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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 3 quantitative methods

Management 3Quantitative Methods

The Time Value of Money

Part 1

slide2

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.
slide3

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

slide4

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”

slide5
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
slide6

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.
slide7

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

slide8

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”.
slide9

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
Notice that …
  • FVF’s are > 1
  • PVF’s are < 1

And the Tables are inverses of each other.

slide16

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
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.
slide18

Derive “r” and “t” from the basic equation

[3] r = [FV/PV] 1/t – 1

and

[4] t = ln (FV/PV) / ln(1+r)

slide19

Solving for “r”

What annual interest rate will double money in: 8 years

use

= 2(0.125) -1

= 9.05 percent

slide20

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
The Rule of 72

Doubling Time approximately = 72 / annual interest rate

returns on investment what s the r
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
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

slide24

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

slide25

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%

slide26
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
slide27

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

slide28

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