FINA Exam 2 Chps 3, 4, 5
periodic rate
- APR / # of compounding periods per year -- APR / m
time value of money
- a dollar received today is worth more than a dollar received in the future
annuity cash flows
- a stream of equal cash flows that are received at regular intervals - must be the same amount received at regular intervals (yearly, semiannually, quarterly, monthly, weekly, etc.)
present value of a lump sum
PV = FV / (1 + r)^n - is the current dollar value of a future amount - the amount of money that would have to be invested today at a given interest rate over a specified period of time to equal a future amount - discounting the cash flows - answers the question, "If I can earn r percent on my money, wha tis the most I would be willing to pay now for an opportunity to receive FV dollars N periods from today?"
simple vs. compound interest
Simple Interest - Interest Earned: -- Year 1 - $5.00 -- Year 2 - $5.00 -- Year 3 - $5.00 - Value at Year-end: -- $105.00 -- $110.00 -- $115.00 Compound Interest - Interest Earned: -- Year 1 - $5.00 -- Year 2 - $5.25 -- Year 3 - $5.51 - Value at Year-end: -- $105.00 -- $110.25 -- $115.76
more than one time
TVM calculations when interest is compounded _______ _______ _____ ________ per year - you can calculate the present value or future value of a lump sum, annuity, or mixed stream of cash flows using the standard formula after making these two adjustments: -- convert the annual interest rate to the periodic rate --- periodic rate = r/m --- r = annual interest rate --- m = # compounding periods per yr -- convert the number of years to total number of compounding periods --- total # compounding periods = m * n --- m = # compounding periods per yr --- n = #yrs
variables
five _______________ in Time Value of Money (TVM) Equations: - Future Value (FV) - Present Value (PV) - Interest Rate (r or I/Y) - Payment (PMT) - Number of Periods (n)
interest rate
is simply the compound growth rate - If this is 4%, that simply means that each year you will have 4% more money than you did the year before - you can apply this concept to lots of situations
natural log
ln means ____________ _____
views
two common __________ of time value of money - future value - present value
annual percentage yield (APY)
- banks and other borrowers are required by law to quote this on things like savings accounts and other investments - this is the same thing as EAR - so rates quoted to borrowers (APR) and rates quoted to savers (APY) are not computed the same way - when you BORROW money, you are quoted the APR, which is not the true interest rate when taking compounding frequency into account - but when you INVEST money, the financial institutions are required to quote you theAPY, which is the same as the APR and DOES take compounding frequency into account
FV of an ordinary annuity
- can be found by summing the FV of each year's cash flow HOWEVER - instead of doing each individually, you can use this formula -- FVA = PMT [(1 + r)^n - 1]/r
amortized loans
- car loans, mortgages, and most other consumer loans are these - the borrower typically makes a fixed payment every period - some of the payment is for interest on the principal, and the remainder is a payment on the principal of the loan - so, you are paying off the principal over the life of the loan - interest is based on the principal balance at the beginning of the period - whatever portion of the payment is NOT paid as interest is applied to the principal, which reduces the loan balance - as the principal balance declines, the interest portion of each payment decreases, and the payment on the principal increases, until the principal is completely paid off - the value of the loan (loan amount) is the present value of the payments (present value of an annuity)
monthly
- compounding over 12 periods with the year - m = 12
daily
- compounding over 365 periods within the year
weekly
- compounding over 52 periods within the year
quarterly
- compounding over four periods within the year - m = 4
semi-annually
- compounding over two periods within the year - m = 2
compound interest example
- compute this on a $100 deposit that earns 5% per year -- Year 1 - $100 * .05 = $5.00 --- At the end of year 1, you have $105 -- Year 2 - $105 * .05 = $5.25 --- At the end of year 2, you have $110.25 -- Year 3 - $110.25 * .05 = $5.51 --- At the end of year 3, you have $115.76
simple interest example
- compute this on a $100 deposit that earns 5% per year -- Year 1 - $100 * .05 = $5.00 --- At the end of year 1, you have 100 + 5 = $105 -- Year 2 - $100 * .05 = $5.00 --- At the end of year 2, you have 105 + 5 = $110 -- Year 3 - $100 * .05 = $5.00 --- At the end of year 3, you have 110 + 5 = $115
continuously
- constant compounding over infinite periods
time lines
- depict the investment cash flows, both positive and negative, on a horizontal line with time zero (today) at the left and future periods shown moving from left to right -- what is the future value of $100 invested today at 3% at the end of seven years? 0 PV -$100 |----|----|----|----|----|----|----| 7 FV?
FV of an annuity due
- earns interest for one more year than an ordinary annuity because the payments are made at the beginning of the year instead of the end of the year - formula: -- FVAdue = [PMT [(1 + r)^n - 1]/r] (1 + r) - the "(1 + r)" at the end of the formula takes into account the extra year's interest - so, you are just multiplying the FV of an ordinary annuity by (1 +r) to take into account the extra year's interest earned - can also change a calculator setting...
rule of 72
- for reasonable rates of return (3%-20%), you can approximate the time it will take you to double your money using this formula: - approximate time to double your money = 72/r - for 6% interest rate, the time to double your money = 72/6 = 12 years - using calculator: -- PV = -1, FV = 2, I/Y = 6, PMT = 0, CPT N = 11.90 years - for 13% interest rate, time to double your money = 72/13 = 5.54 years -- using calculator: N = 5.67 years
compound interest
- interest earned on both the initial principal and the interest received from prior periods - "interest earned on interest"
simple interest
- interest earned only on the original principal amount invested
compounding more frequently than annually
- interest is often compounded more than once per year: -- semi-annually -- quarterly -- monthly -- weekly -- daily -- continuously
annuity
- is a stream of equal cash flows paid out at regular intervals - these cashflows can be inflows of cash (monthly pension check) or outflows of cash (monthly mortgage payment)
perpetuity
- is an annuity with an infinite life - it never stops providing the holder with payment dollars at the end of the year - for example, if you endow a scholarship at ECU, you donate some money (the principal) - the ECU Foundation invests that money and pays the scholarship funds out of the interest it earns each year, never touching the principal -- PVAperpetuity = PMT / r
interest
- is the price you pay for borrowing money -the price of "renting" money
compounding
- is the process of accumulating interest in an investment over time to earn more interest (FUTURE VALUE)
discounting
- is the process of computing the PRESENT VALUE - you are bringing the cash flows back to today's dollars (________________ them back to today's dollar)
annuity due
- payments occur at the beginning of the period instead of the end of the period
ordinary annuity
- payments occur at the end of the period
present value
- represents the dollar value today of a future amount, or the amount you would invest today at a given interest rate for a specified period of time to equal a future amount -- how much would you have to put into a saving account today if it earned 5% compounded annually and you wanted to have $300 five years from now?
types of interest
- simple interest - compound interest
cash stream patterns
- so far, we have only calculated the future value and present value of a lump sum (one cash flow) - many times, however, there are more than one cash flow involved: -- Mixed Stream of Cash flows -- Annuity Cash flows
borrowing, investing
- so, if you are ________________ money, you would prefer to have a 10% loan that is compounded annually, but if you are ______________ money, you would prefer to have interest compounded more frequently - truth-in-lending laws in the U.S. require lenders to disclose the APR on a loan - the APR is NOT the effective annual rate, though, and if interest is compounded more than once per year, the true rate for the loan is higher than the APR - APR = periodic interest rate X number of periods per year
solving for interest rate
- sometimes you know the PV, FV, and n, but you need to solve for this - you can do this using your calculator, using Excel, or using the following formula: -- r = (FV/PV)^1/n - 1
solving for number of periods
- sometimes you need to know how long it will take some present amount to reach a future level (solve for this) - you can solve for this using a financial calculator, using Excel, or using the following formula: n = ln(FV/PV) / ln (1 +r) *ln means natural log
increases
- the EAR ________________ with increasing compounding frequency - also, the stated rate and the EAR are equal when you have annual compounding - so, the EAR is the interest rate expressed as if it were compounded once per year
more
- the _______ frequently interest is compounded, the higher the future value
higher
- the ____________ the discount rate, the lower the present value - also, the longer the period of time, the lower the present value - this is the exact opposite of future value - if the discount rate is zero the present value will equal the future value
annual percentage rate (APR)
- the annual rate of interest based on interest being compounded once per year - = periodic interest rate x number of periods per year - = periodic rate x m
mixed stream of cash flows
- the cash flows are not all the same amount and/or they are not received at regular intervals
always greater
- the future value of an annuity due is _____________ _____________ than the future value of an otherwise identical ordinary annuity (because you are earning an extra year's interest)
effective annual rate (EAR)
- the true rate of return to the lender and the true cost of borrowing to the borrower based on compounding frequency - is the rate of interest actually paid or earned - it differs from the stated rate of interest in that it reflects the impact of compounding frequency - this is the rate you would need to earn with annual compounding to be as well off as you are with multiple compounding periods per year -- EAR = (1 + APR/m)^m - 1 - in the formula write the APR as a decimal - the more times interest is compounded per year, the higher this is - this is the interest rate expressed as if it were compounded once per year - in the formula write the APR as a decimal
future value
- the value of a present amount at a future date found by applying compound interest over a specified period of time -- how much money will you have at the end of five years if you invest $100 today in a savings account earning 5% compounded annually?
value of any financial asset
- the value of any financial asset is the present value of the cash flows that you expect to receive from that asset -- share of stock -- bond -- real estate investments -- project for a business -- equipment for a business
two types of annuities
- there are ______ ________ ____ _______________: -- ordinary annuity -- annuity due - always assume that payments are made at the end of the period unless otherwise specified
solve
- there are five variables associated with time value of money calculations: -- present value = PV -- future value = FV -- interest rate = I/Y or r -- number of periods = N or n -- payment = PMT - we can ________ for any of these variables using a formula, calculator, or Excel
loans
- there are three major types of these: -- discount loan -- interest-only loan -- amortized loan
interest-only loans
- these pay the annual interest each year and then repay the principal in full at the maturity date - if you have a $25,000 six-year loan with an 8% interest rate, you would pay: -- 8% x $25,000 = $2,000 per year in interest every year for six years -- pay back the $25,000 principal at the end of the sixth year - this is how bonds work
FV of a mixed stream of cash flows
- to calculate this, you simply take the future value of each cash flow, and then add all them all together - time lines are very important when calculating the FV or PV of a mixed stream of cash flows - you can use your financial calculator to find the FV of each cash flow, but there is no calculator "shortcut" for calculating this -- you have to calculate the FV of each cash flow individually, and then sum them
PV of a mixed stream of cash flows
- to find this, simply calculate the present value of each cash flow and then add the all together to get this - the value of any financial asset is equal to the present value of the cash flows you expect to receive from that asset
PV of an annuity due
- when the payments are made/received at the beginning of the year instead of the end of the year, you have an annuity due - to calculate the this, simply multiply the PVA for an ordinary annuity by (1+ r) -- PVAdue = PMT * [1 - (1/(1 + r)^n ]/ r * (1 + r)
PV of an ordinary annuity
- when you calculate this (equal cash flows received or paid at regular intervals), you are discounting all of the cash flows back to today's dollar at the given interest rate - when the cash flows constitute an annuity (same cash flows at regular intervals), you can use the following formula instead of calculating the PV of each individual cash flow: -- PVA = PMT * 1 - [1/(1+r)^n]/r
discount loan
- with this, you pay off the principal (the original loan amount that you borrowed) AND all of the interest at one time at the maturity date of the loan - this is how U.S. Treasury Bills and U.S. Savings Bonds work Y -- you borrowed $25,000 from the bank and agree to repay the principal and interest (rate is 8%) at the end of six years -- FV = 25,000 (1.08)^6 = 25,000 (1.5869)= $39,671.86 -- so, the bank will loan you $25,000. For six years, you will not make any payments, but at the end of the sixth year, when the loan matures, you will payback the principal ($25,000) plus all of the interest that has accrued over the six years ($14,671.86)
adjusting
- you can calculate the FV or PV of any lump sum, annuity, or mixed stream of cash flows using the standard formulas and ________________ for periodic interest rate and total # of compounding periods
using a financial calculator
- you can solve time value of money problems quickly and easily ________ ___ _____________ _______________. you will use the following five keys: - I/Y = annual interest rate. enter it as a whole number (i.e. enter 9.8% as "9.8."). When you use the formula, you use decimal form (.098), but when you use a financial calculator, use the whole number - N = number of periods/years - PV = present value. Enter this as a negative. For example, if you are depositing $400 today and looking for the future value, you will enter "-400" as the present value - PMT = payment. You don't have any payments with lump sum, so enter "0" for the payment - FV = future value. When you are looking for the FV, enter "CPT FV"
future value of a lump sum
FV = PV * (1 + r)^n where: - FV = the future value at the end of the nth period - PV = the initial principal or present value - r = the annual rate of interest paid - n = the number of periods, typically years
