Chapter5. Time Value of Money* The Money Rules #1: The more, the better => $2 today better than $1 today #2: The sooner, the better => $1 today better than $1 tomorrow #3: Tradeoff between the amount and the time => $1 today is equivalent to say, $1.06 tomorrow or $.96 today may be equivalent to $1 tom. #4: Choose the point in time first, then try to come up with the equivalent $ amount => $100 today vs $109 tomorrow? Today (Present Value): $100 today vs say $104.64 today’s value (PV) of $109 tom. OR Tomorrow (FV): say, $106 FV of $100 today vs. $109 tomorrow. => NEED PV->FV or FV->PV conversions just like any conversions to compare, add, subtract. e.g, 25 inches vs 2 feet, note an inverse relationship (note the inverse relationship between the two conversion factors, x12 or x(1/12))* TVM Fundamental RelationshipsNote: (1) Interest rate per period vs. APR (nominal rate, rate quoted)e.g., 6% for six months vs. 10% for a year? Which one is more? That is why we want to have the interest rates quoted on the same time length, a year. However, what is the actualinterest earned? For all of our TVM calculations, we use interest rate per period (=periodic rate). How can I get the interest rate per period? Answer=”APR/# of periods ayear” Ex: Quarterly Compounding, 10% APR, 10%/4=2.5% per period(2) Simple interest vs. compound interest (interest on interest) in case of multiple prds? Ex: Semi-annual example, 2 periods (two six months) with 12% APR? What is the interest rate per period=12%/2=6%, What is your total interest for a year based on simple interest calculation method? 6% x 2 =12%. Based on compounding? (1+0.12/2)2 = (1.06)2=1.1236 => 12.36% actual rate of return from compounding method. Yield (APY) =actual rate of return vs. APR (Rate) (3) FV = PV*{(1+I)N} => Given any 3 variables out of 4 variables (FV, PV, I, and N), you can solve the equation for the 4th Sometimes, we use i or r instead of I, and n instead of N, { }=FVIF(I%, N)(4) Timeline! I=interest per period!, N=#periods! (p.124 – p.125)1) Finding FV: $100 deposited for 3 years at 10%, semi-annual compounding? (p.125 – p.130)(1) Using the formula (timeline!)(2) Using the TVM(FV) table, FVIF(I%, N)=Conversion factor of[1$ today->FV] : note!(3) Financial calculator: Check p/Y=1. Clear TVM each time before calculation. Aside: Rule of 72! How long it takes to double your money, #of periods to double my money=72/interest rate per period, Alternatively, you can use this to determine the interest rate required for you to double the moneyEx. 6% per year => 72/6=12 years, 10 years? Approx 7.2%. 7.18% ExactlyEx. How long does it take to triple your money?2) Finding PV (discounting): How much to deposit to have $100 3 years later at 10%, semi-annual compounding? (p.131 – p.133)(1) Using the formula, Note that the inverse relationship, PV=FV*{1/(1+I)N}(2) Using the TVM(PV) table, PVIF(I%, N) = 1/(1+I)N(3) Financial calculator:3) Finding a time period: $100 deposited today and wish to have $370 at 16% with semi-annual compounding, # years? => get # of periods, then 8.5 years (p.134)(1) Using the formula => 370=100*(1+.08)N, # of years=N/2=17/2=8.5(2) Using the TVM(maybe FV) table(3) FC: Note “-“ . Presee CPT, then N4) Finding an interest rate: $100 today and wish to have $1,700 in 12.5 years with semi-annual compounding, interest rate (APR)? => APR=12%*2=24% (p.133 – p.134)(1) Using the formula and Trial & Error, 100=1700*{1/(1+I)25} or 1700=100*(1+I)2525(2) Using the TVM (maybe FV) table(3) FC: Note “-“ . Press CPT, then and i/Y* Multiple Cash Flows 1) Uneven Flows=> Brutal Force (p.143 – p.146): Determine the PV( or FV) of each payment. Then add all the PVs to determine the PV of all the payments.FC: Enter CFo=0, CF1=100, CF2=200, CF3=300, CF4=400, CF5=600 together with I/Y=5%, then determine NPVExcel is an excellent choice in this case.Enter each cell the payment, 100, 200, 300, 400, 600 then enter =npv(5%, address of 100: addressof 600), hit “enter” to get 1,335We can also Excel to back-figure the interest rate to produce the PV given. In other words, determine I/Y given PV and cash flows. In this example, have -1,335 on top of the cash flows (right above 100), then “=IRR(address of -1,335: address of 600)” and hit “enter” to get 5%. 2) Annuity: equal amount for some consecutive periods (p.134 – p.141) (1) Different types – Ordinary (p.135)=each payment is made at the end of each period, Annuity Due (p.135)=each payment is made at the beginning of each period, Perpetuity(p.141 – p.142)=ordinary annuity, but forever (2) Annuity examples (annuity, mortgage payments, lease payments, etc. vs. apartment rent)(2) Analogous example: 3 yards(1760), 3 feet(5280), 3 inches(63360) on the line(3) How to compute FV, PV of an ordinary annuity? (p.135, p.138)Ex1: Save $100 each month for 10 years, at 6%, FV at the end of the 10th year? $16,388How much extra money you have from this = 16,388 – 12,000 =4,388 due to interest.Ex; I would like to receive $100 each month for the next 10 years at 6%. How much do I have to save today? $9,007.35a) Brutal Force = even for annuity, you can deal with each payment, then sum them up to get the total PV or FV. This is inefficient.b) FVIFA(I%, N), PVIFA(I%,N): In our example above, I=0.5%, N=120c) FC: note “-“ PMT, N, I (4) How to compute FV, PV of an annuity due? (p. 137: FV of an annuity due)a) Brutal Forceb) FVIFA*(1+I), PVIFA*(1+I) => since you have each period one period earlierc) FC: Change the setting to BEG => maybe, don’t do itd) Ex2: everything is the same as Ex1, except you pay (or get) $100 at the beginning of each month? PV=100*PVIFA(.5%, 120)*(1.05)=9,052.38. Alternatively, you can look at this as an ordinary annuity with 119 periods and $100 extra today =100*(.5%, 119)+100=9,052.38. How about FV? FV=100*FVIFA(.5%, 120)*(1.05). Alternatively, you can determine the FVbased on the PV you have already determined using the relationship between PV and FV, FV=PV*(1.05)120.(5) How to compute FV, PV of perpetuity? (p.141 – p.142: PV of perpetuity)PV of perpetuity = C/I, C=payment per period, I=interest rate per periodEx: PV of $100 monthly perpetuity at 12% APR= 100/0.01=10,0003) Constant growth rate model(1) What is it? Cn=Cn-1*(1+g), Timeline(2) PV equation and the formula =C1/(I-g)(3) Perpetuity is a special case with g=0% = C/I as we saw earlierEX: Co=$2, g=5%. Then C1=Co*1.05=$2.10, C2=C1*(1+g)=Co*(1+g)2, -
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