# Week 4. Chonga Zangpo, DFB

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1 Week 4 Time Value of Money Chonga Zangpo, DFB

2 What is time value of money? It is based on the belief that people have a positive time preference for consumption. It reflects the notion that people prefer to consume things today than at some time in future. A dollar today is worth more than a dollar in the future as the value of money changes with time. The difference in value between a dollar in had today and a dollar promised in the future is the time value of money. Therefore, it is very important for us to know what is a dollar worth in future time.

3 Time value of money Two views on time value: Present value Future value Compounding Future Value year 5% - \$1000 \$5000 \$4000 \$3000 \$2000 \$1000 Present Value Discounting

4 Time Lines A time line is a horizontal line that starts at time zero (today) and shows cash flows as they occur overtime. Time lines are an important tool for analyzing problems that involve cash overtime. Provide an easy way to visualize the cash flow associated with the investment. Time lines help us to correctly identify the size and timing of the cash flows critical tasks in solving time value problems Year 5% - \$10,000 \$6,000 \$5,000 \$4,000 \$3,000 \$2,000 \$1,000 Cash flow at the end of each year

5 Future value & Compounding The future value of an investment is what the investment will be worth after earning interest. The process of converting the initial amount into future value is called compounding. Single amounts: FV 1 = Principal + Interest earned = PV x (1+ i) FV n = PV x (1+i) n = PV x (FVIF i,n ) OR Note: to solve a future value problem, we need to know the future value factor, (1 +i) n

6 Future value & Compounding Example: Suppose you have an opportunity to make a \$5000 investment that pays 15% per year. How much money will you have at the end of 10 years? Solution: FV 10 = \$5,000 x ( ) 10 = \$5000 x = \$20, FV 10 = \$20, PV = \$

7 Present value & discounting Present value is the current value of future cash flow discounted at the appropriate discount rate. PV = what is the value today of a cash flow promised in the future? The process of converting future value into present value is known as discounting and the interest rate is known as discount rate. Present value equation: PV = FV n /(1 + i) n OR = FV n x (PVIF i,n )

8 Present value & discounting Example: Suppose you plan to take a graduation vacation to Europe when you finish college in two years. If your savings account at the bank pays you 6% per annum, how much money you need to put aside today to have \$8000 when you leave for Europe? Solution: PV = FV n /(1 + i) n OR FV n x (PVIF i,n ) = FV 2 /(1+0.06) 2 = \$8000 x 1/(1.06) 2 = \$8000 x = \$7, PV = \$7, FV 2 = \$8000

9 Annuities An annuityis a stream of equal periodic cash flows, over a specified time period. Types of annuities: 1. Ordinary annuity An annuity for which the cash flow occurs at the endof each period. 2. Annuity due An annuity for which the cash flow occurs at the beginningof each period. Note: In general, both the present value and future value of an annuity due will be higher than the present value and future value of an otherwise identical ordinary annuity.

10 Comparison of cash flows in annuities Following table represents a 5 year \$1000 annuities of cash inflow with two options: ordinary annuity and annuity due. Which of this two options would you choose? End of years Annuity A (ordinary) Annuity B(annuity due) 0 \$ 0 \$ TOTAL \$ 5000 \$ 5000

11 1. Present value of Annuities Present value of an ordinary annuity: You would like to know what would be the present value of future cash flows to be received in next five years at the endof each year. You require annuity to provide a minimum return of 8% per annum *PVIF 8%,n \$1000 \$1000 \$1000 \$1000 \$1000 \$ PV = \$3,993 2.Short Method: PVA n = PMT x (PVIFA i,n ) = 1000 * (PVIFA 8%,5 ) = 1000 * = \$3,993

12 1. Present value of Annuities Present value of an annuity due: 1000*PVIF 8%,n You would like to know what would be the present value of future cash flows to be received in next five years at the beginningof each year. You require annuity to provide a minimum return of 8% per annum. \$ PV = \$4, \$1000 \$1000 \$1000 \$1000 \$ Short Method: PVA n (due) = PMT x ((PVIFA i,n ) x (1+i)) = 1000 * (PVIFA 8%,5 ) x (1+0.08) = 1000 * (3.993 * 1.08) = \$4,312

13 1. Future value of Annuities Future value of an ordinary annuity: If you wish to determine how much money you would have at the end of fifth year if you have decided to collect the cash flows at the endof the year for next five years and deposit the same into a saving account paying you 8% interest per annum Here n = 5-1 =4 \$ * FVIF 8%,n FV = 5,866 (i.eno. of years earning interest) 2.Short Method: FVA n = PMT x (FVIFA 8,n ) = 1000 * = \$5,867

14 1. Future value of Annuities Future value of an annuity due: If you wish to determine how much money you would have at the end of fifth year if you have decided to collect the cash flows at the beginningof the year for next five years and deposit the same into a saving account paying you 8% interest per annum \$ * FVIF 8%,n FV = 6,335 2.Short Method: FVA n (due) = PMT x (FVIFA 8,n x (1+i)) = 1000 * (5.867 * 1.080) = \$6,336

15 Mixed streams (multiple cash flows) Two basic types of cash flow streams are possible: the annuity and the mixed stream. A mixed stream is a stream of unequal periodic cash flows that reflect no particular pattern. The present values and future values of a mixed stream cash flows are calculated in the same way as the long process of calculating present values and future values of annuities.

16 EXAMPLE: Present value of a mixed stream Frey Ltd, a shoe manufacturer, has been offered an opportunity to receive the following mixed stream of cash flows over the next five years. If the firm must earn at least9%onitsinvestments,whatisthemostitshouldpay for this opportunity? End of year Cash flow 1 \$

17 Solution: Year (n) Present value of a mixed Cash flow (1) stream PVIF 9%,n (2) Present value [(1)x(2)] 1 \$ \$ Present value of mixed stream \$ \$ \$ \$ \$ \$ \$ \$400 \$800 \$500 \$400 \$300

18 Example: Future value of a mixed stream Shrell Industries, a cabinet manufacturer, expects to receive the following mixed stream of cash flows over the next five years fromoneof itssmallcustomers.ifshrellexpectstoearn8%on its investments, how much will it accumulate by the end of year 5 if it immediately invests cash flows when they are received? End of year Cash flow 1 \$

19 Solution: Year Cash flow (1) Future value of a mixed No.of years earning interest (n) (2) streams FVIF 8%,n (3) Future value [(1)x (3)] 1 \$11, = \$15, , = , , = , , = , , = , Future value of mixed stream \$83, \$ \$ \$11500 \$14000 \$12900 \$16000 \$ \$ \$ \$ \$83, Future value

20 Compounding effect on interest rates Interest is often compounded more frequently than once a year. Following are some of the more frequent compounding intervals: 1. Semi-annual compounding Two compounding periods in a year One half of the stated interest rate is paid twice a year 2. Quarterly compounding Four compounding periods in a year One-fourth of the stated interest is paid four times a year

21 Compounding effect on interest rates A general equation for compounding more frequently than annually: FV n = PV x (1+i/m) mxn Example: Fred More has decided to invest \$100 in a saving account paying 8% pa. If he leaves his money in the account for 2 years, what will be his future value in following circumstances? 1. Interest being compounded semi-annually: FV 2 = \$100 x [1+ (0.08/2) 2x2 ] = 100x(1+0.04) 4 =\$ Interest being compounded quarterly: FV 2 = \$100 x [1+ (0.08/4) 2x4 ] = 100x(1+0.02) 8 = \$117.16

22 Nominal and effective rates of interest When interest are compounded more frequently, the real interest that we pay (earn) could be more than the stated nominal rate of interest. The effective, or real annual rate (EAR) can be calculated as follows: EAR = [1+(i/m)] m 1

23 Nominal and effective rates of interest Considering the Fred Moore example: 1. For annual compounding: EAR= [1 + (0.08/1)] 1 1= (1+0.08) 1 1 = = 0.08 = 8% 2. For semi-annual compounding: EAR= [1 + (0.08/2)] 2 1 = (1+0.04) 2 1 = = = 8.16% 3. For quarterly compounding: EAR= [1 + (0.08/4)] 4 1 = (1+0.02) 4 1 = = = 8.24%

24 Special application of time value Future and present value techniques have a number of important applications in finance. Under this section, we look at following four: 1. Deposit needed to accumulate a future sum 2. Loan amortization 3. Interest or growth rates 4. Finding an unknown number of periods

25 Special application of time value 1. Deposit needed to accumulate a future sum: Suppose you want to buy a house five years from now, and you estimate that an initial down-payment of \$20,000 will be required at that time. To accumulate the \$20,000, you wish to make an equal annual end-of-year deposit into an account paying annual interest of 6%. HOW MUCH SHOULD YOU DEPOSIT? FVA n =PMTx(FVIFA i,n ) SolvingforPMT: \$20,000=PMTx(FVIFA 6%,5yrs ) \$20,000=PMTx5.637 PMT = \$20,000/5.637 = \$3,547.99

26 Special application of time value 2. Loan amortization: Loan amortization refers to the calculation of equal periodic loan. Amortizing a loan actually involves creating an annuity out of a present amount. Example: Suppose you borrow \$6000 at 10% pa and agree to make equal annual end-of year payments over four years. How much do you have to pay at the end of every year? PVA n = PMT x (PVIFA i,n ) Solving for PMT: \$6000 = PMT x (PVIFA 10%,4 ) \$6000 = PMT x PMT = \$6000/3.170 = \$

27 Special application of time value 3. Interest or growth rates: It is often necessary to calculate the compound annual interest or growth rate (i.e. the annual rate of change in values) of series of cash flows. We can use ether present value or future value interest factor to find the growth rate. Example: Ray Noble wishes to find the rate of interest or growth reflected in the stream of cash flows he received from a real estate investment over the period 2003 to The table in next slide lists those cash flows:

28 Special application of time value Year Cash flow 2007 \$ By using 2003 as a base year Ray can see that interest has earned or growth experienced for four years. How? Step: 1. Divide the amount received in the earliest year (PV) by the amount received in the latest year (FV n ) Step: 2. Refer the financial table PVIF i,4yrs and compare with the above. The corresponding interest rate is the growth rate. Ray s growth rate = 5% ( 1250/1520) = PVIF i,4yrs = or close 5%

29 Special application of time value 4. Finding an unknown number of periods: There will be circumstances when you will need to know the time period needed to generate/accumulate/payoff a given amount of cash flow from an initial amount. Scenario 1:You wish to determine the number of years it will take for your initial \$1000 deposit, earning 8% annual interest, to grow to equal \$2500. PV = FV n x (PVIF i,n ) (PVIF i,n ) = PV/ FVn (PVIF 8%,n ) = 1000/2500 = 0.400, refer the present value interest factor table at 8% At 8% column in the table, is closest to value at n = 12 ; therefore it will take 12 years to grow you initial deposit \$1000 to equal \$ 2500

30 Special application of time value Scenario 2 (with annuity): You wish to know how many years it will take to repay \$25000 loan at 11% if you make an annual payment of \$4800 at the end of each year? Using Table to solve: PVA n = PMT x (PVIFA i,n ) PMT = PVA n /(PVIFA i,n ) PVIFA i,n = PVA n / PMT Therefore, PVIFA 11%,n = 25000/4800 PVIFA 11%,n = look for number of periods for an 11% interest rate associated with annuity factor close to in the table therefore, the number of years necessary to repay the loan fully is approx (to the nearest year) is 8 years.

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