ICFI Net Present Value and Other Investment Criteria By : Else Fernanda, SE.Ak., M.Sc.
Economic Value of Time • Shariah admits the concept of money’s time value to the extent of pricing in a credit sale, it does not endorse placing “rent” on money advances, as interest does in the case of credit and advances. • While money’s time value is acceptable in the case of pricing assets and their usufruct, it is not acceptable in the case of any addition to the loan’s or debt’s principal. • It is better to use term economic value of time instead of time value of money.
Economic Value of Time • Future Values and Compound Interest • Present Values • Multiple Cash Flows • Level Cash Flows: Perpetuities and Annuities • Non-Annual Compounding Periods • APR versus EAR • Calculating an Annuity Payment • Calculating Number of Payment Periods • Calculating Interest Rate in an Annuity • Nominal versus Real Rates • Deferred Annuities
Future Values • Future Values - Amount to which an investment is expected to grow by a certain growth rate or price to which a commodity sale by deferred payment should be applied. • Compound Margin - Margin earned on Margin. • Simple Margin - Margin earned only on the original investment.
Future Values • The general formula for any one-year investment is: FV = PV + PV * (r) Or: FV = PV * (1 + r) • For the case of 2 periods: FV2 = FV1 * (1 + r) = PV * (1 + r) * (1 + r) = PV * (1 + r)2 • We can generalize the future value equation: FVt = PV * (1 + r)t Where t = number of periods
Future Values with Compounding Margin Rate
Solving for Future Value • There are several ways to solve time value of money problems. • Plug into the equation manually • Use a Financial Calculator (textbook provides some instructions) • Use Excel’s built-in formulas
Using Excel to find Future Values • You could simply create a formula to perform the future value equation: FVt = PV * (1 + r)t Or you can use the built-in Future Value function (FV): FV (RATE, NPER, PMT, PV, TYPE) Margin Rate (%) Total # Periods Present Value • PMT and TYPE are included to handle annuities (a series of equal payments equally spaced over time) -> for now we will set them to 0 (or leave blank).
Using Excel to find Future Values • You must make the Present Value negative to represent a cash outflow. Excel requires a cash inflow in order for there to be a cash inflow Margin rate
Using Excel to find Future Values • To see just how powerful compounding can be, try changing the # of years from 5 to 30: • A one time investment of $10,000 invested at 10% per year will grow to $174,494.02 (or 17.449 times it’s value) after 30 years! Doubling the initial investment will double the future value. Margin Rate
Using Excel to find Present Values • Suppose that the problem is restated as, “What initial investment is required so that you will accumulate$16,105.10 after 5 years if you earn 10% interest per year?”. We want to solve for present value. • PV = FVt /(1 + r)t • PV = 16105.10 / (1 + 0.10)2 = $10,000 • Now, we must discount the cash flows back in time.
Another way to look at Present Value • PV = FVt = FV x 1 (1 + r)t (1 + r)t Discount Factor • Discount Factor measures the present value of $1 received in time t. • $16,105.10 x 1 = $0.62 (1 + 10%)5
Using Excel to find Present Values • You can use the built-in Present Value function (PV): PV (RATE, NPER, PMT, FV, TYPE) Margin Rate (%) Total # Periods Future Value • PMT and TYPE are included to handle annuities (a series of equal payments equally spaced over time) -> for now we will set them to 0 (or leave blank).
Using Excel to find Present Values • The present value will appear negative if the future value is positive (Excel requires • a cash inflow in order for there to be a cash inflow). Margin Rate
Using Excel to find Present Values By setting up the future value and present value next to one another and making the future value in B2 = E5, you can see that present value and future value are inverse functions. Margin Rate Margin Rate
PV of Multiple Cash Flows • Your auto dealer gives you the choice to pay $15,500 cash now, or make three payments: $8,000 now and $4,000 at the end of the following two years. If your accepted margin rate is 8%, which do you prefer? Immediate Payment $ 8,000.00 PV1 4,000 / (1+ o.08)1 $ 3,703.70 PV2 4,000/ (1+0.08)2 $ 3,429.36 Total $ 15,133.06
Perpetuities & Annuities • Perpetuity A stream of level cash payments that never ends. • Annuity Equally spaced level stream of cash flows for a limited period of time.
Perpetuities • Most annuities call for payments to be made over some finite period of time. Some annuities go on indefinitely (perpetually) and these are called perpetuities. PV(Perpetuity) = Payment margin rate
Annuities • Up to now, we have looked at the present and future values of single cash flows (lump sums). • Annuities are a series of nominally equal cash flows, equally spaced in time. Examples of annuities are: car payments, rent payments, mortgage payments, which all involve cash outflows. You might someday own annuities as part of your retirement program (in the form of a cash inflow).
Present Value of an Annuity • One way to find the present value of an annuity is to find the present value of the cash flows separately and add them together. • Where PVA is the present value of the annuity, t is the time period, N is the total number of payments, Pmtt is the payment in period t and r is the discount rate.
Present Value of an Annuity • Where PVA is the present value of the annuity, t is the time period, N is the total number of payments, Pmtt is the payment in period t and r is the discount rate. • Let’s say that you are thinking of buying a new car. Your car dealer is offering financing of annual payments of $3600 at a 5% interest rate for 5 years. • How much would it cost you if instead you bought the car outright?
Using Excel to find the Present Value of an Annuity • Or you can use the built-in Present Value function (PV): FV (RATE, NPER, PMT, FV, TYPE) Margin Rate (%) Total # Periods Future Value • Recall: when calculating a lump sum PV we set PMT and TYPE to 0. • However for annuities we will set PMT to the dollar amount of the periodic payment. FV will be 0. • TYPE is an optional binary variable (0 or 1) which controls whether Excel assumes payment occurs at the end (0) or the beginning (1) of the period. When the payment occurs at the beginning of the period (called an ‘annuity due’) as opposed to the end (called an ‘ordinary annuity’), each payment is compounded for an extra year.
Future Value of an Annuity • Imagine that you have graduated and are working your first year and have decided to participate in the company’s 401k plan (or similarly an Individual Retirement Account (IRA)). What makes these retirement plans so attractive is that you can contribute pre-tax income into them and your investment will grow tax-free until you are retirement age (59 ½) and begin making withdrawals. • Say you want to invest say $3000 each year and see what the future value of that money will be when you retire. You would use the future value of an annuity formula: • Where FVA is the future value of the annuity, t is the time period, N is the total number of payments, Pmtt is the payment in period t and r is the interest rate.
Using Excel to find the Future Value of an Annuity • You can use the built-in Future Value function (FV): FV (RATE, NPER, PMT, PV, TYPE) Margin Rate (%) Total # Periods Present Value • Recall: when calculating a lump sum FV we set PMT and TYPE to 0. • However for annuities we will set PMT to the dollar amount of the periodic payment. TYPE is an optional binary variable (0 or 1) which controls whether Excel assumes payment occurs at the end (0) or the beginning (1) of the period. PV will be 0.
Non-Annual Compounding Periods Some investments make interest payments semiannually, monthly, daily, or even more frequently. Recall our basic time value of money formula: FVt = PV * (1 + r)t r is the periodic rate of interest and t is the total number of periods. r might be the weekly interest rate and t the number of weeks. Since rates are usually quoted in terms of simple (or nominal) annual percentage rates (APRs), we can restate the formula: FVt = PV * (1 + r/m)tm Where r is the annual rate, t is the number of years, and m is the number of periods per year.
Non-Annual Compounding Periods • We need to convert the simple rate to the rate per period by dividing the annual rate by the number of periods per year and then the number of periods is the periods per year multiplied by the number of years.
Continuous Compounding We could go further than daily compounding to compound every hour, every minute or even every second. This concept can be extended to the smallest imaginable time period; referred to as continuous compounding. Recall our time value of money formula: FVt = PV * (1 + r/m)tm Where r is the annual rate, t is the number of years, and m is the number of periods per year. If we make m equal to infinity, r would be equal to 0, so instead we would have to take the limit as m approaches infinity. We would get: lim FVt = PVert m -> ∞ • e is the base of a natural logarithm (approximately = 2.718). • In Excel, we use the EXP function which raises e to a specific power: • =PV*exp(rate * number of years)
Effective Annual Rate • Another way to compare investments with different compounding periods, is to convert the simple rate to an effective annual rate (EAR). This means converting a rate that may be compounded monthly or daily to an equivalent rate with only one compounding period.
Effective Margin Rate • Effective Annual Margin Rate - Margin rate that is annualized using compound margin. • Annual Percentage Rate – Margin rate that is annualized using simple margin.
Effective Margin Rate Example • Given a monthly rate of 1%, what is the Effective Annual Rate (EAR)? What is the Annual Percentage Rate (APR)? EAR = (1+ .01)12 – 1 = r = 12.68% APR = .01 x 12 = 12%
Solving for the Annuity Payment Margin Rate
Solving for the Number of Periods in an Annuity Margin Rate
Solving for the Margin Rate in an Annuity Margin Rate
Deferred Annuities • Not all annuities begin their payouts in the year following the analysis. For example, if you are planning your retirement you will probably start with the amount of income you will need each year during retirement. If you are a student, you will not need this retirement income for many years, so it is known as a deferred annuity. • You can use Excel’s PV function to shift time, transporting you to the year before retirement in order to calculate the lump sum goal amount required at the time of retirement. • Then you can determine how to reach that goal, either through a lump sum payment now or an annual payment.
Deferred Annuities • Step 1 – find out how much we will need:
Deferred Annuities • Step 2: What would it take to reach that goal in a lump-sum payment?
Deferred Annuities • Step 3: What would it take in terms of annual payments to reach that goal?
Deferred Annuities • We have ignored the effects of inflation and taxes on retirement savings in this worksheet, but you can see how important it is to start saving early for retirement.