Business mathematics gmb 105 pres

Business Mathematics GMB 105

Victor Gumbo - PhD, CBiiPro

NUST

28 August - 1 Sept 2014 (HRE) and 2 - 6 Sept 2014 (BYO)

Victor Gumbo - PhD, CBiiPro Business Mathematics GMB 105

Data and their presentation

There are two types of statistics:• Descriptive Statistics• Inferential Statistics - makes possible theestimation of a characteristic of a population basedonly on sample results.


Sampling and Data Collection

Types of data

I Categorical random variables: yield categoricalresponses - yes/no, agree/disagree, etc.

I Numerical RV: can be discrete or continuous.


Data Collection

Typically data are collected through sampling:• Simple Random Sample - every individual in thepopulation has the same probability of beingchosen. This can be done with replacement orwithout replacement.


Data Collection

• Stratified Sample - The N individuals are firstsubdivided into separate sub-populations, or strata,according to some common characteristic. Thisallows for over-sampling of some groups to ensurerepresentation.


Data Collection

• Cluster Sample - The N individuals are dividedinto several clusters so that each cluster isrepresentative of the entire population. Then arandom sample of clusters are taken and allindividuals in the selected clusters are studied.Counties, election districts, families, etc.


Data and their Accuracy

Problems with Sampling1. Selection Bias - occurs when certain groups inthe population were not properly included, e.g., inblack/white wage differential, medical trials.2. Non-response Bias - occurs when certain groupsin the population do not respond to the survey, e.g.,in Mail or Telephone surveys.


Data and their Accuracy

3. Measurement Error - reflects inaccuracies in therecorded responses, due to weakness in wording ofsurvey, interviewer’s effect on respondent, effortmade by respondent.4. Sampling Error - reflects the heterogeneity orchance differences from sample to sample, +- 4percentage points, margin of error.


Frequency Distributions and Charts

I The frequency of a particular event is thenumber of times that the event occurs.

I The relative frequency is the proportion ofobserved responses in the category.


Example:

We asked a group of students what country theircar is from (or no car) and made a tally of theanswers. Then we computed the frequency andrelative frequency of each category. The relativefrequency is computed by dividing the frequency bythe total number of respondents. The collecteddata was as follows.US = 6; Japan = 7; UK = 2; Korea = 1; None =4.


Example:

(a) Produce a relative frequency table.(b) A bar graph is called a Pareto chart since theheight represents the frequency. The widths of thebars are always the same. Produce a bar chart ofthe data given.


Example:

(c) We make a circle graph often called a pie chartof this data by placing wedges in the circle ofproportionate size to the frequencies. To find theangles of each of the slices we use the formula

A =Freq

Total× 360 (1)


Histograms

Histograms are bar graphs whose vertical coordinateis the frequency count and whose horizontalcoordinate corresponds to a numerical interval.


Example: Histogram

The annual salaries of a certain number ofemployees in different companies was recorded withthe results in thousands of US dollars as follows:15.4, 16.7, 16.9, 17.0, 20.2, 25.3, 28.8, 29.1, 30.4,34.5,36.7, 39.1, 39.4, 39.6, 39.8, 40.1, 42.3, 43.5, 45.6,45.9,48.3, 48.5, 48.7, 49.0, 49.1, 49.3, 49.5, 50.1, 50.2,52.3.


Example: Histogram

Using a frequency distribution table with classintervals of length 5, produce a table of ClassInterval, Frequency, Relative Frequency andCumulative Relative Frequency.


The Shape of a Histogram

I A histogram is unimodal if there is one hump,

I Bimodal if there are two humps andmultimodal if there are many humps.

I A nonsymmetric histogram is called skewed ifit is not symmetric.


Positive or right skewed

I The right tail is longer; the mass of thedistribution is concentrated on the left of thefigure. It has relatively few high values.

I The distribution is said to be right-skewed or”skewed to the right”.

I Example (observations): 1,2,3,4,100 (the meanis greater than the median).


Negative or left skewed

I The left tail is longer; the mass of thedistribution is concentrated on the right of thefigure. It has relatively few low values.

I The distribution is said to be left-skewed or”skewed to the left”.

I Example (observations): 1,1000,1001,1002. (themean is less than the median).


Statistical measures

I In this section we recap the statistical measuresmean, median, mode and range.

I The mean, median and mode give an indicationof the ‘average’ value of a set of data, i.e. someidea of a typical value.

I The range, however, provides information onhow spread out the data is, i.e. how varied it is.


Statistical measures: Arithmetic Mean, Median, Mode andRange

Mean =sum of all data

number of values

=

∑xi

n

I Mode = most common value

I Median = middle value when data is arranged inorder

I Range = largest value smallest value.


Example 1

Shoe Size vs Frequency4 25 46 77 58 69 310 3.


Example

For tCompound interesthe above data, calculate(a) the mode;(b) the median;(c) the mean;(d) the range.


Example

Data on the number of minutes that a particulartrain service was late have been summarized in thetable. (Times are given to the nearest minute.)Minutes Late vs Frequencyon time 191-5 126-10 911-20 421-40 441-60 2over 60 0


Example 2

(a) How many journeys have been included?(b) What is the modal group?(c) Estimate the mean number of minutes the trainis late for these journeys.(d) Which of the two averages, mode and mean,would the train company like to use in advertisingits service? Why does this give a false impression ofthe likelihood of being late?(e) Estimate the probability of a train being morethan 20 minutes late on this service.


Descriptive Stats

Use the relevant Excel file.


Other Measures

QuartilesQuartiles split the observations into four equal parts.The first quartile Q1 = 1(n + 1)/4 orderedobservations.The second quartile Q2 = 2(n + 1)/4 orderedobservations.The Third quartile Q3 = 3(n + 1)/4 orderedobservations.


Other Measures

Example: Given64.8 130.5 159.2 358.6 395.3 414.6 439.8So Q1 = 8/4 = 2, Q2 = 4, Q3 = 24/4 = 6.


The Interquartile Range

The interquartile range is obtained by subtractingthe first quartile from the third quartile, IQR =Q3 − Q1.


Variance, Standard Deviation and Volatility

I These are measures of how the values fluctuateabout the mean.

I The sample variance is roughly the average ofthe squared differences between each of theobservations in a set of data and the mean:

S2 =

∑ni (xi − x̄)

n − 1(2)

The Sample Standard Deviation is just thesquare root of the variance.


Volatility Estimation

Calculate the volatility Oil Price Data given in theExcel file.


Regression and correlation

I In modern science, regression analysis is anecessary part of virtually almost any datareduction process.

I Popular spreadsheet programs, such as QuattroPro, Microsoft Excel, Matlab, SAS, etc providecomprehensive statistical program packages,which include a regression tool among manyothers.


Classification of linear regression models

In a regression analysis we study the relationship,called the regression function, between one variabley , called the dependent variable, and severalothers xi , called the independent variables.


Classification of linear regression models

• Regression function also involves a set of unknownparameters βi .• If a regression function is linear in the parameters(but not necessarily in the independent variables!)we term it a linear regression model. • Otherwise,the model is called non-linear. Linear regressionmodels with more than one independent variable arereferred to as multiple linear models, as opposed tosimple linear models with one independent variable.


Main Objectives of Multiple Linear Regression Analysis

• Our primary goal is to determine the best set ofparameters βi , such that the model predictsexperimental values of the dependent variable asaccurately as possible (i.e. calculated values yjshould be close to experimental values y ∗j ).


Main Objectives of Multiple Linear Regression Analysis

• We also wish to judge whether our model itself isadequate to fit the observed experimental data (i.e.whether we chose the correct mathematical form ofit). We need to check whether all terms in ourmodel are significant (i.e. is the improvement in“goodness” of fit due to the addition of a certainterm to the model bigger than the noise inexperimental data)


Multiple linear model

General formula:y = β0 + β1x1 + β2x2 + · · ·+ βnxn + µ


Multi-Collinearity

• Multi-collinearity tests for existence of correlationamong two or more explanatory variables.• It arises from the perfect linear relation amongexplanatory variables.• Multi-collinearity increases the standard errors ofthe coefficients.


Multi-Collinearity

• Increased standard errors in turn mean thatcoefficients for some independent variables may befound not to be significantly different from zero,whereas without multi-collinearity and with lowerstandard errors, these same coefficients might havebeen found to be significant.


Multi-Collinearity

• In other words, multi-collinearity misleadinglyinflates the standard errors and consequently resultsin inaccurate parameter estimations. Thus, it makessome variables statistically insignificant while theyshould otherwise be significant and will also inflatethe log-likelihood ratio of the logistic model.However, if proven to be present, multi-collinearitycan be remedied by either adding or dropping somevariables.


Example

Consider a salaries model for some HR. Theindependent variables are Experience (in years),Educational Level (in number of degreescertificates), Old Salary (in US dollars) and Age.


Compounding, discounting and annuities

Interest: It is the additional money besides theoriginal money paid by the borrower to the moneylender in lieu of the money used.Principal: The money borrowed (or the moneylent) is called principal.Amount: The sum of the principal and the interestis called amount.


Compounding, discounting and annuities

Thus, amount = principal + interest.Rate: It is the interest paid on principal for aspecified period.Time: It is the time for which the money isborrowed.Simple Interest: It is the interest calculated onthe original money (principal) for any given timeand rate.Formula: Simple Interest = (Principal x Rate xtime)/100


Compound Interest

Compound interest (abbreviated C.I.) can be easilycalculated by the following formula:

A = P(1 +

r

100

)n

where A is the final amount, P is the principal, r isthe rate of interest compounded yearly and n is thenumber of years.

C .I . = A− P = P(1 +

r

100

)n

− P

= P[(

1 +r

100

)n

− 1]


Exercise

Example(a) Find the simple interest amount of 1500 for 6years at 5.2% annually. Ans: 1968(b) Find the compound amount of 1500 for 6 yearsat 5.2% compounded annually. Ans: A = 2033.


Compound Interest: Remark

I Interest may be converted into principalannually, semiannually, quarterly, monthly etc.

I The number of times interest is converted intoprincipal in a year is called the frequency ofconversion, and the period of time betweentwo conversions is called the conversion periodor interest period.


Compound Interest

Thus “rate of 5%” means a rate of 5%compounded annually; 12% compoundedsemi-annually means that each interest period of 6months earns an interest of 6%. Thus the rate ofinterest per interest period isr = (annual rate of interest)/(frequency ofconversion)and the number of interest periods isn = (given number of years) x (frequency ofconversion).


In solving problems on compound interest, remember thefollowing:

1.

A = P(1 +

r

100

)n

, and

C .I = P[(

1 +r

100

)n

− 1]

where A is the final amount, P is the principal, r isthe rate of interest compounded yearly (or everyinterest period) and n is the number of years (orterms of the interest period).



2. When the interest rates for the successive fixedperiods are r1%, r2%, r3%, ..., then the finalamount A is given by

A = P(1 +

r1100

)(1 +

r2100

)(1 +

r3100

). . .



3. S.I. (simple interest) and C.I. are equal for thefirst year (or the first term of the interest period) onthe same sum and at the same rate.



4. C.I. of 2nd year (or the second term of theinterest period) is more than the C.I. of 1st year (orthe first term of the interest period), and (C.I. of2nd year - C.I. of 1st year) = S.I. on the interest ofthe first year.



5. Equivalent, nominal and effective rates ofinterestTwo annual rates of interest with differentconversion periods are called equivalent if they yieldthe same compound amount at the end of the year.



For example, consider an amount of 10 000 investedat 4% interest compounded quarterly. So, theamount at the end of one year =10000(1.01)4 = 10406. This is equivalent tointerest of 4.06% compounded annually because10000(1.0406) = 10406.



When interest is compounded more than once in ayear, the given annual rate is called nominal rateor nominal annual rate. The rate of interest actuallyearned is called effective rate. In the aboveexample, nominal rate is 4% while effective rate is4.06%.



If nominal rate is r% compounded p times in year,then effective rate of interest is(1 + r

100p

)p

− 1.



6. Present value or present worth of a sum of Pdue n years at r% compound interest is

PV =P(

1 + r100

)n


Example

What is the PV of 2000 US dollars due in 2 years at5 percent compound interest?


Depreciation

I All fixed assets such as machinery, building,furniture etc. gradually diminish in value as theyget older and become worn out by constant usein business.

I Depreciation is the term used to describe thisdecrease in book value of an asset.


Depreciation

There are a number of methods for calculatingdepreciation.However, the most commonmethod which is also approved by income taxauthorities, is the Diminishing BalanceMethod.


Depreciation

II Here each year’s depreciation is calculated onthe book value (i.e., depreciated value) of theasset at the beginning of the year rather thanoriginal cost.

I Note that as the book value decreases everyyear, the amount of depreciation also decreasesevery year. Therefore, this method is also calledReducing Instalment Method or “WrittenDown Value Method”.


Depreciation

I If the rate of depreciation is i% per year and theinitial value of the asset is P , the depreciatedvalue at the end of n years is P

(1− i

100

)nand

the amount of depreciation isP[1−

(1− i

100

)n].

I If n is large, log tables should be used forcalculation.


Exercises

ExampleA car is currently valued at 6000 US dollars. If thecar model depreciates at 7% per year, what will bethe value of the car after 3 years?


Exercises

ExampleThe population of an industrial town is increasingby 5% every year. If the present population is 1million, estimate the population five years hence.Also estimate the population three years ago. Ans:1276 280 and 863 838


Cash Flows: DCF Method

I Capital investment decisions are long-termcorporate finance decisions relating to fixedassets and capital structure.

I Investment decisions are based on severalinter-related criteria.



I In general, each project’s value will be estimatedusing a discounted cash flow (DCF) valuation,and the opportunity with the highest value, asmeasured by the resultant net present value(NPV) will be selected.

I This requires estimating the size and timing ofall of the incremental cash flows resulting fromthe project.



I These future cash flows are then discounted totoday to determine their present value takinginto account the time value of money.

I These present values are then summed, and thissum net of the initial investment outlay is theNPV.


NPV, IRR, WACC

In DCF valuation, the value of an asset is thepresent value (PV) of the expected future cashflows on the asset.

PV =n∑

t=1

CFt(1 + r)t

, discrete state (3)

=n∑

t=1

CFte−rt , continuous state (4)


NPV, IRR, WACC

where CFt is the future cash flow in period t, r isthe appropriate discount rate given the riskiness ofthe cash flow, and n is the life of the asset. So, touse the DCF, you need to:• estimate n the life of the asset,• estimate the cash flows CF during the life of theasset,• estimate the discount rate r to apply to thesecash flows in order to get the PV.


NPV, IRR, WACC

The NPV is thus given by the formula:

NPV = −(Investment Cost) +

PV︷︸︸︷n∑

t=1

CFt(1 + r)t

(5)


NPV, IRR, WACC

I The generally accepted discount rate is theweighted average cost of capital (WACC).

I Practitioners use a rate that is slightly higherthan the risk free rate or a rate that iscommensurate with the organization’s WACC,which represents risk of “business as usual”.


NPV, IRR, WACC

I Cost of capital represents the cost of financingan organization’s activities which is normallydone through some combination of debt andequity.

I Since debt and equity carry different costs ofcapital, a weighted average is required.


NPV, IRR, WACC

The WACC of different cost components of issuingdebt, preferred stock, and common equity is:

WACC = WdCd(1− tr) +WpCp +WeCe , (6)

where W represents the respective weights, C is thecost corresponding to debt d , preferred stock c andcommon stock e; tr is the effective corporate taxrate.


NPV, IRR, WACC

I The internal rate of return (IRR) is a capitalbudgeting metric used by firms to decidewhether they should make investments.

I It is an indicator of the efficiency of aninvestment, as opposed to net present value(NPV), which indicates value or magnitude.

I The IRR is the annualized effective compoundedreturn rate which can be earned on the investedcapital, i.e., the yield on the investment.


NPV, IRR, WACC

I A project is a good investment proposition if itsIRR is greater than the rate of return that couldbe earned by alternate investments (investing inother projects, buying bonds, even putting themoney in a bank account).

I Thus, the IRR should be compared to anyalternate costs of capital including anappropriate risk premium.


NPV, IRR, WACC

I In general, if the IRR is greater than theproject’s cost of capital, or hurdle rate, theproject will add value for the company.

I Given a collection of pairs (time, cash flow)involved in a project, the internal rate of returnfollows from the net present value as a functionof the rate of return.

I A rate of return for which this function is zero isan internal rate of return.


NPV, IRR, WACC

Thus, in the case of cash flows at whole numbers ofyears, to find the internal rate of return, find thevalue(s) of r that satisfies the following equation:

NPV = −(Investment Cost) +n∑

t=1

CFt(1 + r)t

= 0 (7)


Generic DCF Model

You want to start a chicken business and you wantto assess the success prospects as well as apply for aloan from a bank, if it is a GO project.


Annuities

I An annuity is a series of regular, equally spaced,payments over a defined period of time (oftencalled the term) at a constant rate of interest.

I The payments may occur weekly, fortnightly,monthly, quarterly or yearly.


Annuities - Examples

Examples of annuities include:

I regular payments into a savings account orsuperannuation fund,

I loan payments and periodic payments to aperson from a retirement fund.


Ordinary Annuity

An Ordinary annuity is an annuity where the regularpayment is made at the end of the successive timeperiods.


Ordinary Annuity

• A loan from a bank is an example of an annuitywith a present value and repayments for the term ofthe loan.• In other words, the bank gives you the lump now(at present) and the repayments are made inperiodic payments after this.


Ordinary Annuity - PV

Each repayment must be changed to its presentvalue. Doing this and then considering the sum ofthe series, the sum of the series is

A = R1− (1 + r)−n

r(8)

where $A is the present value of an ordinaryannuity, $R is the amount of each payment, r is therate per period (payment) and n is the number ofperiods (payments).


Example 1

Mr. and Mrs. Gozho wish to have an annuity forwhen their daughter goes to university. They wishto invest into an annuity that will pay theirdaughter $1000 per month for 4 years. What is thepresent value of the annuity given that currentinterest rates are 8% p.a?


Solution

The information given is: R = 1000, r = 0.08/12per month, n = 12×4 = 48 months. Ans: 40961.91


Example 2

Pride borrows $20000 to buy a car. She wishes tomake monthly payments for 4 years. The interestrate she is charged is 10.5% p.a. What is the size ofeach monthly payment?


Solution

The lump sum of $20000 is given at the beginningof the loan, this indicates it is the present value ofan annuity: A = 20000, r = 0.105/12 per month,= 12× 4 = 48 months. We need to find R . Ans:512.07


The Future Value formula of an Ordinary Annuity

Where regular payments are made with a lump sumat the end, the lump sum at the end is called theFuture Value of an annuity.A good example of this is a saving scheme whereregular payments are made to build to a lump sumat the end of a period of time.In business, this is called a sinking fund. It is usedto save for the future replacement of major capitalitems


The Future Value formula of an Ordinary Annuity

The future value of an annuity is the value of allpayments at the end of the term. It is given by theequation:

S = R(1 + r)n − 1

r(9)

where S is the future value of the ordinary annuity,R is the periodic payment, r is the interest rate perperiod and n is the number of payments.


Example 3

Mr Sithole deposits $150 into a bank account at theend of the month for 5 years at a rate of 7%compounded monthly. Find the future value of theannuity. Hint:R = 150, r = 0.07/12, n = 5× 12 = 60.Ans:10738.94


Example 4

Evans is planning for his retirement 20 years away.When he retires he wants a lump sum of $300000.His financial advisor suggested that 5% p.a. was asuitable interest rate to consider. How much will hehave to pay per month into his retirement fund(assume ordinary annuity).Hint:S = 300000, r = 0.05/12, n = 12×20 = 240months. Ans:729.87


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