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MATH260. FINANCIAL MATHEMATICS
创建日期：2024-05-24 15:49:05
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标签： MATH260
FINANCIAL MATHEMATICS

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MATH260. FINANCIAL MATHEMATICS

PART I:
DEPARTMENT OF MATHEMATICAL SCIENCES
DR SIMON A. FAIRFAX
1
2PART I: Modern Portfolio Theory
1. Time value of money
.The time value of money (TVM) is an economic principle that suggests present day money
is worth less than money in the future because of its earning power over time. Put simply a
pound today is worth more than a pound next year because money can be invested today and
earn interest. The time value of money relates to three basic parameters: inflation, opportunity
cost and risk.
.Inflation is reducing what is known as the purchasing power of money because it increases the
prices of goods and services. Therefore, over time the same amount of money can purchase fewer
goods and services. Opportunity cost refers to the potential gain or loss on an investment that
someone gives up by taking alternative action. Risk relates to the investment risk that investors
undertake when putting their money into investment assets. We will study investment risk and
returns in subsection 4.
1.1. Future Value.
.Money value fluctuates over time hence has different values in the future. This is because one
can invest today in an interest-bearing bank account or any other investment and that money will
grow/shrink due to the rate of return.
.To evaluate the real worthiness of an amount of money today after a given period of time,
economic agents compound the amount of money at a given interest rate. Compounding at the
risk-free interest rate would correspond to the minimum guaranteed future cash flow. If one wants
to compare their change in purchasing power, then they should use the real interest rate which
incorporates inflation.
.Future value (FV) is the amount to which a current investment will grow over time when placed
in an account that pays compound interest. The process of going from today’s values, known as
the present values (PV) to the future value (FVs) is called compounding.
Suppose you deposit £100 in a bank that pays 10% interest each year. What is the
future value of this deposit at the end of years 1, 2 and 3?
Example 1.1
Solution 1.1. At the end of each the first three years the investment will grow to £110, £121
and £132.10 respectively. This information is best displayed in a time-line.
Years
0 1 2 3
Initial deposit £100
Interest earnt £10 £11 £12.10
Total future value £110 £121 £132.10
MATH260. FINANCIAL MATHEMATICS PART I: DEPARTMENT OF MATHEMATICAL SCIENCES 3
Mathematical formulas for calculating future values
.In general, to find how much the investor earns at the end of year n we using the compounding
formula
FVn = PV (1 + r)
n.(1.1)
where
• FVn = the future value at the end of year n,
• PV = the initial investment or initial value of your account,
• r = interest rate paid by the bank in the account holder, and
• n = the number of years.
A company invests £2M to clear a tract of land and plants some palm trees. The trees
will mature in 5 years, at which time the farm will have a market value of £5M . What
is the expected annual rate of return for the company’s investment?
Example 1.2
Solution 1.2. We are given n = 5, FVn = 5M and PV = 2M . We aim at finding r we know
from Equation (1.1) that
FVn = PV (1 + r)
n.
Hence FVnPV = (1 + r)
n i.e., r = n
√
FVn
PV − 1.
r = 5
√
5
2 − 1 = 0.2011
.Care is needed with the units. The rate of interest should be represented as a decimal in the
compounding formula. The investment offers an opportunity to gain 20.11% per annum.
How long will it take to double a capital investment of £c attracting interest at 6%
compounded yearly?
Example 1.3
Solution 1.3. We are given PV = c, FV = 2c and r = 6%, where c is the initial capital. We
aim at finding n. We have that
FVn = PV (1 + r)
n,
i.e. ln
(
FVn
PV
)
= n ln(1 + r) and therefore
n =
ln
(
FVn
PV
)
ln(1 + r)
=
ln 2
ln(1 + r)
.
n = 11.89. Hence it would take 12 years.
1.2. Present Value.
.Present value, often called the discounted value, is a financial formula that calculates how much
a given amount of money received on a future date is worth in today’s dollars. In other words, it
computes the amount of money that must be invested today to equal the payment or amount of
cash received on a future date.
.The process of finding present values is called discounting and the interest rate used to calculate
present values is called the discount rate.
4 DR SIMON A. FAIRFAX
Consider a riskless investment opportunity that will pay £133.10 at the end of 3 years.
Suppose your local bank is currently offering 10 percent interest on 3-year Certificates
of Deposit (CDs), and you regard the security as being exactly as safe as a CD. How
much should you be willing to pay for investment? What conclusions can we draw?
Example 1.4
Solution 1.4.
Representing cash flows Let’s set up a time line for the cash flows.
Years
0 1 2 3
FV3 = £133.10PV =?
Conclusion From the future value example, an initial amount of £100 invested at 10% per
year would be worth
£100× 1.103 = £133.10
at the end of 3 years. Hence, £100 is defined as the present value of the £133.10 due in 3 years
when the opportunity cost rate is 10%.
Interpretation of the fair value Using this defined value, we can make several conclusions.
• If the cost of the investment was less than £100, you should buy it, because its price would
then be less than the £100 you would have to spend on a similar-risk alternative to end
up with £133.10 after 3 years.
• If the investment cost more than £100, you should not buy it, because you would have to
invest only £100 in a similar-risk alternative to end up with £133.10 after 3 years.
• If the cost was exactly £100, then you should be indifferent. Therefore, £100 is defined as
the fair value of the investment.
Mathematical formulas for calculating present values
In general, the present value of a future cash flow given the discount rate and number of years
in the future that the cash flow occurs can be computed using the following equation.
PV =
CFn
(1 + r)n
,(1.2)
where
• CFn = the future cash flow occurring at the end of year n,
• PV = the present value,
• r = the interest or discount rate, and
• n = the number of years.
Theresa will retire in 15 years. This year she wants to fund an amount of 18, 000 GBP
to become available in 15 years. How much does she have to deposit into a pension plan
earning 8% annually?
Example 1.5
Solution 1.5. We are given n = 15, CFn = 18000GBP and r = 8%. We aim at finding PV
We know from Equation (1.2) that
PV =
CFn
(1 + r)n
.
Hence PV = 18.000(1+0.08)15 = 5674.35
MATH260. FINANCIAL MATHEMATICS PART I: DEPARTMENT OF MATHEMATICAL SCIENCES 5
2. Interest rates
.An interest rate is a promised rate of return denominated in some unit of account (GBP, USD,
EURO, etc) over some time period (a month, a trimester, a semester, a year, 10 years, or longer).
The time period and the unit of the account are indicted.
2.1. Nominal rates.
.The nominal rate rnom is the rate that is quoted by banks, brokers, and other financial institutions.
However, to be meaningful, the quoted nominal rate must also include the number of compounding
periods per year. The quoted rate does not account for the effects of compounding.
.An interest rate in GBP terms might be risky when estimated as a function of inflation. Suppose
you deposit 100 GBP in a bank that pays 10% interest each year. In one year’s time, you are
guaranteed to collect 110 GBP in cash, but at this time costs will be higher than one year ago.
2.2. Real rates.
.The real return on your investment, studied by Fisher, will depend on what your money can buy
in one year relative to what it could buy today. Using the following notation
• R the real interest rate
• i the inflation rate in a period
• r the interest rate received in that period
then the relationship between R and r is described by compounding income using the rate r
(numerator) and then discounting outgoings using the rate of inflation (denominator):
1 +R =
1 + r
1 + i
⇐⇒ R = 1 + r
1 + i
− 1 = r − i
1 + i
.
Hence, the real rate of return is the rate of interest received above inflation discounted to remove
the effects of inflation over that period,
(2.1) R =
r − i
1 + i
.
Simple example Let i be 4% in a year. With 10% nominal interest rate paid yearly, after netting
out the 4% depreciation in purchasing power of money, you are left with a net growth in purchasing
power of approximately 6% in the future. Here 6%/(1 + 0.04) = 5.77% is called the real rate of
interest for an investment made today.
2.3. Periodic Rate.
.The periodic rate is the rate charged by a lender or paid by a borrower each period. It can be
a rate per day, per week, per three-month, per six-month period, per year, or per any other time
interval.
.We find the periodic rate as using
rPER =
rnom
m
,(2.2)
where m denote the number of compounding periods a year.
.The future value of an initial investment at a given interest rate compounded m times per year
at any point in the future can be found by applying the following equation:
FVn = PV (1 +
rnom
m
)nm ,(2.3)
where n is the number of years.
6 DR SIMON A. FAIRFAX
Suppose that you invest 100 GBP in an account that pays a nominal rate of 10%,
compounded monthly. How much would you be paid after 3 years?
Example 2.1
Solution 2.1. We are given n = 3, m = 12, PV = £100 and r = 10%. We know from (2.3) that
FVn = PV (1 +
rNom
m
)nm
Hence FV3 = 100(1 +
0.1
12 )
3×12 = £134.82.
Calculate the point in time at which some initial capital c has doubled, if interest is
compounded (a) monthly or (b) weekly, using an interest rate of r% per annum (p.a.).
In particular give a numerical answer to the above for r = 5.
Example 2.2
Solution 2.2. We have FVn = 2c, PV = c, m = 12 or 52. We aim at finding n given in terms
of the compounding periods. It follows from (2.3) that
FVn = PV (1 +
r
m
)nm.
Hence ln
(
FVn
PV
)
= mn ln(1 + r%m ) and therefore
n =
ln 2
m ln(1 + rm )
(a): In this case, m = 12 and hence
n =
ln 2
12 ln(1 + r%12 )
(b): In this case, m = 52 and hence
n =
ln 2
52 ln(1 + r%52 )
Numerical solutions for r = 5
(a): n = ln 2
12 ln(1+ 0.0512 )
= 13.9. The initial capital would have doubled after 13.9 years. But
if you are only allowed to take your earning at the end of each month then it would have
doubled (and even a little) after 13 years and 11 months. (Assuming 0.9 years corresponds
to 0.9× 12 = 10.8 months)
(b): n = ln 252 ln(1+ r52 )
= 13.87. The initial capital would have doubled after 13.87 years. But
if you are only allowed to take your earning at the end of each week then it would have
doubled (and even a little) after 13 years and 46 weeks. (Assuming 0.87 years corresponds
to 0.87× 52 = 45.24 weeks)
MATH260. FINANCIAL MATHEMATICS PART I: DEPARTMENT OF MATHEMATICAL SCIENCES 7
Assume that one year from now, you will deposit £1, 000 into a savings account that
pays 8%. How much will you have in your account five years from now (a) If the bank
compounds interest annually? (b) if the bank used quarterly compounding rather than
annual compounding?
Example 2.3
Solution 2.3. We simply need to know that the time to maturity is 4 years (5 years from now
minus 1 year from now)
(a): With annual compounding, the investor will have:
1000(1 + 0.08)4 = 1360.49.
(b): With quarterly compounding, the investor will have:
1000(1 +
0.08
4
)4×4 = 1372.79.
2.4. Effective annual rate.
.The Effective Annual Rate rEAR is the annual rate that produces the same result as if we had
compounded at a given periodic rate m times per year.
.To calculate the EAR we use
rEAR = (1 +
rnom
m
)m − 1.(2.4)
.The present value of a future cash flow given the discount rate (compounded m times per year)
and the n number of years in the future that the cash flow occurs can be computed as follows
PV = CFn(1 +
rNom
m
)−nm,(2.5)
.The number (1 + rNomm )
−nm is called the discount factor.
8 DR SIMON A. FAIRFAX
3. Continuous compounding
.The formula of the future value FVn at time n of a present value PV attracting interest at a
rate r > 0 compounded m times a year is given by
FVn = PV (1 +
r
m
)nm.
.Assume that the compounded period becomes shorter and shorter i.e., m becomes bigger and
bigger. Then in the limit as n goes to infinity, we obtain:
lim
m→∞FVn =PV limm→∞(1 +
r
m
)nm
=PV lim
m→∞
[
(1 +
r
m
)
m
r
]nr
=PV enr,
where we have used e = lim
x→∞(1 +
1
x )
x.
.This is known as the continuous compounding with corresponding growth factor enr.
3.1. Continuous rates of interest.
.The future value M(t) with a nominal rate r with continuous compounding and initial or present
value M(0) is given by
M(t) = M(0)ert.(3.1)
An investor receives £1100 in one year in return for an investment of £1000 now. Cal-
culate the percentage return per annum with
(a) Annual compounding
(b) Semi-annual compounding
(c) Monthly compounding
(d) Continuous compounding.
Example 3.1
Solution 3.1. (a): With the annual compounding, using Equation (1.1), we have that:
1000(1 + r) = 1100.
Then, the return is
r =
1100
1000
− 1 = 0.1
or 10% per annum.
(b): With the Semiannual compounding, using Equation (2.3), we have that:
1000(1 +
r
2
)2 = 1100.
Then, the return is
r = 2
(√1100
1000
− 1
)
= 0.0976
or 9.76% per annum.
MATH260. FINANCIAL MATHEMATICS PART I: DEPARTMENT OF MATHEMATICAL SCIENCES 9
(c): With monthly compounding, using Equation (2.3), we get
1000(1 +
r
12
)12 = 1100.
Hence, the return is
r = 12
(
12
√
1100
1000
− 1
)
= 0.0957
or 9.57% per annum.
(d): With continuous compounding, using Equation (3.1), we get
1000er = 1100.
Hence, the return is
r = ln
1100
1000
= 0.0953
or 9.53% per annum.
3.2. Relation between continuous and nominal compounded interest rates.
.We want to give the relation between continuous and nominal compounded interest rates. Let rc
be the interest rate with continuous compounding and rnom be the equivalent nominal rate with
compounding m times per annum. We have that
FVn = PV (1 +
rNom
m
)nm = PV ercn,
i.e.,
(1 +
rNom
m
)nm = ercn.
This means that
rc = m ln
(
1 +
rNom
m
)
(3.2)
and
rNom = m
(
erc/m − 1
)
(3.3)
Consider an interest that is quoted as 10% per annum with semiannual compounding.
Find the equivalent rate with continuous compounding.
Example 3.2
Solution 3.2. We are given m = 2 and rNom = 0.1 We aim at finding rc.
We know from Equation (3.2) that
rc = m ln
(
1 +
rNom
m
)
.
Hence rc = 2 ln
(
1 + 0.12
)
= 0.09758 or rc = 9.758% per annum.
10 DR SIMON A. FAIRFAX
A bank quotes you an interest rate of 14% per annum with quarterly compounding. What
is the equivalent rate with (a) continuous compounding? (b) annual compounding?
Example 3.3
Solution 3.3. We have that m = 4 and rNom = 0.14 We aim at finding rc and rEAR.
(a): The rate with continuous compounding is given according to Equation (3.2) by:
4 ln
(
1 +
0.14
4
)
= 0.1376
or 13.76% per annum.
(b): The rate with annual compounding is given according to Equation (2.4) by(
1 +
0.14
4
)4
− 1 = 0.1475
or 14.75%
Paul has 1500 GBP and wishes to invest this money in order to use it in six years. What
advice would you give him considering that he has the following two possibilities?
A: Put the 1500 GBP in a bank that offers to pay interest (compounded annually)
of 4% in the first year, 4.4% in the second year and so on, increasing by 0.4%
each year and hence paying 6% in the sixth year.
B: Invest the 1500 GBP in an instant access account which pays a fixed continu-
ously compounded rate of interest 4.3%.
EXERCISE
MATH260. FINANCIAL MATHEMATICS PART I: DEPARTMENT OF MATHEMATICAL SCIENCES 11
4. Risk and return
.The basic premise of an investor is that he likes returns and dislikes risk. An investor will purchase
a financial asset because he wants to increase his wealth, i.e., to earn a positive rate of return on
his investments. Due to uncertainty, he does not know what rate of return his investments will
bring.
.In finance, people will invest in risky assets only if they expect to receive higher returns. When
evaluating potential investments in financial assets, there are two dimensions of the decision making
process namely expected return and risk.
4.1. Investment returns.
.Returns.The concept of return gives to investors an appropriate way of expressing the financial
performance of an investment. There are two concepts of returns:
(1) The currency (GBP, USD, EURO,...) return which is given by
Currency return = Amount received− Amount invested
(2) Rates of return, or percentage returns which is given by
Rate of return =
Amount received− Amount invested
Amount invested
× 100%
=
Currency return
Amount invested
× 100%
.Investment in shares. Shares, also known as equities, provide you with part-ownership of a
company so when you invest in shares; you are buying ‘a share’ of that business. Companies issue
shares to raise money and investors buy shares in a business because they believe the company
will do well and they want to ‘share’ in its success.
.Dividends. Shareholders can be financially rewarded in two primary ways; firstly through the
appreciation of value in a share’s price (we will revisit valuation later in the course) and, secondly,
through dividends. Dividends are amounts of monies given to shareholders from company profits.
.The rate of return r from an investment in a share over time t ∈ [0, T ] whose beginning and
ending share price are S(0) and S(T ) respectively, which paid a dividend of D at t = T , is
r =
S(T )− S(0) +D
S(0)
× 100%.
In this case, the value S(T ) − S(0) represents the currency return in the investment through
appreciation in the share price. The amount D is comparable to interest received from a bank
account. The rate of return for the dividend income only is D/S(0)× 100% over time T .
Suppose you buy 10 shares of a stock for £100. At the end of one year, you sell the stock
for £110 after receiving a dividend of £3. What is the return on your £100 investment?
What is the rate of return?
Example 4.1
Solution 4.1.
• The currency return is £110−£100 + £3 = £13.
• The rate of return is:
£110−£100 + £3
£100
× 100% = 13%.
12 DR SIMON A. FAIRFAX
.Mathematical interpretation
(1) A negative rate of return indicates the original investment was not recovered.
(2) Rates of return are a better measure of relative returns. A £10 return on a £100 invest-
ment is a good return, but a £30 return on a £3, 000 investment is poor.
(3) Taking into account the timing of the return, a £50 return from a £100 investment is
a quite good return if it occurs after one year, however, the same return after 20 years
would not be so good.
4.2. Measuring individual asset return.
.The return on an investment will depend the outcome of a series of future events. For example,
the returns on banking shares will depend on economic factors such as individual and commercial
activity, political aspects, global markets and so on. The outcome of key future events will
determine individual returns. We use the concepts of events and probability distributions to
define and measure risks and rewards.
Definition 4.1. Given a probability distribution of rate of returns, the expected rate of return
on the individual stock or investment is defined as:
r̂ = E[r] =P1r1 + P2r2 + . . .+ Pnrn
r̂ =
n∑
i=1
Piri,
where
• r̂ = E[r] =the expected return on the stock,
• n = the number of events/states used to determine returns,
• Pi = the probability of state i occurring, and
• ri = the return on the investment in state i.
This weighted average is viewed as the expected reward for investing in an investment.
The table below provides a probability distribution for the returns on stocks A, B, C
and D. What is the expected rate of return stocks A, B, C and D?
State Probability Return on A Return on B Return on C Return on D
1 0.10 15% 10% 25% 25%
2 0.20 20% 40% 15% 50%
3 0.40 5% 30% 5% 20%
4 0.30 10% -10% 10% 65.4%
In this example of probability distribution, there are four possible states of the world.
For example, each state may represent the behaviour of the economy in the UK.
Example 4.2
Solution 4.2. The expected rate of return on stock A, stock B, stock C and stock D are given by:
r̂A = E[rA] =P1r1 + P2r2 + P3r3 + P4r4
=0.10× 15% + 0.20× 20% + 0.40× 5% + 0.30× 10%
=10.5%
and similarly r̂B = E[rB ] = 18%, r̂B = E[rC ] = 10.5% and r̂B = E[rD] = 40.12%.
Decision making: Stock D offers a higher expected return than stocks A, B and C which makes it
the most attractive from a returns point of view. What about the risk of each stock?
MATH260. FINANCIAL MATHEMATICS PART I: DEPARTMENT OF MATHEMATICAL SCIENCES 13
Probability distributions for returns on stocks A,B,C and D.
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EMESTER1 MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 EARN 5 FE 825 IERG 5130 STAT 8178 EE161 ADS2 G54FUZ CS326 113B ECN 226 QBUS6860 ELEC3202 COMP9334 MGT11B MET CS 777 STA238 00099F Python SMM306 CMT206 EC116 SP MAST90084 CS480/680 BCPM0091 CMPT 300 ECON2209 FY20 STAT 432 CVEN9625 MATH1712 CMP9771M CMP3750M CIS 325 AI506 MAT344H1S MGT7180 PSTAT 160B CSYS5010 1D STAT 453/558 ME152B STA304H1F CSCI-128 ME428 CO3002 FINE 7640 COMP3234 ECSE 223 EE497 – 3D MATH5905 ECS648U ENG5009 MATH10242 EC665 A CPI221 ISOM5220 MACE40402 MGT B456F COMP9315 AFM 113 STAT 778 CS 564 MA231 CS 230 CS 124 COMP5511 CSC317 ITEC1010 ELEC207 INFT 3019 7CCMFM06 EE 161 5G ECLT5810 MATH 7349 CS526 O2 COMP90054 ELE00063M ACCT5034 ARC1015 CSCI 40 CSCI E40 FIN 524B CS 462 389COM CMT309 MFIN703 COMP4250 STA304 STA304H1S STA 1003 FIN 448 BUFN 732 CS520 CVEN3701 ECMT6006 PHAS0100 CSIT314 MDIA 5028 CSCI 2134 MAT 451 CMPT 459 STAT 457 INFO20003 COMP4141 IELE8066 F540 ECM9 CSE 6321 FIN 430 MATH3911 BFC5916 COMP1011 EEE 6209 STATS 240 CW3 COMP206 COMP1003 4331W ISE 530 CMPT 295 MATH 239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 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COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 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FN1024 MATH0094 MANG 6554 ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 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