Time Value of Money

• Present value is a concept that is simple to compute.

• It is useful in decision making ranging from simple personal decisions— buying a house, saving for a child’s education, and estimating income in retirement—to more complex corporate financial decisions—picking projects in which to invest as well as the right financing mix for these projects.

Present Value

1. Individuals prefer present consumption to future consumption. (real interest rate)

2. When there is monetary inflation, the value of currency decreases over time (inflation).

3. A promised cash flow might not be delivered for a number of reasons (risk)

Why is cash worth more now?

• Real Interest rate plus Inflation Rate = Interest rate demanded in the market for a 10year government bond. (Risk free rate of return).

• Today’s Risk free rate of return = US 10 year bond = 2.78% = Real Interest rate (Treasury inflation protected securities, TIPS) 0.63% + Expected inflation 2.15%

• Discount rate = Risk free rate of return + risk premium.

Important Equations

Notation Stands ForPV Present valueFV Future value

CFt

Cash flow at the end of period t

AAnnuity: constant cash flows over several periods

r Discount rate

gExpected growth rate in cash flows

nNumber of years over which cash flows are received or paid

• Say you could receive either $15,000 today or $18,000 in four years. Which would you choose?

• Let's find the future value of $15,000 and

present value of $18,000 if interest rates are currently 4%.

• Future Value of Simple Cash Flow = CF0 (1+ r)t

• Present Value of Simple Cash Flow =

Compound and discount

=     

From the above calculation we now know

our choice is between receiving $15,000 or $15,386.48 today. Or $17,547 or $18,000 in 4 years. Of course we should choose to postpone payment for four years!

Compound and discount

Holding Period Stocks Treasury Treasury

(Years) Bonds Bills

1 $112.40 $105.20 $103.60

5 $179.40 $128.85 $119.34

10 $321.86 $166.02 $142.43

20 $1,035.92 $275.62 $202.86

30 $3,334.18 $457.59 $288.93

40 $10,731.30 $759.68 $411.52

• Effective Interest Rate =

• Say for the above example we get paid 12 monthly payment of a 4% interest rate, what is the effective interest rate?

• 4.07%

Monthly Payments

Effective interest rate

Frequency Rate Days FormulaEffective rate

Annual 10% 1 0.10 10%

Semi- 10% 2 (1 + 0.10/2)2 – 1 10.25%

annual

Monthly 10% 12 (1 + 0.10/12)12 – 1 10.47%

Daily 10% 365(1 + 0.10/365)365 – 1 10.5156%

• 1. What is the present value of $1,000 we expect to receive in 3 years with a discount rate of 10%?

• 2. What is the future value of $10,000 today in 10years with a compound interest rate of 3%, paid yearly?

• What if the interest is paid monthly?

Questions

Cash flow

Payment of $ 3000 at the end of each of next 5 years

$3000 $3000 $3000 $3000 $3000

PV 0 1 2 3 4 5

Which is better?

A. Receiving $1,000 every year for the next five years?

B. Receiving $4,000 today?

C. Receiving $5,500 after 5 years?

The interest rate is 5%.

Present Value of Annuity – Same cash flow every Year

Present Value of Annuity – Same cash flow every Year

• Future Value= $1000*[5.53]= $5525.63

• Present Value

• = $1000*[4.33]= $4329.48

• You get a mortgage loan of 200m won. The interest rate is 0.3% a month. The repayment period is 30 years. Calculate your monthly repayment?

• 909,290 won a month.

Calculating Annuity payments - Mortgage

• You want to save 400m won for your retirement by the time you are 65. How much do you need to save every month if you start saving at 35? The interest rate is 0.3% a month.

• 618,500

Calculating Annuity payments - Pension

Growing Annuities

A(1+g) A(1+g)2 A(1+g)3A(1+g)n

0 1 2 3 n

• A growing annuity is a cash flow that grows at a constant rate for a specified period of time.

Growing Annuity

• Suppose you have the rights to a gold mine for the next twenty years, over which time you plan to extract 5,000 ounces of gold every year. The price of gold is $300 an ounce and you expect it to increase 3 percent a year. The discount rate is 4% What is the present value of all your gold?

Question