four years

$N 1 2 3 F4 1o V Calculating 0w the FV of a Simple Annuity Due FV = Sum of the above FV’s = 0 Or one formula to get $463.71 Basically it is =) FV OSA * (1+i) Applying the FV formula for the previous example:

If $100 payment at 6% compounded annually made at beginning of the year for four years: FV (due)= 100*[(1+0.06)4-1]/0.06 *(1.06)= 100*4.3746*1.06=463.70

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Calculating the FV of a Simple Annuity Due: If you deposit $100 at the beginning of each month for five years in an account paying 4.2% compounded monthly. Find the balance at the end of five years: FV (due)= 100*[(1.0035)60-1]/0.0035 *(1.0035) =100*66.6359*1.0035 = 6686.92 I=42%/12=0.35% n=5*12=60

Calculating Interest Earned How much interest will the account have earned over the five years? Since the value of the contributions to the account totaling = $100 x 60 = $6,000 (Principal only) Accumulated value (as per calculated above) = 6,686.92 (P+I) Thus, interest earned = (P+I) – P = 6,686.92 – 6,000

=$686.92

Find the PV of $100 payments at 6% compounded annually made at beginning of the year for four years

$N o 1 0w P 0 V 1 2 3 PV = SUM of above PV’s calculation = $ Formula for Present Value of Simple Annuity Due

Again, same as FV formula (above), it is = Present Value of ordinary simple annuity x (1+i) Applying the PV Formula for the Previous Example = 100*[1-(1.06)-4]/0.06*(1.06)=367.30

$100 payments at 6% compounded annually made at beginning of the year for four years

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PV = same as above = $367.30 Calculating PV of an Annuity Due Payments of $425 are made at the beginning of each quarter for 10 years. If money is worth 6% compounded quarterly, find the present value of the payments: Components: Beginning = Annuity Due Payment = quarterly and interest is also quarterly = Simple Look for PV? PV of Simple Annuity Due PMT =425, I = 6%/4 = 1.5%, N=10 x 4 = 40 PV of simple annuity due = 425*[(1-(1.015)^-40]/0.015*1.015 =425*29.9158*1.015=12904.95

Finding PMT for a Simple Annuity Due PMT is found the same way as we did for Simple Annuities. Just remember to include the (1+i) in your calculations. Finding n for a Simple Annuity Due The is also done the same way as we did for Simple Annuities. Include the (1+i) additional factor in your calculations. E g.: What is the balance of beginning of quarterly deposit of $300 at 5% compounded quarterly after 16 yrs.? interest&payment: both quarterly after 16 years=FV FV due: PMT=300, I=5%/4=1.25% N=16*4=64 FV due = 300*(1.0125)^64-1/0.0125 * (1.0125) =300*97.16259*1.0125 = 29513.14

CLASS EXAMPLE: Exercise 13.1 #2: Accumulate Value = FV interest = quarter vs payment = quarter FV simple (due) PMT = 360, I=7%/4=1.75, N=12*4=48 FV (due)=360*[(1.0175)^48-1/0.0175]*(1.0175) =360*74.262784*1.0175 = 27202.46

#3: a) FV, beginning of quarter = due PMT=530, I=3.92%/4=0.98%, N=4*4=16 FV (due)=530*[(1.0098)^16-1]/0.0098 *(1.0098) = 530*17.23*1.0098 = 9221.39 b) interest? total payment (P+I)-total contributions ℗ 9221.39-(530*16)=9,221.39-8480=741.39

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c) wait for additional year: N now = 16+4=20 530*[(1.0098)^20-1]/0.0098 *(1.0098) = 530*21.97619*1.0098 = 11,761.53 cuanto mas 11,761.53-9,221.39 = 2,539.32

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#8: Self-study for final exam. a) $116,235.83; b) $55,800 and c) $60,435.83 Up to now: Simple annuity due is done, next for GENERAL annuity due General Annuities Due

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General Annuities Due are identical to Simple Annuities Due, the only exception being that the interest conversion periods are not the same as the frequency of payments. Because of this you use the same formula you used for General Annuities, adding ____________ The solving for general annuiteis due is the same as it was for General Annuities. Just remember the additional (1+p) in the formula. This applies to solving for Future Value, Present Value, PMT, n and i. CLASS EXAMPLES EXERCISE 13.2 #1: after 5 years: FV - beginning: due - payments at 6 most vs interests quarterly = general FV general annuity due: PMT = 5000, I = 6%/4=1.5%, N= 5*2= 10 C=4/2 = 2 P=(1+0.015)^2-1 = 1.030225 - 1 = 0.030225 FV (general due) = 5000*(1.030225)^10-1/0.030225 *(1.030225) =5000*11.475765*1.030225 = 59,113.10

#4:

Self-study for final exam….= $26,632.40

Deferred Annuity A deferred annuity is one in which the first payment is made at a time ___l8r than_________________ the end of the first payment interval. Time period from “now” to starting point of term of annuity is called the period of deferment. The

symbol d is used to represent the number of compounding periods in the period of deferment.

Deferred Annuity Types As discussed earlier, we can have ordinary deferred annuities (ODA) (both the simple and the general cases), and deferred annuities due (DAD)(again with both the simple and general cases being possible) or simple (I+P: same interval) or general (I&P: not same interval) therefore: (SODA) or GODA or SDAD or GDAD (in each case: FV or PV) Present Value of a Deferred Annuity Step 1: Find the PV of the annuity at this point. You can consider it as an ordinary annuity (in this case 3 payments). Step 2: Find the PV of the value found in Step 1 (the PV of the annuity). Treat this value as a lump sum.

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Finding the PV of a Deferred Annuity:

CLASS EXAMPLES: EXERCISE 13.3 #3: how much borrow $ now = PV End of every 3 months = ordinary Payment/quarter vs interest/quarter PMT = 8500, I=9%/4=2.25%, N = 8*4 = 32 PV of simple ordinary annuity = 8500*(1+(1.0225)^-32)/0.0225 = 8500*22.637674 = 192,420.23 But payments are deferred for 3 yrs = D = 3*4=12 PV = 192,420.23*(1.0225)-12=192,420.23*0.7657 = 147,329.92

pv oda

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look for PMT

FV: next 4 years Deferrel? = D = (5.5*12) = 66 I = 4.2%/12 = 0.35%, N = 4(12)=48 PV now =8000 She saves $ for 66 periods, then start monthly withdrawal FV=8000*(1.0035)^66 = 8000*1.259351 = 10,074.81 After 66 periods = PV = 10,074.81 for monthly withdrawal 10,074.80983 = PMT [1-(1.0035)^-48]/0.0035 *PMT

PMT = 228.38

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Looking for N - how long? PMT= 500, end = not annuity due (no (1+i), payment at quarterly vs 4% quarterly PV = $4000, I=4%/4 = 1% N= 5(4)=20 FV = 4000(1.001)^20 = 4000(1.220190) = 4880.76016 for withdrawal PV = 4880.76016, PMT = 500, N =?, I = 1% 4880.76016=500[(1-1.01)^-n]/0.01 9.761520 = (1-1.01^-n)/0.01 0.902385 = 1.01^-n or ln0.0902385 = -n ln1.01 or -0.102714 = -n 0.009950 or n= 10.322697/4 = 2.581 yrs

(deferrls + PV of OSA)

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CLASS EXAMPLES: GENERAL DEFERRED ANNUITY DUE – EXERCISE 13.4 payment: beginning + int/payment intervals are not the same, D is given on either PV or FV either PVGDAD or FVGDAD

#2. size of payment = PMT PV = 14,000, i = 6.5%/2 = 3.25%, N = 3(2) = 6, 1st payment due 4 yrs from now : D = 4(2) = 8 borrow 14,000 deferred to 8 periods = FV = 14,000*(1.0325)^8 = 14,000 * 1.291578 = 18,082.085 after 4 yrs: 18802.085 starts semi annual payments: 1st payment is due 4 yrs = beginning of 5th year = annuity due FV (due) = 18,082.085 = PMT * [1-1.0325)^-6]/0.0325*(1.0325) 18,082.085 = PMT*1.0325*5.372590 or PMT = $3259.68

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#5 looking for n = 13.660591 quarters or 3.42 yrs

#10: due, deferral a) today: PV? PMT = 400, I=6%/12 = 0.5%, N = 4(12)= 48 I&P: both monthly: simple = PV of simple annuity due PV (due) = 400*(1.005)*[(1-1.005)^-48]/0.005 = 400*1.005*42.5803 = 17,117.29 Since money deposit today will have withdrawal 7.5 yrs later, deferment (D) = 7.5(12) =90 PV now = 17,117.29*(1.005)^-90 = 17,117.29*0.638344 = 10,926.71 b) total withdrawals: 400*48 = 19,200 c) Interest = b - a = 19200-10926.71 = 8273.29

Perpetuity Periodic payments begin on a fixed date and continue indefinitely. Since there is no end to the term, it is not possible to determine the FV of a perpetuity. For example: PV of a sum of $ in future (with VERY long period) = close to $0 e.g. $1,000 in 1,000 years later….PV = 1,000/(1.10)1000 = close to $0 Value of perpetuity = PV of continuous PMT

Examples of Perpetuities · · ·

preferred shares

Dividends on Interest payments on _______ permanent basis______________________________(e.g. your scholarship = interest $) Consolidated annuities

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Graph of Ordinary Perpetuity · PV = the present value of the perpetuity · PMT = the periodic rent (or perpetuity payment) · i = the rate of interest per conversion period · p = the effective rate of interest per payment period.

PV of Ordinary Simple Perpetuity > not in final

As n in the formula increases and approaches infinity, the factor (1+i)-n approaches 0.

Present Value of Ordinary General Perpetuity

Finding the PV of a Perpetuity E.g. Find the amount of money invested today at 6% compounded annually which will provide a scholarship of $1,200 at the end of every year: PV? PV of Ordinary Simple Annuity

=

Finding the PV of a General Perpetuity E.g. What sum of money invested today at 6% compounded quarterly will proved a scholarship of $1,000 at the end of each year? General annuity when I at Quarterly but P at annually

PV of Simple Perpetuity Due

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PV of General Perpetuity Due

CLASS EXAMPLES PERPETUTITIES DUE: EXERCISE 9.5 #2

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SUMMARY:

ASSIGNMENT FOR CHAPTER 9: REVIEW EXERCISES QUESTIONS #2 & 5

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