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Maths (Statistics (RESIDUAL PLOTS: (From Residual Plot: Data is linear if…
Maths
Statistics
Co-Dependant:
Variables rely equally on each other
RESIDUAL PLOTS:
From Residual Plot: Data is linear if
:
Same number of point above + below line
If the numbers are small
If there is no pattern
POSITIVE RESIDUAL:
Above least squares regression line
NEGATIVE RESIDUAL:
below least squares regression line
2nd-STATPLOT-Change Ylist (L2) to RESID by:
2nd-STAT-RESID-Trace
To get scatter plot:
:
STAT-EDIT-L1(IV),L2(DV)-Y=make sure you clear-2nd -STAT PLOT- ENTER-ON-ZOOM
r and r^2
Pearson correlation coefficient: r
r= -1, Perfect negative correlation
r= 0, No Correlation
r=1, Perfect Positive correlation
r= Correlation
r= 0.7 Positive-strong
Anything below is weak
Coefficient coefficient (direction); r^2
r^2=0,No correlation
r^2=1, perfect correlation
r2= How close the data is fitted to the regression line
r2=0%= Model explains none of the variability of the response data around its mean.
r2=100%= Model explains all the variability
Find the r2 and interpret the meaning:
r2= 0.598
59.8% of the variation in the money lost can be explained by the time spent gambling
Rogue Values: Outliers
To find curve; exponential model:
STAT-CALC-Expreg
Explanatory Variable:
Indépendant Variable(Xaxis)
Response Variable:
Dependant Variable (Yaxis)
EQUATION OF LINE:
2nd-Catalogue-Diagnostics ON
STAT-CALC-4.LinReg-Xlist:L1-Ylist:L2-Storereg-VARS-Yvars-Function-Y1-Calc
NORMAL DISTRIBUTION
How many are expected to score within 10% of mean?
Mean: 45 Stdev:8.2
P(35<x<55)
NormalCDF (35,55,45,8.2)
=0.777
For 200 students:
0.777x200=155students
P(x>K) =3.5%
Find P(x<K) =100-3.5=0.968
Suppose x is normally distributed with mean= 80 STDEV:10
Find P(x<72)
NormalCDF(-E99,72,80,10)
=0.312
Hence, find K, P(72<x<K)=0.1
0.312+0.1=0.312
INVNORM(0.312,80,10)=75.1
ASSUME THERE ARE NO HALF POINTS IN STATISTICS
DESCRIBING SHAPE OF CURVE:
As x increases, Y also increases
Y is always greater than zero
The y intercept is...
As x increases, Y approaches zero
INTERPOLATION:
Prediction within the data
EXRTAPOLATION:
Prediction outside data range, unreliable
Financial Modelling
Loans
PV= -
Total Paid To Lender:
Interest + actual loan amount
SINKING FUNDS AND INTEREST ONLY LOANS
Interest only loan $2500 000, 6% compounded half yearly for 5 years.
Sinking fund earns 4.5% compounded monthly
a)
find half yearly interest payments
2500 000x0.06x(1/2) =$75000
b)
find sinking fund repayments
N= 60
I= 4.5
PV= 0
PMT= ? = -37232.55
FV= 2500 000
P/Y= 12
C/Y= 12
c)
find value after 3 yrs
N= 36
I= 4.5
PV= 0
PMT= -37232.55
FV=? =1432190.57
P/Y= 12
C/Y= 12
d)
find book value after 3yrs
2500000-1432190.57 =1067809.43
e)
find total cost of loan
37232.55x60 =2233953
75000x10 = 750 000
2233953+750000
=2983953
COMPARISON RATES
$75000, 14yrs
a)
7.85% compounded monthly, no set up fees or monthly fees
Comparison rate= 7.85%
b)
7.78% compounded monthly, $6 monthly fee
STEP 1: No Initial Fees PV= 75 000
STEP 2:
N= 168
I= 7.78
PV= 75000
PMT= ? = -75000
FV= 0
P/Y= 12
C/Y= 12
STEP 3: (ADD MONTHLY)
734.15+6= 740.15
STEP 4:
N= 168
I= ?? =7.92%
PV= 75000
PMT= -740.15
FV= 0
P/Y= 12
C/Y= 12
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WHEN THERE IS INITIAL FEES, USE ORIGINAL PRINCIPLE W/O FEES AT STEP 4
Investments
PV= +
Sinking Funds Benefit to Investors:
They can claim as a tax deduction
Invests
$4000, earns 5%, compounded monthly for 6 years
a)
find final value of investment
N= 72
I= 5
PV= -4000
PMT= 0
FV= ? =5396.07
P/Y= 12
C/Y= 12
b)
how much interest earnt?
5396.07-4000 = 1396.07
c)
marginal tax rate is 45%, calc tax paid
0.45x1396.07 = 628.23
d)
inflation 3%, index for inflation
4000x(1.03)^6 =4776.21
Invests
$20000 for 3yrs, earns 3.85% compounded monthly, Inflation 3.4%
a)
find value after 3yrs
N= 36
I= 3.85
PV= -20000
PMT= 0
FV= ? =22444.54
P/Y= 12
C/Y= 12
b)
Index for inflation
20000x(1.034)^3 = 22110.15
c)
find real rate return
N= 3 (NUM OF YEARS)
I=?? =0.502%
PV= -22110.15 (INFLATION)
PMT= 0
FV= 22444.54 (VALUE)
P/Y= 1
C/Y= 1 (ONE)
TAX
Only interest, not payments
INTEREST
Simple Interest:
P x i x n
P= I/(ixn)
i=I/(Pxn)
n=I/(Pxn)
Compound Interest
FV=PV(1+i)^n
When compounded monthly times n by months/quarters
When given PV:
FV=4418.67, PV(1.06)^4
= 4418.67/(1.06)^4
EFFECTIVE RATE
EFF(6,2)
6= Compound interest %
2= Num of Compounding periods
Assumptions
Afford repayments throughout duration
Size of repayments and Interest rate stays same/fixed
Wage stays same
Contributions stay the same
Limitations:
Interest may go up
May want more expensive/better later on
Discrete models
Forward Scan:
Largest number
Backwards Scan:
Smallest number
Dummy Links:
HUNGARIAN ALGORITHM STEPS
EXPLAINING WHY OPTIMUM ALLOCATION CANNOT BE DONE WHEN ONLY TWO LINES:
There are a minimum of two lines to cover the zeroes, but
there needs to be a minimum of 4 lines o cover the zeroes.
Fortnights: 26
Weeks: 52
Days: 365
To get years:
244/12
=20.33
0.33x12= 4
20 years and 4 months