Financial Math Final Flashcards
Simple Interest Formula
I = Pit
Interest charges = PrincipalInterest rate per yearterm of the loan, measured in years
Approximate Time (definition)
Pretend that every month has exactly 30 days
Exact Time (definition)
Find the exact number of days using a serial table which gives numbers to every day of the year
Promissory Note (definition)
document which physically represents a loan
** often SOLD on the basis of DISCOUNT INTEREST, though they usually ACCRUE interest on a Simple Interest basis.
Leap Year (definition)
significant because it means that we’d have to divide the number of days (in exact time) by 366 instead of 365
Future Value (definition)
the amount that money will grow into at some specified point in the future (represented by S)
Simple Interest Factor
(1 + it) – multiplying by this factor moves money forward in time
Future Value formula
S = P(1 + it)
Prompt Payment Discounts (definition)
many contractors offer discounts for early payment of invoices for large amounts of money.
think of paying early as loaning the money to the contractor and the discount as interest earned on that “loan.”
Prompt Payment Discount formula
x/y, n/z = x% discount if paid within the first y days, otherwise the Net amount (n) is due in z days.
Present Value (definition)
the amount of money which must be invested now to grow into a given amount at some point in the future. Represented by “P.”
Present Value Formula
P = S
——-
(1 + it)
“discounting the note” (definition)
this happens when promissory notes (which represent loans) are sold to a third party. The price of the loan is always less than the maturity value.
How to find the discounted note value (“sale price”)
step 1. find the maturity value of the note (future value formula) if not already given.
step 2. find the present value of this maturity value at the sale date (present value formula).
**usually, the interest rates for these 2 steps will be different.
Rate of Return (R.O.R.) formula
i = I d = D 360 (100 - B)
—- ——- d= —— * ————-
Pt St t (B)
*simple *discount disc. for treas. bids
** t meas. in days
(S) i = ------ ^(1/n) - 1 (P) *compound
Focal Date (definition)
In order to reconcile payments and obligations made at different points in time, we have to move all money to a single point in time – the focal date.
“The Golden Rule of Finance” (definition)
monies cannot be added or reconciled unless they are valued at the same point in time.
Net Present Value (definition)
the difference between income and expenses after everything has been moved to the present.
**if we think of expenses as “negative income,” then NPV just means “total present value.”
** depends on interest rate
Internal Rate of Return (I.R.R.) (definition)
the interest rate that makes the net present value come out to ZERO.
Merchant’s Rule (definition)
the maturity date of the loan is used as the focal point. The final payment is equal to the original future value of the loan MINUS the future value of all partial payments.
The United States Rule (definition)
a new computation is done each time a payment is made. We compute the future value of the loan, subtract the payment, and then use the balance as the principle for the new loan.
Annuity (definition)
a sequence of payments made at regular intervals in time
- the payments are usually equal in size
- period of time between payments is called a RENT PERIOD
** if payments of R are made at the end of n consecutive interest periods, we can find the total future value at the day of the final payment
Future Value of an Ordinary Annuity formula
Sn = R * [(1 + i)^n - 1]
—————–
[ i ]
where:
R = rent
n = total # of payments
i = rate per period
Fence Post Principle (definition)
if the problem gives starting and ending dates, and the payments are made at both of these dates, then the total number of payments is ONE MORE than the NUMBER OF INTEREST PERIODS
“a fence of length n has n+1 posts”
Ordinary Annuity (definition)
payments occur at the END of each rent period
Annuity Due (definition)
payments occur at the BEGINNING of each rent period
Present Value of an Annuity (definition)
total present value of all payments one interest period BEFORE first payment (at the beginning of first rent period)
Present Value of an Ordinary Annuity formula
[ 1 - (1 + i)^-n ]
An= R * ——————-
[ i ]
Finding R (formulas)
R = (Sn) / [((1 + i)^n - 1) / i ] R = (An) / [(1 - (1 + i)^-n) / i ]
The Sale Price of an Annuity (definition)
the TOTAL PRESENT VALUE of the remaining payments
Three options when finding “n” (definition)
- to APPROXIMATE the # of payments, we must round to the nearest WHOLE NUMBER
- if we want to save AT LEAST the target value, we need to ROUND UP
- to save the target value EXACTLY, we round DOWN to get a # of full payments, and then do a separate calculation to find the amount of the final, partial payment
TVM Solver PMT setting for Ordinary Annuity?
“ “ “ “ for Annuity Due?
PMT @ END for ORDINARY annuities
PMT @ BEGIN for annuities DUE
** PMT= is always NEGATIVE
Truth in Lending Act (July 1969) (definition)
requires the annual percentage rate to be stated on all loan papers or contract documents issued by another lender. The stated finance charge or interest rate must include all costs included in obtaining a loan. Allows the borrower to change his mind within 3 business days.
APR (definition)
the nominal rate at which the cash value of the loan equals the present value of the payments
Add-on loan (definition)
a simple or discount interest loan paid back on a payment schedule instead of with a lump sum
Rule of 78s (definition)
used for rebates given on add-on loans, repaid early
Rebate formula
n(n + 1)
rebate = ————- * (finance charge)
m(m + 1)
where:
n= # of REMAINING PAYMENTS
m= TOTAL # of payments
finance charge= INTEREST or DISCOUNT
Future Value of an Annuity Due formula
Sn(due)= R * [(1 + i)^n - 1]
—————– * (1 + i)
[ i ]
Present Value of an Annuity Due Formula
[ 1 - (1 + i)^-n ]
An(due)= R * ——————- * (1 + i)
[ i ]
Finding the Partial Payment with the TVM Solver formula
to find An(due), go back to the TVM solver, change N to be exactly the time at which the partial payment occurs, enter normal rent, and solve for PV. (can do the same thing to solve for FV.)
- *** THEN, (this PV) + (x / [(1 + i)^n]) will equal the Future Value!
- *(WHERE: PV is the present value obtained from TVM and X is the partial payment to be found)
discount interest (definition)
based on the future value with interest paid at the beginning of the term
discount interest formula
D=Sdt
WHERE:
D=discount=cost for the use of borrowed $
S=amount= money being borrowed in the loan (future value of money)
d=discount interest rate (/yr)
t= term= length of the loan in years
discount “proceeds” (present value) formula
P = S (1 - dt)
discount “amount” (future value) formula
P
S = ———
(1 - dt)
Discount Interest Factor
multiplying by (1 - dt) moves money backwards in time and dividing by (1 - dt) moves money forward in time.
Coupon Equivalent Formulas
d i
i = ———- d= ———
(1 - dt) (1 + it)
Discounting Promissory Notes Formula
step 1. find the future value of the note if it is not given already
step 2. find the present value when the note is moved back to the sale date
Treasury Bills “T-bills” (definition)
the government borrows money by selling treasury bills at competitive public auctions via bank discount bids given as percentages.
Money Market (definition)
an informal network for buying and selling short-term credit instruments
Secondary Market (definition)
organized exchanges located in a specific place (i. e. NYSE)
Commercial Paper (definition)
promissory notes sold by a corporation to raise money
Federal Funds (definition)
loans between commercial banks (Large banks borrow from small banks to maintain their minimum legal reserve funds)
Banker’s Acceptances (definition)
loan instruments commonly used to expedite transaction between importers and exporters. The importer’s bank acts as a middle man and assumes the legal obligation to make a payment
Discounted Loans/Securities - REMEMBER
when working with discounted loans/securities, the amount given is typically the FUTURE VALUE
Compound Interest (definition)
simple interest applied repeatedly.
Interest Period (definition)
if interest is earned or computed m times a year, then we apply the simple interest formula every 1/mth of a year. this period of time is called an interest period
Nominal Rate (definition)
The yearly interest rate, denoted i(nom)
Rate per Period formula
i = i(nom) / m
Compound Interest Future Value Formula
S = P (1 + i)^n
n = m*t
where m= # of interest period per year
& t= # of years in the term
Compound Interest Present Value Formula
S
P= ————-
(1 + i)^n
Simple v. Compound Interest
simple – only earn interest on principal
compound – also earn interest on interest (this is the better option in the long term)
Annual Effective Rate (definition)
a.k.a. Annual Percentage Yield, APY
the equivalent simple interest rate earned by investment over the course of one year.
Annual Effective Rate Formula
i(eff) = [1 + (i nom/ m)]^m - 1
Finding i(nom) given i(eff)
i(nom) = m * [(1 + i eff)^1/m - 1]
Finding the Compound Interest Rate (i) formula
( S )^1/n
i = ( —– ) - 1
( P )
Finding “n” in a Compound Interest Loan formula
ln (S/P)
n= ———–
ln (1 + i)
Finding “n” in a compound interest Loan
*** if we need the exact number of days
step 1. future value calculation
step 2. t = Int Charges / Present Value * Interes
*** will give a decimal answer
step 3. t = 360 * (decimal from step 2)
Fortune Annuity (definition)
when using an annuity to save $, we often wait several interest periods after the last payment is made before using the money – called a fortune annuity.
Fortune Annuity formula
Sn(for) = R * [(1 + i)^n - 1]
—————– (1 + i) ^p
[ i ]
*where:
p = the number of interest periods between the end of the last payment and when you take out the money.
Deferred Annuity (definition)
when using an annuity to pay off a debt, we often wait several interest periods before making the first payment. This is called a deferred annuity.
Deferred Annuity fomula
[ 1 - (1 + i)^-n ]
An(def) = R * ——————- * (1 + i) ^(m-1)
[ i ]
Perpetuity (definition)
if we invest money and never withdraw the principal, we can generate an infinite sequence of payments. This is called a perpetuity.
Perpetuity formula
R = i * A(infinity) and A(infinity) = R/i A(infinity) = principal = total present value of all payments
Simple Annuity (definition)
interest is compounded at the same frequency that payments are made
General annuity (definition)
payments occur more or less frequently than interest is compounded
turning a general annuity into a simple annuity
if we have a general annuity, we can turn this into a simple annuity by pretending that interest is computed p times per year at a rate per period j that is equivalent to i.
j = (1 + i)^k - 1
WHERE:
# of interest periods per yr
k = ——————————————
# of rent periods per yr
