TIME VALUE OF MONEY Prepared by Lucky Yona

Introduction Time value of money is one of the most important concept in Finance. Money has in possession today is more valuable than money in the future because the money it has now can be invested and can earn positive returns. Time value of money is based on the assumption that 1$ today is worth than more than 1$ that will be received at some future date.

Introduction The study of time value of money will involve considering two major views 1. Future Value 2. Present Value

Future Value Future value is the cash you will receive at a given future date. Present Value is just like cash in hand today. Present value measure cash flows at the start of a project’s life ( time zero). Both techniques will give the same answer but managers because they make decision at time zero- they tend to rely on present values techniques.

Computational Tools The calculation of present values and future values might be time consuming. There are computational tools which can streamline the exercise These include Financial Tables- These include various future and present values interest factors that simplify time value calculations. Financial Calculators - They include numerous preprogrammed financial routines.

Computational Tools Electronic Spreadsheets They perform complex analyses They automate the choice of best option They allow linkages which makes possible to do sensitivity analysis. They encourage teamwork They enhance learning.

Basic Pattern of Cash Flow Single amount : A lump-sum amount either currently held or expected at some future date. Examples include $ 1,000 today and $ 650 to be received at the end of 10 years. Annuity: A level of periodic stream of cash flow. Example include either paying out or receiving $ 800 at the end of each of the next years. Mixed Stream- A stream of cash flow that is not an annuity; a stream of unequal periodic cash flows that reflect no particular pattern.

Future Value of Single Amount Future value is the value at a given future date of a present amount placed on a deposit today and earning interest at a specified rate. It depends on the rate of the interest earned and the length of time a given amount is left on the deposit. It is found by applying the compound interest over a specified period of time.

Formula FV= PV (1 + r) N Where FVn = Future Value at the end of period n P = Initial Principal or present value r = annual rate of interest paid n= number of periods ( typically years) that the money is left on deposit

Example Suppose you place a $100 in an 8% p.a. interest investment at the end of one year you will have $108 The future value at the end of year one= $100 x (1+.08)=$ 108 Future value at the end of year two is $ 108 x (1+.08)= $116.64 or $100x(1+.08) x(1+.08)= $116.64. We can also use the future value tables guided by the formula Fv=PV x(1+K) to the power of n.

Example 2 Timothy Lucky places $ 800 in a saving account paying 6% interest compounded annually. He wants to know how much money will be in the account at the end of 5 years.

Solution FV5 = $ 800 * ( 1+ 0.06 ) 5 = 800 * 1.338 = $ 1070.40

Present Value of Single Amount The process of finding present values is often referred to a discounting cash flows. Discounting cash flows is the process of finding present values ; the inverse of compounding interest Discounting determines the present value of a future amount, assuming an opportunity to earn a certain return on the money. The annual rate of return is variously referred as discount rate, required return, cost of capital and opportunity cost.

Formula PV = FVn (1 + r) N = FVn * ( 1 ) (1+ r )N

Example Faith Lucky wishes to find the present value of $ 1,700 that will be received 8 years from now. Her cost of capital is 8%. Determine the Present Value. Answer = $ 918.42

Annuities An annuity is a stream of equal periodic cash flows over a specified time period. These cash flows can be inflows of returns earned on investments or outflows of funds invested to earn future returns. These cash flows are usually annual but can occur at other intervals, such as monthly ( rent, car payments)

Types of Annuities Ordinary Annuity Annuity Due An annuity for which the cash flow occurs at the end of each period Annuity Due An annuity for which the cash flow occurs at the beginning of each period

Future Value of Ordinary Annuity The formula for the future value interest factor for an ordinary annuity FVIFAr = ( 1 + r ) T-1

Example Faith Lucky wishes to determine how much money she will have at the end of 5 Years if she chooses annuity A, the ordinary annuity. It represents deposits of $ 1,000 annually, at the end of each of the next 5 years, into a savings account paying 7% annual interest.

Calculating Future Value ------------------------------------$1,311 ------------------------------$1,225 -----------------------$1,145 -----------------$1,070 ---------$1,000 $1000 $ 1,000 $1,000 $1,000 $1,000 5,751 1. Future Value = $ 5,751

Present Value of Ordinary Annuity The formula for calculating the present value in this case is PVIFAr,n = 1 (1+ R )T

Future Value of Annuity Due Remember that cash flow of an annuity due occur at the start of the period. The Formula is FVIFAr,n ( annuity due)= FVIFAr,n * (1+r)

PRESENT VALUE OF MIXED STREAMS Mixed streams are streams of cash flows that reflect no particular pattern. This can be calculated by using the relevant present value interest factor for the mixed stream as shown in the financial tables. In the case of a perpetuity that is an annuity with an infinite life that is an annuity that never stops.

EXERCISES 1. Vicky has $ 10,000 that she can deposit in any of three savings accounts for a three year period.Bank A compounds interest on annual basis at 4%, Bank B at 8% and Bank C at 12%. What amount would Vicky have at the end of the third year leaving all interest paid on deposit in each bank?

Solution Bank A= $ 11,248.64 Bank B = $ 12,597.12 Bank C= $ 14, 049.28

EXERCISES 2. You have a choice of accepting either of two five year cash flow streams or lump sum amounts. One cash flow is an annuity and the other is a mixed stream.You may accept either one.Assuming a 9% opportunity cost of capital, which cash flow form would you prefer? The cash flows for the lump sum at time zero is $2825 for alternative A and $2800 for alternative B, The other cash flow streams are as shown below.

EXERCISES End of yr. A B 1 $700 $1,100 2 700 900 3. 700 700 4. 700 500 1 $700 $1,100 2 700 900 3. 700 700 4. 700 500 5. 700 300

Solution CASE A Lump sum = 2825 ( 1+0.09)p5= $ 4346.6 Annuity = $ 4189.3 Year 5 = 700(1+0.09)4 Year 4 = 700( 1+0.09)3 Year 3 = 700 ( 1+0.09)2 Year 2 = 700 (1+0.09) 1 Year 1 = 700(1+0.09)0

CASE B Lump sum FV = 2,800 ( 1+0.09)5= $ 4,309 Mixed Streams ( Fv) = $ 4395