PDF Capital Budgeting and Corporate Objectives

Capital Budgeting and Corporate Objectives

Discounting

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Lecture 0. The Time Value of Money Would you rather receive $1000 today -- or next year? Obviously, you would want the money today. You could bank the money today and it would be worth more than $1000 next year. The most basic principle of Finance is that a dollar today is worth more than a dollar in the future.

We will explore the answer to two questions: "What is the value next year of $1000 received today?" and "What is the value today of $1000 received next year?" The first question asks us to find the future value of money received now. The second question asks us to find the present value of money received in the future.

Before answering these questions, we need to establish some notation:

V0 Present Value

Vn Future Value at the end of n periods

n The number of periods.

R

Nominal interest rate, also called the "annual percentage rate" (APR) or the "stated interest rate"

m The number of compounding periods per year

i Effective periodic interest rate, or effective interest rate per compounding period (i=R/m).

T The number of years; T = n/m

r The effective annual interest rate, also called the true annual interest rate

The effective periodic interest rate, i, equals the nominal interest rate R divided by the number of compounding periods per year.

1

i R

(1)

m

The effective (or true) annual interest rate, r, is the annually compounded interest rate which is equal to the effective periodic rate, i, compounded for m periods.

r

1

im

1

1

R m

m

1

(2)

When interest is compounded continuously.

r eR 1

(3)

Where e is the natural exponent, which is approximately 2.718.

0.1 Future Value -- Single Cash Flow The future value formula is derived by example. Suppose that you have a deposit of $1000 that pays 4% annually. What is the ending balance?

Year Beginning Balance Interest Payment Ending Balance Formula

1

1000.00

2

1040.00

3

1081.60

4

1124.86

5

1169.85

6

1216.64

40.00 41.60 43.26 44.99 46.79 48.67

1040.00 1081.60 1124.86 1169.85 1216.64 1265.31

1000 (1.04) 1000 (1.04)2 1000 (1.04)3 1000 (1.04)4 1000 (1.04)5 1000 (1.04)6

The formula for future value when interest is compounded annually is straightforward from this

example:

Vn V 01 Rn

(4)

2

0.2 Continuous and Discrete Time Compounding Suppose that another bank offers you the same nominal interest rate of 4%, but offers to compound the interest every six months. Would you prefer this deal?

The deposit will pay 2% every six months. The following table shows how this investment will grow:

Year Beginning Balance Interest Payment Ending Balance Formula

0.5

1000.00

1.0

1020.00

1.5

1040.40

2.0

1061.21

2.5

1082.43

3.0

1104.08

3.5

1126.16

4.0

1148.68

4.5

1171.65

5.0

1195.08

5.5

1218.98

6.0

1243.36

20.00 20.40 20.81 21.22 21.65 22.08 22.52 22.97 23.43 23.90 24.38 24.87

1020.00 1040.40 1061.21 1082.43 1104.08 1126.16 1148.68 1171.65 1195.08 1218.98 1243.36 1268.23

1000 (1.02) 1000 (1.02)2 1000 (1.02)3 1000 (1.02)4 1000 (1.02)5 1000 (1.02)6 1000 (1.02)7 1000 (1.02)8 1000 (1.02)9 1000 (1.02)10 1000 (1.02)11 1000 (1.02)12

This investment will be worth an extra $0.40 in the first year, and an extra $2.92 at the end of the

sixth year, because of compounding twice per year. From this table, we can also postulate a

generalized formula for future value, when there are m compounding periods per year.

Vn

V

01

Rn m

V 01

in

(5)

If the interest is compounded continuously, then the formula becomes:

Vn V 0e RT

(6)

3

While the nominal rate on these deposits remains the same, the effective annual rate changes. The effective annual rate can be used to compare investments with different compounding periods.

Example 1 Suppose a bank offers a nominal interest rate of 4% (R = 0.04) on your savings deposit. The following table illustrates the different effective or true interest rates depending on how many times the interest is compounded each year.

Compounding

Formula

Annually

r = (1+0.04/1)1-1

Semiannually r = (1+0.04/2)2-1

Quarterly

r = (1+0.04/4)4-1

Monthly

r = (1+0.04/12)12-1

Weekly

r = (1+0.04/52)52-1

Daily

r = (1+0.04/365)365-1

Hourly

r = (1+0.04/8760)8760-1

Continuous r = e0.04 -1

Effective Annual Rate 4.00000% 4.04000% 4.06040% 4.07415% 4.07948% 4.08085% 4.08107% 4.08108%

So, the investor will always prefer more compounding periods to less. The continuous

time rate of interest is always higher than the periodic interest rate.

0.3 Money Multiplier The term

Rn

Mn 1 m

(7)

for discrete compounding, or

M n eRT

(8)

for continuous compounding, is sometimes referred to as the money multiplier. As the name implies, the money multiplier measures the factor by which your money multiplies in the future,

4

given a nominal rate R and a maturity of n periods. Often, the return on investment depends upon the length of time the money is tied up. Consider a schedule of bank interest rates. The 1-5 year rates were quoted from Wachovia.

Investment Period 1 Year 2 Years 3 Years 4 Years 5 Years 10 Years 15 Years 20 Years 30 Years

Money Rate (R) 8.150 8.200 8.350 8.400 8.500 9.000 9.000 9.000 9.000

Multiplier (Mt) 1.0815000 1.1707240 1.2719989 1.3807566 1.5036567 2.3673637 3.6424824 5.6055107 13.267678

Note that the money market multiplier increases exponentially with longer time to maturity.

Furthermore, the rate of growth depends upon the interest rate.

0.4 Present Value -- Single Cash Flow The Present Value formula can be derived from the formula for the future value. Suppose that we know the future value (Vn) of an investment. The present value of that investment (V0) is easily calculated. From the formula for future value, we know:

Rn

Vn V 01 m

(9)

Divide both sides by the money multiplier to get the present value:

R n

V0 Vn1 m

(10)

For continuous compounding, we have the following formula:

V 0 Vne RT

5

(11)

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