Future Value

1 -Ahmad plans to retire in fifteen years. Can he afford a $250,000 condominium when he retires if he invests $100,000 in a fifteen-year Mellon CD (certificate of deposit) which pays 7.5%

interest, compounded annually?

Solution:

Yes, he could afford to purchase the condominium since he should have $295,887.74

when he retires .

$295,887.74 = $100,000) 1.075(15 .

2 -Can Ali afford the condominium if he purchases three consecutive five-year CD’s? The current five-year rate is 6%. Rates for the second and third five-year periods and expected to be 6.5% and

7.5%, respectively .

Yes, he can still afford it :

FV = 100,000 (1.06)5(1.065)5(1.075)5

FV = 100,000 (1.3382)(1.37009)(1.43563)

FV = 100,000 (2.6322)

FV = 263,216 .

3 .What is the future value of $26 billion invested by UAE for 325 years at an average rate of return of 7%? (In this context, did the UAE make a poor decision to sell Jabal-Ali Area to the Saudi Arabia?)FV = 26(1.07)325

FV = 9.2194 x 1010 = $92.194 billion

If the UAE had invested at a average annual rate of 7%, they would have over $92

billion after 325 years .

Time Value of Money Review

Future Value of an AnnuityThe Future Value of an Annuity tells you how large a sum a stream of even payments will accumulate to in a given time given an investment rate and a compounding frequency. The formula is:

mrmr nm

/

11*Payment Annuity of Value Future

*

Time Value of Money Review

Future Value of an AnnuityExample: Assume that you want to save for a house. You plan on depositing $250 each month into an account which pays 8% per year with annual compounding. How much will you have after 10

years?

51.736,4512/08.

11208.

1*$250 Annuity of Value Future

120

Sinking Funds

* Money regularly set aside by a company to redeem its bonds, debentures or preferred stock from time to time as specified in the indenture or charter

**A sinking fund can be defined as an annuity invested in an order to meet a known commitment at some future date. Sinking funds are usually used for the following purposes:-Repayment of debts.

-To provide funds to purchase a new asset when the existing asset is fully depreciated.

Time Value of Money ReviewSinking Fund Factor

The sinking fund factor tells you how much you put aside each month to have a fixed amount at the end of a given time period, assuming an interest rate and compounding frequency. The formula is:

1r/m1

r/m * RequiredAmount Future Payment *

mn

Example of debt repayment using a sinking fund

Let’s say that you want to save 5 years and at the end of that time you want to have $20,000. If you can invest at 10% with monthly deposits and compounding, how much must you deposit each month?

27.258$

1.10/121

.10/12 * 20,000 Payment 60

Time Value of Money Review

Present Value of a Lump SumThe future value of a lump sum tells us how much we have to invest today to receive a fixed amount in the future. This essentially tells us what we should be willing to pay today for a fixed amount in the future. That is, the present value of a lump sum is the amount we should be willing to pay for the right to receive a certain cash flow in the future.

Time Value of Money Review

Present Value of a Lump Sum

The formula for this is just future value of a lump sum formula rearranged:

mn*

mr

1

Value Future ValuePresent

Present Value of a Lump Sum (single)

The formula is simply a rearrangement of the future value of a lump sum:

Ex. 3: What is the present value of $50,000 received in 10 years with a 10% discount rate and monthly compounding?

mn

m

r

1

Value Future ValuePresent

35.470,18

1210.

1

$50,000 ValuePresent 120

Present Value of a Lump Sum

Ex. 4: What is the present value of $50,000 received in 10 years with a 10% discount rate and yearly compounding

16.277,1910.1

$50,000 ValuePresent

10

Time Value of Money Review

Present Value of a SeriesPresent values are additive. This means that the present value of a stream of cash flows is simply the sum of the present values for each of the individual cash flows. Thus, for a series of T cash flows:

T

ii

mr1

i

1

Value Future ValuePresent

Time Value of Money Review

Present Value of a Series•Example 5: A bond will pay you $60 every six

months for the next 2 years. At the end of the third year you will also receive principal of $100. If your discount rate is 8%, how much should you pay for

this bond?

60.072,1$

208.

1

1060

208.

1

60

208.

1

60

208.

1

604321

PV

Time Value of Money Review

Present Value of an AnnuityIf the future cash flows are all of equal amounts, you can use a shortcut equation, known as the present value of an annuity equation:

mr

nm

/mr

1

1-1

*Payment ValuePresent

*

Present Value of an Annuity

Example 6: What is the present value of winning a 20,000,000 lottery if you receive payments of $1,000,000 annually, your discount rate is 10% (annually compounded), and your first payment is

in exactly one year?

72.563,513,8

10.1.10

1-1

* 1,000,000 ValuePresent 20

Time Value of Money Review

Present Value of an AnnuityNote that you sometimes have to combine

these formulas .For example, assume that you will receive $250 each month for 10 years, starting in 5 years. Your discount rate is 6% with monthly compounding.

You can use the annuity formula to get the present value at time 5 years of the 10 years worth of payments:

36.518,2212/06.

12.06

1

1-1

* 250$ ValuePresent

120

Time Value of Money Review

Present Value of an Annuity

--To determine the present value today of that annuity, you have to discount back present value of the annuity:

48.694,16

1206.

1

36.518,22 ValuePresent 60

48.694,16

1206.

1

12/06.12.06

1

1-1

* 250$ ValuePresent 60

120

-Combining into one equation gives:

Capital Recovery Factor• A capital recovery factor is the ratio of a

constant annuity to the present value of receiving that annuity for a given length of time. Using an interest rate i, the capital recovery factor is:

• where n is the number of annuities received. This is related to the annuity formula, which gives the present value in terms of the annuity, the interest rate, and the number of annuities.

• If n = 1, the CRF reduces to 1+i. As n goes to infinity, the CRF goes to i.