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Introduction to The Time Value of Money

Introduction to The Time Value of Money. Econ4 173A. The Starting Point. Dollar$ in-hand today are worth more than the same dollar$ a day in the future. In other words, no one would trade $100 today for $100 tomorrow. Discounting & Compounding. We discount promises to pay later .

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Introduction to The Time Value of Money

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  1. Introduction toThe Time Value of Money Econ4 173A

  2. The Starting Point Dollar$ in-hand today are worth more than the same dollar$ a day in the future. In other words, no one would trade $100 today for $100 tomorrow.

  3. Discounting & Compounding • We discountpromises to pay later. • We compound $money we have today. • A dollar now (a) involves no risk of actually receiving that dollar and (b) can be invested to earn a return or lent to earn interest.

  4. Why? A dollar now – is a present dollar “PV”: Involves no risk – you have it; It can be invested to earn a return; You can consume w/ it now, no waiting.

  5. Context Some people have more present dollar$ than they need so they can lend them to People who don’t have present dollars but need them.

  6. The Markets Lenders (banks) & Investors - sell present dollars for promises to be paid something in the future. Borrowers buy present dollars. The interest rate is the annual cost of this.

  7. The Interest Rate • How much buyers of PV’s pay FV’s is the price of consumption today versus waiting to consule in the future. • The price for $1 today is going to be at least $1 plus something. The something is the interest rate. • So we can write that: $ PV x (Something > 1) = $ Future Dollar PV x (1 + r) = FV for every year,

  8. The Basic Relationship “compounding” 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 values and two parameters: 1. FV = future value in $$ 2. PV = present value in $$ 3. r = an annual rate of return, an interest rate 4. t = number of years between today and the “future”

  9. Derive Present Value “discounting” The earning power of a dollar in-hand today is based on the Compounding process, written: [1] FV = PV x (1+r) t FV = PV x FVF To get a future value, you use a FV Factor [2] PV = FV x (1+r) -t PV = FV x PVF To get a present value, you use a PV Factor

  10. Some symmetry We can reverse the Compounding process by inverting the equation [1] to get: [2] PV = FV x 1/(1+r)t so that we can define two factors: [1a] the FVF = (1+r) t and [2a] the PVF = 1/(1+r) t

  11. The Tables We can make a set of Tables – a Matrix on t and r, Future Value Factors = FVFs = (1+r)t and Present Value Factors = PVFs = (1+r) –t You can print these Tables from the webpage.

  12. Notation Formula Factor FVF (10%, 7)‏ (1.10)^7 1.9487 FVF ( 7%, 10)‏ (1.07)^10 1.9672 PVF (10%, 7)‏ 1 / (1.10)^7 0.5132 PVF (7%, 10)‏ 1 / (1.07)^10 0.5083 Find these on the Tables

  13. 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?

  14. 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.

  15. 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

  16. Simple Scenarios The simple borrowing/lending scenario involves trading single sums of money across discrete time periods for single sums in return: • How much should someone pay me in 10 years if I give them $ 1,000 today? Use r= 6%. You know the PV, so you use equation [1] to find the FV. • How much do I need to deposit today to have $100,000 in 10 years. Use 6%. This gives you the FV, so you use equation [2] to find the PV.

  17. Solutions • How much should someone pay me in 10 years if I give them $ 1,000 today? Use 6%. You know the PV, so you use equation [1] to find the FV. FV = PV x (1+r)t FV = 1,000 x (1.06)10 FV = 1,000 x 1.79084 $1,790.84 is the answer

  18. Solutions • How much do I need to deposit today to have $100,000 in 10 years. Use 6%. This gives you the FV, so you use equation [2] to find the PV. PV = FV x (1+r)-t PV = 100,000 x (1.06)-10 PV = 100,000 x 0.558398 $ 55,839.90 is the answer

  19. 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”.

  20. Intra - Period Compounding: more than once per year. • Quarterly compounding. use tQ = 4x and rQ = r/4 • Monthly compounding. use tM = 12x and rM = r/12 • Daily compounding. use tD = 360x and rD = r/360

  21. 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% Repeat this using two years

  22. If you put $100 in the bank at 5 percent interest but compounded daily, then how much will you have your will have after one-year?

  23. You will have $ 105.13 i.e. you will earn an extra 13 cents

  24. A.P.R. versus A.P.Y. Annual Percentage Return is the basic r% annual 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

  25. Derive 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)

  26. Finding “r” What annual interest rate will double money in 8 years; this is a rate question, so use equation [3] r =(FV/PV)^(1/t) -1 You want the “r” that makes FV = 2xPV, so r = 2(1/8) -1 = 9.05 percent

  27. Solving for “t” How long will it take to double your money at an “r” = 9 percent; this is a “t”, or a time question, so Use equation [4] t = ln (FV/PV) / ln(1+r) = ln (2) / ln (1.09)‏ = 0.6931 / 0.0861 = 8.04 years

  28. The rule of 72 Doubling Time approximately = 72 / annual interest rate

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