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Management 3 Quantitative Methods. The Time Value of Money Part 1. The Problem. … is a pricing problem . How much does the buyer of today’s dollars pay the lender with future dollars?
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Management 3Quantitative Methods The Time Value of Money Part 1
The Problem • … is a pricing problem. • How much does the buyer of today’s dollars pay the lender with future dollars? • Obviously, the price for $1 present dollar is going to be at least $1 future dollar plus some a few ¢ cents each year.
The Formulation $ Future Dollar = $ Present Dollar x (Something > 1) because the Present Dollar can be lent or invested $ Future Dollar = $ Present Dollar x (1 + the rate of return) $ Future Dollar = $ Present Dollar x (1 + r) for each period $ Future Dollar = $ Present Dollar x (1 + r)tyears
The Basic Relationship The earning power of a dollar in-hand today is based on the Compounding process, written: [1] FV = PV x (1+r)t where there are two $ valuesand two parameters: 1. FV = future value in $dollars 2. PV = present value, the value of $ dollars in-hand today 3. r = an annualrate of return or an interest rate 4. t = number of years between today and the “future”
I have $100 (PV) and if I lend it for 3 years at 10 percent interest. What will I be paid by the borrower - the FV? • $ 130.00 • $ 133.10 • $ 300.00
The answer is $ 133.10. • $ 100.00 is your Principal. • $ 30.00 is simple interest = 10% x $100 3 times. • $ 3.10 is compound interest = • $ 2.00 on the first $10 of simple interest • $ 1.00 on the second $10 of simple interest • $ 0.10 on the first $1.00 of interest.
I have $100 (PV) and I lend it for 3 years at 10 percent interest. Note that the FVF for t = 3 and r = 0.10 is = 1.331
Simple versus compound interest • Simple interest is interest only on the original principal, not on any accrued interest. • Compound interest is interest on simple interest – called interest-on-interest – thus “compounding”.
Symmetry We can reverse the Compounding process by inverting the equation [1] to get: [2] PV = FV x 1/(1+r)t And we can isolate two factors: [1a] from the FV, we get the FVF = (1+r) t [2a] from the PV we get the PVF = 1/(1+r) t The Factor Tables publish these for ranges of “r” and “t”.
Notice that … • FVF’s are > 1 • PVF’s are < 1 And the Tables are inverses of each other.
Derive FV, PV, r, t from the basic equation [1] FV = PV x (1+r)t where (1+r)t is a FVF [2] PV = FV x 1/(1+r)t where 1/ (1+r)t is a PVF [3] r = [FV/PV] 1/t – 1 [4] t = ln (FV/PV) / ln(1+r)
Analyzing Single Amounts • Determine FV given you have a PV, time, and a rate. • Determine PV given a promised FV, time, and a rate. • Determine return “r” for a certain ending FV, starting PV, and duration “t” between the two. • Determine time “t” it will take to earn a return “r”, on a certain ending FV and starting PV.
Derive “r” and “t” from the basic equation [3] r = [FV/PV] 1/t – 1 and [4] t = ln (FV/PV) / ln(1+r)
Solving for “r” What annual interest rate will double money in: 8 years use = 2(0.125) -1 = 9.05 percent
Solving for “t” How long will it take to double your money at an interest rate of: 9 percent Use = ln (2) / ln (1.09) = 0.6931 / 0.0861 = 8.04 years
The Rule of 72 Doubling Time approximately = 72 / annual interest rate
Returns on Investment – what’s the “r”? • Invest $100 and receive $175 three years from now. 1. What is your return? 2. What is your “annualized” return? • Return = (FV/PV)-1 = (175/100) -1 = 1.75 -1 = 75% • Annualized Return = (FV/PV)^(1/t) -1 = (175/100)^(1/3) -1 1.75 ^(0.33) -1 = 1.205 -1 = 20.5%
Check the Calculation • Starting with $100, check the 20% return: $ 100.00 x (1.20) = $ 120.00 in one year $ 120.00 x (1.20) = $ 144.00 in two years $ 144.00 x (1.20) = $ 172.80 in three years Confirming the “r” is about 20 percent
Intra - Period Compounding: more than once per year. • Quarterly compounding. use t = 4x and r = r/4 • Monthly compounding. use t = 12x and r = r/12 • Daily compounding. use t = 360x and r = r/360
One year FVF’s using 12 percent • Quarterly compounding. (1 + .12/4) ^4 = 1.03^4 or 12.55% • Monthly compounding. (1 + .12/12) ^12 = 1.01^12 or 12.68% • Daily compounding. (1 + .12/360) ^360 = 1.00033^360 or 12.75%
If you put $100 in the bank at 5 percent interest compounded daily, then how much will you have your will have after one-year? • $ 105.00 • $ 105.13 • $ 105.36 • $ 105.50
If you put $100 in the bank at 5 percent interest compounded daily, then how much will you have your will have after one-year? I will have $ 105.13 or 13 cents more than with annual compounding
A.P.R. versus A.P.Y. Annual Percentage Rate is the basic r % published for the compounding process. Annual Percentage Yield is the r % resulting from the actual daily compounding therefore, If APR = 5% then APY = {(1+APR/360)^360 }-1 = {(1.000139)^360} -1 = 1.051267 -1 = 0.0513