# By the definition of regular annuities where the

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By the definition of regular annuities, where the first paymentstarts one period in the future or at the end of the current year, by reverse logic, thisimplies that the PVA is one period from the first payment.What this means is that thePVA is in period 2, i.e., one period prior to the first payment.This implies we have todiscount this annuity from period 2 to period 0 as follows:PV = 100 / (1.12) + 150 / (1.12)2+ 325[1 – (1.12)-6/ .12] / (1.12)2= \$1274.08Note that21.1336\$12.)12.1(13256and we then discount this two periods.Now,we can combine the two terms, the 150 and the annuity value of 1336.21 over onecommon denominator discounting for 2 periods:08.1274\$12121.13361501211002..PVThis will be used later in the semester, not now, but you should try to understand this.2) Financial Calculator:Use the cash flow keys (make sure you hit 2ndand C ALL andhave no BEGIN in screen)PeriodCF0123-8This last step means I am putting in a \$325 Cash Flow and the Number (Nj) of these cashflows are 6.Note, that I could have just hit the CF button 6 times after typing in the 325.This is what I normally do because I usually forget the sequencing of these things andhave to experiment.After entering the cash flows enter the interest rate: 12, i/yrNow hit the Net Present Value (NPV) key, which is the 2ndand then theNPVPRCbuttonNPV = \$1274.08Calculating Interest Rates16
1) Lump Sum PV and FVFrom PV and FV we know the basic equations:n11rPVVorFrFVPVnnnNow we need to solve for r in one of these, so in FV we divide through by PV:by PVrPVFVnn1nnrPVFV1Raise both sides to the power ofn1rPVFVnn1/1Now get r:BLr = [FVn/ PV]1/n- 1eg. \$100 deposited for 5 years, we end up with \$248.9, what is the rate?r = [ 248.9 / 100]1/5– 1 = 1.20 – 1 = .20Note: This is also the way to get a rate of growth of a PV into a FV.In the Financial Calculator:N= 5PV= -100FV=248.9i=??=20%Finding the Interest Rate in AnnuitiesrrAPVAn110Trial and Error17
We cannot solve the annuity formula for r because there is an r in both the numerator anddenominator, so we have to use trial and error, i.e., by just plugging in different valuesand seeing if the PVA equals the PV of the cash flows in the formula.eg. \$500 per year for 5 years which has a1676PVA1) Formula:From the annuity formula:4.1895\$1.37907.500;1.1.11500??1676:10%Try115rrAPVAnSo4.1895\$PVAbut we were trying to get to 1676.So, we conclude thatris too low,we did not discount the 500 cash flows enough.Therefore, we need to try a largerdiscount rate.Try 20%:3.1495\$99.25002.2.11500??16765So now we discounted the 500 cash flows too much and we conclude thatris too high.So again we need to try something in between 10% and 20%.Try 15%:1676352.350015.15.11500??16765So this 3rdtime we luckily narrow in on the answer.2) Fin. Calculator(Note: either the PV or the PMT must be entered as a negative)N= 5PV= -1676FV= 0PMT= 500i=??= you will see that your calculator is “running” but then gives 15Findingrin uneven cash flows:Using the original uneven cash flow example above, we know the PV and the cash flowsand we want to find the interest rate.YearCFPV= 440.19110018
215033251) Formula

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