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LEVEL 1: QUANTITATIVE METHODSReading 1 (1st out of 7): TIME VALUE OF MONEYDifficulty:mediumBenchmark Study Time:2022https://soleadea.org/3.75h

L1, QM, R1: TIME VALUE OF MONEYSINGLE CASH FLOWFuture value of single cash flow (lump sum)FVN PV (1 r)NPV – present value of the investment,FVN – future value of the investment N periods from today,r – rate of interest per period (periodic interest rate).CompoundingCompound interest is about adding interest accrued after the end of each compounding period to the principal. Eachconsecutive period we invest a greater amount and so we receive greater interest.the more frequent the compounding, the higher the future value,the higher the present value, the higher the future value,the higher the number of periods, the higher the future value,the higher the interest rate, the higher the future value.Present value of single cash flow (lump sum)PV FVN(1 r)NPV – present value of the investment,FVN – future value of the investment N periods from today,r – rate of interest per period (periodic interest rate).the more frequent the compounding, the lower the present value,the higher the future value, the higher the present value,the higher the number of periods, the lower the present value,the higher the interest rate, the lower the present value.6Copyright soleadea.org

L1, QM, R1: TIME VALUE OF MONEYTypes of interest ratesStated annual interest rate (rs ) quoted interest rate,rPeriodic interest rate (ms) stated annual interest rate divided by the number of compounding periods peryear,Effective annual rate (EAR).Effective annual rate (EAR)EAR 1 rsmm 1rs – stated annual interest rate,m – number of compounding periods in one year,rs– periodic interest rate.m8Copyright soleadea.org

L1, QM, R1: TIME VALUE OF MONEYANNUITIESDefinitionsAn annuity is a series of cash flows of the same value occurring at equal intervals. One type of annuity is, i.e. an annuity in arrears, where the first payment is madeannuity is. Annuity due is when cash flows occurat the beginning of each month. The third type of annuity isof the first period. Another type ofof each period of an investment, e.g., aka. a never-ending sequence of futurepayments. Perpetuity can be perceived as a perpetual ordinary annuity because payments are made at the end ofeach period.Future value of ordinary annuityFVO A (1 r)N 1 A (1 r)N 2 . . A (1 r) AN 1(1 r)i ) A (FVO A (i 0(1 r)N 1)rFVO – future value of ordinary annuity,r – periodic interest rate,N – number of periods,A – annuity amount (annuity payment),(1 r)N 1(r)– future value annuity factor.Future value of annuity dueFVD A (1 r)N A (1 r)N 1 . . A (1 r)N(1 r)i ) A (FVD A (i 1(1 r)N 1) (1 r) FVO (1 r)rFVD – future value of annuity due,r – periodic interest rate,N – number of periods,A – annuity amount (annuity payment).10Copyright soleadea.org

L1, QM, R1: TIME VALUE OF MONEYPresent value of ordinary annuityPVO AAA . . 2(1 r) (1 r)(1 r)NNPVO A (i 111 1(1 r)N) A ()(1 r)irPVO – present value of ordinary annuity,r – periodic interest rate,N – number of periods,A – annuity amount (annuity payment),1 (1(1 r)Nr) – present value annuity factor.Present value of annuity duePVD A N 1PVD A (i 0AAA . . 2(1 r) (1 r)(1 r)N 111 1(1 r)N) A () (1 r) PVO (1 r)(1 r)irPVD – future value of annuity due,r – periodic interest rate,N – number of periods,A – annuity amount (annuity payment).12Copyright soleadea.org

L1, QM, R1: TIME VALUE OF MONEYPresent value of perpetuityPVP ArPVP – present value of perpetuity,A – annuity amount (annuity payment).When using the above formula, we get the present value of a perpetuity one period before the first annuitypayment. For example, if the perpetuity consists of regular annual payments and the first payment is one year fromnow, the present value we get is for today. But if the perpetuity consists of regular annual payments and the firstpayment is three years from now, the present value we get is two years from nowif the perpetuity consists ofregular annual payments and the first payment is ten years from now, the present value we get is nine years fromnow.PV of perpetuity is always one period before the first payment!So, if you are asked to compute the present value of perpetuity on T 0 (now) and the perpetuity consists of regularannual payments and the first payment is three years from now, you’ll use the following adjusted formulaA(adjustment discount the PV of perpetuity r two years back):PV(T 0)Ar (1 r)2Algorithm for solving TVM problemsAfter reading a question:determine what you need to calculate,establish whether you are dealing with a single payment or an annuity or a series of unequal cash flows,in the case of annuity, check if it is an ordinary annuity or annuity due (payment at the end or at the beginningof periods) or perpetuity,,apply the formulas and – using a calculator – solve the problem (use either direct formulas or TVM orCF NPV IRR worksheets),the combination of CF NPV IRR worksheets is especially useful when dealing with a series of unequal cashflows e.g. there are cash flows in year 1, year 3, and year 6 then: for years 1, 3, and 6 enter the cashflows given and for years 2, 4, and 5 input CF 0 next, use the NPV worksheet to compute the presentvalue (NPV) or the future value (NFV).interpret the results when necessary.14Copyright soleadea.org

L1, QM, R1: TIME VALUE OF MONEYSummarizing key concepts: Interest rates: 3 interpretations Components of interest rates Future value & present value of single cash flow (lump sum) Compounding16Copyright soleadea.org

L1, QM, R1: TIME VALUE OF MONEY Types of interest rates Future value & present value of ordinary annuity Future value & present value of annuity due Present value of perpetuityCopyright soleadea.org17

L1, QM, R1: TIME VALUE OF MONEYReviewing formulas:FVN PV (1 r)NPV FVN(1 r)NEAR 1 rsmm 1FVO A (1 r)N 1 A (1 r)N 2 . . A (1 r) AN 1(1 r)i ) A (FVO A (i 018(1 r)N 1)rCopyright soleadea.org

L1, QM, R1: TIME VALUE OF MONEYFVD A (1 r)N A (1 r)N 1 . . A (1 r)N(1 r)i ) A (FVD A (i 1PVO AAA . . 2(1 r) (1 r)(1 r)NNPVO A (i 1Copyright soleadea.org(1 r)N 1) (1 r) FVO (1 r)r11 1(1 r)N) A ()i(1 r)r19

L1, QM, R1: TIME VALUE OF MONEYPVD A N 1PVD A (i 0AAA . (1 r) (1 r)2(1 r)N 111 1(1 r)N) A () (1 r) PVO (1 r)(1 r)irPVP 20ArCopyright soleadea.org

L1, QM, R1: TIME VALUE OF MONEYKeeping myself accountable:When you sit down to study, you may want toto handle your study sessions: study for 25 minutes,then take a 5-minute break. Repeat this 25 5 study-break sequence all throughout your daily study session.Tick off as you proceed.POMODORO TIMETABLE: study-break sequences (25’ ��25’5’5’5’5’5’5’5’Never ever neglect revision! Though it’s not the most popular thing among CFA candidates, regular revision is what makesthe difference. If you want to pass your exam,REVIEW TIMETABLE: When did I review this tedatedatedateCopyright soleadea.org21

flows e.g. there are cash flows in year 1, year 3, and year 6 then: for years 1, 3, and 6 enter the cash flows given and for years 2, 4, and 5 input CF 0 next, use the NPV worksheet to compute the present value (NPV) or the