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## Time Value of Money

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**Time Value of Money**• Time preference/ Value for money is an individual’s preference for possession of a given amount of money now, rather than the same amount at some future time. • Three reasons may be attributed to the individual’s time preference for money: • Risk • preference for consumption • investment opportunities**Required Rate of Return**• The time value of money is generally expressed by an interest rate. This rate will be positive even in the absence of any risk. It may be therefore called the risk-free rate. • An investor requires compensation for assuming risk, which is called risk premium. • The investor’s required rate of return is: Risk-free rate + Risk premium. • The required rate of return may also be called the opportunity cost of capital of comparable risk**Required Rate of Return**• Would an investor want Rs. 100 today or after one year? • Cash flows occurring in different time periods are not comparable. • It is necessary to adjust cash flows for their differences in timing and risk. • Example : If preference rate =10 percent • An investor can invest Rs. 100 if he is offered Rs 110 after one year. • Rs 110 is the future value of Rs 100 today at 10% interest rate. • Also, Rs 100 today is the present value of Rs 110 after a year at 10% interest rate. • If the investor gets less than Rs. 110 then he will not invest. • Anything above Rs. 110 is favourable.**Time Value Adjustment**• Two most common methods of adjusting cash flows for time value of money: • Compounding—the process of calculating future values of cash flows and • Discounting—the process of calculating present values of cash flows.**Future Value**• Compounding is the process of finding the future values of cash flows by applying the concept of compound interest. • Compound interest is the interest that is received on the original amount (principal) as well as on any interest earned but not withdrawn during earlier periods. • Simple interest is the interest that is calculated only on the original amount (principal), and thus, no compounding of interest takes place.**Future Value**• The general form of equation for calculating the future value of a lump sum after n periods, may therefore be written as: FV = PV (1 + i)n**2. Future value and present value**Future value. (1 + i)ⁿ is a future value factor (fvf) To simplify calculations of FV use table of fvf.**Future Value**• In Microsoft Excel: Use FV function. • FV(rate,nper,pmt,pv,type) Where: rate= interest rate. nper= n periods, pmt= annuity value, pv= present value, Type=1 for beginning of the period and 0 for end for end of period.**Present Value**• Present value of a future cash flow (inflow or outflow) is the amount of current cash that is of equivalent value to the decision-maker. • Discounting is the process of determining present value of a series of future cash flows. • The interest rate used for discounting cash flows is also called the discount rate.**Present Value of a Single Cash Flow**• How much the investor give up now to get the amount of Re.1 at the end of one year, assuming discount rate is 10% ? • Formula to calculate the present value of a lump sum to be received after some future periods**2. Future value and present value**Table of present value factor**Types of Annuities**• Ordinary Annuity: Payments or receipts occur at the end of each period. • Annuity Due: Payments or receipts occur at the beginning of each period. • An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods.**Examples of Annuities**• Student Loan Payments • Car Loan Payments • Insurance Premiums • Mortgage Payments • Retirement Savings**Parts of an annuity**End of Period 2 (Ordinary Annuity) End of Period 1 End of Period 3 0 1 2 3 $100 $100 $100 Equal Cash Flows Each 1 Period Apart Today**Parts of an annuity**Beginning of Period 2 (Annuity Due) Beginning of Period 1 Beginning of Period 3 0 1 2 3 $100 $100 $100 Equal Cash Flows Each 1 Period Apart Today**FVAn= A(1+i)n-1 + A(1+i)n-2 + ... + A(1+i)1 + A(1+i)0**Overview of an Ordinary Annuity Cash flows occur at the end of the period 0 1 2 n n+1 i% A A A A = Periodic Cash Flow**The future compound value of an ordinary annuity is as**follows: FV = A {[(1+i)^n - 1]/ i} The term within the curly brackets {} is the compound value factor for an annuity of Re.1, and A is the annuity. Suppose you have decided to deposit Rs.30,000 per year in your Public Provident Fund Account for 15 years. What will be the accumulated amount in your Public Provident Fund Account at the end of 15 years if the interest rate is 8 percent ?**FVADn = A(1+i)n + A(1+i)n-1 + ... + A(1+i)2 +**A(1+i)1 = FVAn (1+i) Future Value of an Annuity Due Cash flows occur at the beginning of the period 0 1 2 3 n-1n . . . i% A A A A A**The future compound value of an annuity due is as follows:**FV = A {[(1+i)^n - 1]/ i} (1+i) Suppose you have decided to deposit Rs.30,000 in an investment for 15 years at the beginning of each year. What will be the accumulated amount at the end of 15 years if the interest rate is 8 percent ? Hint: On the time line, each payment would be shifted to the left one year, so each payment would be compounded for one extra year.**PVAn = A/(1+i)1 + A/(1+i)2 + ... + A/(1+i)n**Present Value of an Ordinary Annuity -- PVA Cash flows occur at the end of the period 0 1 2 nn+1 . . . i% A A A A= Periodic Cash Flow PVAn**The present value of an annuity hence will be**PV = A {[1 - 1/(1+r)^n]/r} The term within the curly brackets {} is the Present value factor for an annuity of Re.1, and A is the annuity. Suppose you have decided to deposit Rs.30,000 per year in your Public Provident Fund Account for 15 years. What will be the present value of your Public Provident Fund Account if the interest rate is 8 percent ?**PVADn = A/(1+i)0 + A/(1+i)1 + ... + A/(1+i)n-1 =**PVAn(1+i) Present Value of an Annuity Due -- PVAD Cash flows occur at the beginning of the period 0 1 2 n-1n . . . i% A A A A A: Periodic Cash Flow PVADn**The present value of an annuity due hence will be**PV = A {[1 - 1/(1+r)^n]/r} (1+r) Suppose you have decided to deposit Rs.30,000 in an investment for 15 years at the beginning of each year. What will be the present value of your investment if the interest rate is 8 percent ?**How much should you save annually**You want to buy a house after 5 years when it is expected to cost Rs.2 lacs. How much should you save annually if your savings earn a compound return of 12 percent ? Finding the interest rate A finance company advertises that it will pay a lump sum of Rs.10,00,000 at the end of 6 years to investors who deposit annually Rs.1,00,000 for 6 years. What interest rate is implicit in this offer?**Suppose you can lease a computer from its manufacturer for**Rs. 1000 per month. The lease runs for 36 months, with payments due at the end of the month. As an alternative, you can buy it for Rs. 20000. In either case, at the end of the 36 months the computer will be worth zero. • If you save Rs. 2,00,000 per year at an interest rate of 8 percent, how long will it take for you to accumulate Rs. 25,00,000?**If you earn an interest rate of 8 percent, how much must you**save for each of the next 20 years to accumulate the Rs. 25,00,000? • Nirmal had invested Rs. 12000 per year for last three years and has extended his investment for another 15 years but this time he has chosen a less risk fund which is expected to give 12% per annum, earlier investment gave the return of 18% per annum. What will be the amount at the end of 15 years from now?**Important equation to remember**PV(1+i)^n + PMT {[(1+i)^n - 1]/ i} + FV = 0**Perpetuities**• Perpetuity is an annuity that occurs indefinitely. In perpetuity, time period, n, is so large (mathematically n approaches infinity) that the expression (1+r)^n in the present value equation tends to become zero, and the formula for a perpetuity simply condenses into: PV = A/r where A is the annuity amount occurring indefinitely and r is the interest rate.**Present value of a growing annuity**A cash flow that grows at a constant rate for a specified period of time is a growing annuity. The time line of a growing annuity is shown below: A A(1 + g) A(1 + g)n-1 0 1 2 3 n The present value of a growing annuity can be determined using the following formula : (1 + g)n (1 + r)n PV of a Growing Annuity = A r – g The above formula can be used when the growth rate is less than the discount rate (g < r) as well as when the growth rate is more than the discount rate (g > r). However, it does not work when the growth rate is equal to the discount rate (g = r) – in this case, the present value is simply equal to n A/(1+i). 1 –**Present value of a growing annuity**For example, suppose you have the right to harvest a teak plantation for the next 20 years over which you expect to get 100,000 cubic feet of teak per year. The current price per cubic foot of teak is Rs 500, but it is expected to increase at a rate of 8 percent per year. The discount rate is 15 percent. The present value of the teak that you can harvest from the teak forest can be determined as follows: 1.0820 1 – 1.1520 PV of teak = Rs 500 x 100,000 (1.08) 0.15 – 0.08 = Rs.551,736,683**Future value of a growing annuity**Where r = g ,FV = A.n (1+r)^(n-1)**Growing Annuities**• An annuity of Rs.100 is expected to grow at a rate of 2% every year. Assuming the interest rate as 10% per annum • Calculate the present value for this growing annuity for a 5 year duration • Calculate the future value for this growing annuity for a 5 year duration**Present Value of a growing perpetuity**• PV = A/ (r-g)**Uneven Cash Flow Streams**• The PV of an uneven cash flow stream is found as the sum of the PVs of the individual cash flows of the stream.**Present value of an uneven series**A1 A2 An PVn = + + …… + (1 + r) (1 + r)2 (1 + r)n n = At t =1 (1 + r)t Year Cash Flow PVIF12%,n Present Value of Rs. Individual Cash Flow 1 1,000 0.893 893 2 2,000 0.797 1,594 3 2,000 0.712 1,424 4 3,000 0.636 1,908 5 3,000 0.567 1,701 6 4,000 0.507 2,028 7 4,000 0.452 1,808 8 5,000 0.404 2,020 Present Value of the Cash Flow Stream 13,376**Future value of an uneven series**The future value of an uneven cash flow stream (sometimes called the terminal value) is found by compounding each payment to the end of the stream and then summing the future values FVn = A1 (1 + r) + A2 (1 + r)2 +……+ An (1 + r)n-1 n = At (1 + r)n-t t =1**Shorter compounding period**• Future value = Present value 1+ r mxn • m • Where r = nominal annual interest rate • m = number of times compounding is done in a year • n = number of years over which compounding is done • Example : Rs.5000, 12 percent, 4 times a year, 6 years • 5000(1+ 0.12/4)4x6 = 5000 (1.03)24 • = Rs.10,164**EFFECTIVE VERSUS NOMINAL RATE**r = (1+k/m)m –1 r = effective rate of interest k = nominal rate of interest m = frequency of compounding per year Example : k = 8 percent, m=4 r = (1+.08/4)4 – 1 = 0.0824 = 8.24 percent Nominal and Effective Rates of Interest Effective Rate % NominalAnnual Semi-annual Quarterly Monthly Rate % Compounding Compounding Compounding Compounding 8 8.00 8.16 8.24 8.30 12 12.00 12.36 12.55 12.68