FCS 3450 Formula Sheet
Unit 01
1. Relative price and Nominal price
Notations:
RPx = Relative price of commodity x
NPx = Nominal price of commodity x
NPb = Nominal price of base commodity
Formula: RPx =
NPx
NPb
Unit 02
1. Inflation rate and the purchasing power of money
Notations:
Yn = The purchasing power of one dollar after n years
ia = Annual inflation rate
n = Number of years
Formula:
1 n
Yn = (
)
1 + ia
2. Calculate inflation rate from CPI numbers
Notations:
iAB=inflation rate from year A to year B
CPIA=CPI for year A
CPIB=CPI for year B
Formula:
i AB =
CPI B
−1
CPI A
Note: This is just a regular percentage computation formula.
3. Formula for converting one year’s dollar into another year’s dollar:
Notations:
B=year B
A=year A
YB->A = converting money from year B to year A
Formula: YB − > A = YB ×
CPI A
CPI B
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4. Real interest rate and Nominal interest rate
Notations:
nr = nominal interest rate
rr = real interest rate
I = inflation rate
Formula:
nr = rr + i + (rr × i ) or
rr =
nr − i
1+ i
Note: These two formulas depict the same relationship. One can be derived from the other.
5. Expected value computation
Expected value = Sum of (outcome i * probability of outcome i )
Unit 03
Notations:
FV=future value
P = lump sum principle
Pp=amount of periodical payments
r = interest rate for a certain period
n = number of periods
1. Future Value Factor
FVF= (1+r)n
2. Future value for one time lump sum investment
FV=P*FVF
3. Future Value Factor Sum (BOM)
FVFS = (1 + r ) n + (1 + r ) n −1 + ... + (1 + r )1 =
(1 + r ) n +1 − 1
−1
r
4. Future Value Factor Sum (EOM)
FVFS = (1 + r ) n −1 + (1 + r ) n − 2 ... + (1 + r ) 0 =
(1 + r ) n − 1
r
5. Future value for equal periodical investment
FV=Pp*FVFS
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Unit 04
Notations:
PV=present value
P = lump sum amount
Pp=amount of equal periodical payment
r = discount rate for a certain period (often we use interest rate to approximate discount rate)
n = number of periods
1. Present Value Factor
PVF =
1
(1 + r ) n
2. Present value for one time lump sum payment
PV=P*PVF
3. Present Value Factor Sum (BOM)
PVFS =
1
1
1
+
+ ... +
= 1+
0
1
(1 + r )
(1 + r )
(1 + r ) n −1
1−
1
(1 + r ) n −1
r
4. Present Value Factor Sum (EOM)
PVFS =
1
1
+ ... +
=
1
(1 + r )
(1 + r ) n
1−
1
(1 + r ) n
r
5. Present value for equal periodical payments
PV=Pp*PVFS
Unit05
1. Predicting inflation rate
Predicted inflation rate in year A
= Money supply growth rate in year (A-2) - Economy growth rate in year A
Unit06
1. Interest rate spread
The interest rate spread = long-term interest rate - short-term interest rate
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Unit07
1. Conventional mortgage
a. Mortgage Payment
Monthly payment (M) = Loan amount / PVFS (n, monthly r)
n=number of months for the loan, r=mortgage interest rate
Keep in mind that loan = purchasing price – deductable.
b. Mortgage interest payment
I=last month balance * monthly r
c. Mortgage principal payment
P=monthly payment (M) – interest payment (I)
d. Mortgage balance
Mortgage balance = Previous monthly balance – this month’s principal payment (P)
2. Homeowner tax benefit
Homeowner tax benefit = marginal tax rate * (annual interest paid on mortgage loan + annual property
taxes -standard deduction)
3. Computing rate of appreciation (A)
A=(selling price/purchasing price)^(1/n) -1
n=number of years one has the house.
Note often the actual money you will get is (selling price – commission), which is often 6% of the selling price.
However here we are not taking that into consideration.
Unit 08
1. Buy or rent
Step 1. Select a holding period for the comparison: n years. All numbers will be converted to FV
Step 2. Calculate the FV of the net one-time costs of homeownership.
First compute Net one-time costs of homeownership = down payment + closing cost
Then convert it to FV =cost * (1+r)^n
Step 3. Calculate the total FV of "net homeownership periodic cost"
Compute mortgage payment first – see unit07. Remember loan = purchasing price – deductable.
Compute tax benefits – see unit07
Compute property tax, hazard insurance, operating cost, and alternative rent for each year. These are usually
simple math manipulations.
Now you can compute the net annual homeownership cost for each year
Net annual homeownership periodic cost = mortgage payment + property tax
+ hazard insurance + operating and maintenance costs - tax benefits - alternative rent (including utility)
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Then you convert these costs in to FV using FVF
The first year FV=first year cost *(1+r)^n
The second year FV=second year cost *(1+r)^(n-1)
.
.
.
The last year FV= last year cost *(1+r)^0 = last year cost
Total FV of net homeownership periodic cost = sum of (FV for all years)
Step 4. Copy down the net outstanding loan balance at the end of holding period.
This is usually given to you.
Step 5. Sum the results of step 2, 3 and 4, calculate the required breakdown sell price by taking realtor
commission into consideration.
Breakdown sell price= (FV of net one time cost + FV of all net periodical cost + mortgage balance) /
(1-6%)
Step 6. Find the breakdown annual rate of housing value appreciation A.
Initial price * (1+A)^n=break-even selling price
A=(breakeven selling price/initial purchasing price)^(1/n)-1
Step 7.Compare the calculated break-even rate of housing value appreciation to forecast of housing value
appreciation. If the expected annual rate of appreciation is > 12% then buying a house is a better deal. Otherwise,
renting is a better deal.
2. PITI= (monthly mortgage payment + monthly property tax + monthly home insurance
payment)/monthly gross income.
3. PITI+debt ratio = (PITI+monthly debt payment)/monthly gross income
4. ARM computation
For the first period teaser rate is used. Monthly payment computed just like regular mortgage (a PVFS
application)
After rate is adjusted, one needs to figure out the new interest rate. If capped, check the cap as well.
New r without cap =index + spread (margin).
New r with cap = min (new r without cap, cap) – pick the one that is lower.
Remember that your n has changed as well. If originally your n=360 months, now one year has passed when the
interest rate changes, you only have 348 months left.
You also need to figure out the new balance. If it’s simple like one month you can do it (see Unit07). If it’s a long
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time I will give you the number.
Compute mortgage payment using PVFS (new r, new n) and the new balance.
4. Interest only mortgage
Monthly payment = loan * monthly interest rate.
5. Graduated payment mortgage –negative amortization
Balance increase for a month = interest payment – the actual payment
Unit 09
Cost and benefit of human capital investment from a purely financial perspective
(1) Cost
Cost = PV of (tuition + opportunity cost)
Note opportunity cost is usually foregone income you could have earned if you worked full time instead.
Detailed steps:
Step 1. Figure out (tuition + opportunity cost) for each year.
Step 2. Convert all of them to Present values (PV) using the alternative financial market interest rate
given to you.
Note: If the cost occurs next year, then discount it for one year. If the cost occurs two years after, then discount
the cost for two years, so on and so forth. Note that because each year the cost is likely to be different, you
cannot apply PVFS to simply the computation. I usually give you 1 or 2 years so the computation is not too long.
In the exam I have on the slides, it is 4 years.
Step 3. Add all the PV of costs up you have the PV of total costs.
(2) Benefit
Benefit = PV of (all future income increase due to this education)
Note because usually there are many years you can get this benefit (from graduation to retirement), I will
simplify things and assume the benefit you get each year is the same dollar amount. That way you can use PVFS.
If the number is different every year (which is likely to be the case in reality) this computation will be very, very
long unless you use a spreadsheet software.
Detailed steps:
Step 1. Figure out the relevant years.
Remember you need to convert all future benefits into PV, which is THIS YEAR. And you go to school next year
and on.
Use age might be helpful. If you are 25, go to school next year for two years, then you will be in school when
you are 26 and 27. You start working with that education at age 28. You retire at age 65. So the benefit you get is
from age 28 to age 65.
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The first year you will get benefit is at age 28, which is 3 years from now (28-25).
The last year you will get benefit is at age 65, which is 40 years from now (65-25).
I am going to use these numbers in the following formula as a general formula will need more notations that may
cause more confusion.
Step 2. Use PVFS to compute the PV of all future benefits.
PV = annual benefit *(1/(1+r)^3+ 1/(1+r)^40) = annual benefit *([PVFS(n=40, r]-[PVFS(n=2, r])
Note you can think of n=2 as the number of years you are in school if that helps. Those are the years you don’t
get benefits.
Unit 10
(1). Expected loss = Sum of (frequency of loss for outcome i * severity of loss for outcome i)
Note in this case i can be different for different situations. Once can have 20 outcomes or 2 outcomes. Whatever
the i is you just need to add them all up. Also, the sum of all the i’s do not need to add up to 100% (this is
different from expected value covered earlier). This is because there is a large chance you never suffer any loss.
(2) Insurance premium = expected loss + service charge (or administrative cost)
(3) Computing out-of-pocket (OOP) payments
Note here the importance of cap. In this textbook caps are always put on co-insurance. In those cases deductable
still needs to be paid in addition to co-insurance. In reality there might be different configurations but for this
class we stick with this scheme for simplicity as it is good enough to make the conceptual point.
OOP=Deductable + Co-payment
Co-payment = Minimum of [ (Total medical bill – deductible) * co-insurance rate, or cap].
Minimum means you take the smaller number of these two.
(4) Insurance payment
Insurance payment = total medical bill – OOP.
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Unit 11.
1. Annual effective yield (AEY) calculation:
n=number of years (note if time period is given in months you need to convert it to years)
a. On capital gain: AEY= (1+ total proportional gain)^ (1/n) – 1
Proportional gain = ( New Price / Original Price) -1
b. On income distribution: AEY= (1+ percentage gain)^ (1/n) – 1
Percentage gain = Income dividends / Original Price
c. If an investment has both capital gain and income distribution then
Total AEY=AEY on capital gain + AEY on income distribution
2. After-tax AEY
After-tax AEY=AEY*(1-marginal tax rate)
Unit 12.
1. Retirement planning
Step 1. Determine the annual consumption level you want when you retire – be sure to adjust for inflation
Inflation-adjusted dollars= Current dollars *(1+i)^n
i=annual inflation rate, n=number of years
Step 2. Determine how much annual income you will have in retirement from Social Security and/or pension
Step 3. Determine the gap between what you need and what SS provides – this is the amount you will need to get
from your own retirement saving
Gap=Inflation-adjusted dollars you need – (SS income + Pension)
Note these need to be all inflation adjusted to the same year.
Step 4. Determine the retirement saving amount (nest egg) you need in order to generate that annual income
Nest egg = Gap*PVFS(n, r)
n=number of year you will live after retirement (use average life expectancy), r=annual annuity interest rate
Step 5. Determine how much to save each year in order to have that nest egg
Annual saving = Nest egg / FVFS(n, r)
N=number of years you will be saving for retirement, r=annual interest rate for saving for retirement
2. Annual percentage rate (APR)
Monthly payment = (Loan amount – upfront fee)/ PVFS (n, rm=monthly APR)
Numerical methods are needed to solve for monthly APR, and APR=monthly APR*12
Solving APR is not required for this class. However you should know the setup.
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