Time Value of Money

a dollar today is worth more than a dollar tomorrow

PV = Present Value FV = Future Value r = interest rate t = numbers of time periods 1/(1+r)t = PVIF = PV interest factor

Present Value

110

121

133.1

100 x (1.10)1

100 x (1.10)2

100 x (1.10)3

100

Given ; PV = 100 , i = 10% , t = 1 , FV = ?

Given ; PV = 100 , i = 10% , t = 2 , FV = ?

100

Given ; PV = 100 , i = 10% , t = 3 , FV = ?

100

Calculating Annuity Present Value

It’s just a combination set of PV calculations

PV

PV1

PV2

PV n-1

PV n

i %

A A A A

1 2 n-1 n

Calculating Annuity Present Value

Annuity Discount Factor = PVIA =

PV = A x [ PVIA (r,n) ]

= (A / r ) x [ 1 – ( 1/(1+r)n )

A = PMT = Annuity

Example;

Calculating Annuity Present Value

Given ; A = 100 , i = 10% , t = 3 , FV = ?, PV = ?

A A A

110 = 100(1.1)1

121 = 100(1.1)2

100 = 100(1.1)0

331

248.68 = (100/0.1)x[1-(1/(1.1)3]

F = P(1+i)n

P = (A/i)[1-1/(1+i)n]

248.68 x 1.103 =

Example; a Thai government bond (LB06DA), which will mature in 8 Dec 2006 , give 8% coupon. If we buy this bond on 8 Dec2002 (just after the coupon was paid),by using discount rate at 3%, how much we need to pay?

Remember that Govy bond’s features are

1.Par = 1,000

2.Pay coupon semi-annual

Calculating Bond’s Present Value

Bond’s price = PVIA(1.5%,8) + PVIF(1.5%,8)

= (40/0.015)x[1-(1/(1.015)8] + 1000 / (1.015)8

= 299.437 + 887.711

= 1,187.148

Note ; In Excel, use function “PRICE” by input 1.Settlement date

2.Maturity date 3.Coupon 4.Yield 5.Redemption 6.Frequency

Calculating Bond’s Present Value

12/02

12/06

12/03

12/04

12/05

06/03

06/04

06/05

06/06

40 40 40 40 40 40 40 40

1,000

Number of period = n = 8

i = 1.5% per period

Example Time Value

Q1) Joe’s plan

You are an financial consultant for a farmer named Joe. He just celebrated his 35th birthday yesterday on 31Dec2000. After he retire himself at 65 year olds (31Dec2030),he plan to withdraw 100,000 per year starting from his 65th birthday until his last 100,000 withdrawal at his 85th year (31Dec2055). Joe can find a bank who gives fix rate 12% per year for him during his lifetime. With information above, he ask for your recommendation on how much he need to annually deposit his money in order to accomplish his goal? One more thing, he ready to make the first deposit on his next birthday and will make the last deposit on his 65th birthday.

The process of providing for a loan to be paid off by making regular principal reduction

Calculating Amortized Loans

Calculating Amortized Loans

Year

end

Beginning

Balance

Interest

Paid (9%)

Principle

Paid

Amount

Paid

Ending

Balance

1 5,000 450 1,000 1,450 4,000

2 4,000 360 1,000 1,360 3,000

3 3,000 270 1,000 1,270 2,000

4 2,000 180 1,000 1,180 1,000

5 1,000 90 1,000 1,090 0

1,350 5,000 6,350

Example 1 ; Anna take out a $5,000, five-year loan at 9%. She agreed to amortize $1,000 principle each year on her loan. How much she must pay in each year?

Calculating Amortized Loans Example 2; Mary take out a $5,000, five-year loan at

9%. She agreed to pay $1,000 each year until the loan expired. How many years she will pay off all the loan she took? And how much for the last payment?

Year

end

Beginning

Balance

Interest

Paid (9%)

Principle

Paid

Amount

Paid

Ending

Balance

1 5,000 450 550 1,000 4,450

2 4,450 401 600 1,000 3,851

3 3,851 347 653 1,000 3,197

4 3,197 288 712 1,000 2,485

5 2,485 224 776 1,000 1,708

6 1,708 154 846 1,000 862

7 862 78 862 940 0

1,940 5,000 6,940

5000 =(1000/0.09)*[1-(1/(1.09)^6.9375)]