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Management 3 Quantitative Methods

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?

nathan-mcneil
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Management 3 Quantitative Methods

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  1. Management 3Quantitative Methods The Time Value of Money Part 1

  2. 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.

  3. 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

  4. 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”

  5. 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

  6. 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.

  7. 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

  8. 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”.

  9. 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”.

  10. Find these on the Tables

  11. Find these on the Tables

  12. Find these on the Tables

  13. Find these on the Tables

  14. Find these on the Tables

  15. Notice that … • FVF’s are > 1 • PVF’s are < 1 And the Tables are inverses of each other.

  16. 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)

  17. 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.

  18. Derive “r” and “t” from the basic equation [3] r = [FV/PV] 1/t – 1 and [4] t = ln (FV/PV) / ln(1+r)

  19. Solving for “r” What annual interest rate will double money in: 8 years use = 2(0.125) -1 = 9.05 percent

  20. 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

  21. The Rule of 72 Doubling Time approximately = 72 / annual interest rate

  22. 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%

  23. 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

  24. 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

  25. 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%

  26. 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

  27. 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

  28. 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

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