Mathematics Of Finance – CS Foundation Statistics Notes

Mathematics Of Finance – CS Foundation Statistics Notes

1. Interest
The amount charged, expressed as a percentage of the principal, by a lender to a borrower for the use of assets. It involves two persons
Borrower – When you borrow money, you pay interest Moneylender- When you lend money, you earn interest.
Some of the terms used in interest calculation are

• Principal or Capital – sum borrowed
• Interest – Extra money charged by the lender for use of his money
• Moneylender – a person who gives money to the borrower
• Conversion period – the period of time for which interest is calculated
• Simple interest – Simple interest is just the amount of money paid on a loan.
• Compound interest – over here Interest is charged on interest too.

2. Simple interest
Simple interest is called simple because it ignores the effects of compounding. The interest charge is always based on the original principal, so interest on interest is not included.
The formula for this is very simple:
I=PRT,
where I am interested,
P is Principal,
R is the percentage rate expressed as decimal and
T is time, which is generally expressed in years, assuming your rate is an annual rate.
Steps to calculate simple interest

Find the Principal. This is the amount of money borrowed or lent at the start of the year for which the interest will be calculated for.

Find the Rate as a decimal. This is the percentage of the Principal you will pay back each year. Divide the percentage by 100 to give the decimal value.

Specify the Time in years over which you want the interest calculating

Multiply Principle × Rate × Time to calculate the simple interest. This is the money you will pay/be paid on top of what was lent or borrowed.

Example
A sum of money at simple interest amounts to Rs. 815 in 3 years and to Rs. 854 in 4 years. The Principal sum is Rs.
Explanation:
S.I. for 1 year = Rs. (854 – 815) = Rs. 39.
S.I. for 3 years = Rs. (39 × 3) = Rs. 117.
Principal = Rs. (815 – 117) = Rs. 698.
Example
A sum fetched a total simple interest of Rs. 4016.25 at the rate of 9 p.c. per annum, in 5 years. The principal sum is Rs. 8925
Principal = Rs. \(\left(\frac{100 \times 4016.25}{9 \times 5}\right)\)
= Rs. \(\left(\frac{401625}{45}\right)\)
= Rs. 8925.

3. Compound interest
Compound interest is where interest is paid on the amount already earned leading to greater and greater amounts of interest.
Formula
Total Amount = Principal + Cl (Compound Interest)
(a) Formula for Interest Compounded Annually Total Amount = P (1+(R/100))ⁿ
(b) Formula for Interest Compounded Half Yearly Total Amount = P(1+(R/200))²ⁿ
(c) Formulae for Interest Compounded Quarterly Total Amount = P( 1 +(R/400))⁴ⁿ
(d) Formulae for Interest Compounded Annually with fractional years (e.g. 2.5 years)
Total Amount = P(1+(R/100))^a ×(1+(bR/100))
here if year is 2.5 then a =2 and b=0.5
(e) With different interest rates for different years Say x% for year 1, y% for year2, z% for year3
Total Amount = P(1+(x/100))*(1+(y/100))*(1+(z/100))
where,
CI = Compound Interest,
P = Principal or Sum of amount,
R = % Rate per annum,
” n = Time Span in years
Example
An amount of Rs 1,500.00 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. Find the balance after 6.years.
Solution
Use the regular compound interest formula,
A = P (1 + r/n) with P = 1500, r = 4.3/100 = 0.043,
n = 4 (not 1/4), t = 6. Therefore,
A = 1500 \(\left(1+\frac{0.043}{4}\right)^{4(6)}\) = $ 1,938.84

So, the balance after 6 years is approximately Rs. 1,938.84.

4. Annuities
An annuity is a type of investment in which regular payments are made over the course of multiple periods. Annuities help both the creditor and debtor have predictable cash flows, and it spreads payments of the investment out over time.

5. Various types of annuity
1. Contingent annuity – An annuity arrangement in which the beneficiary does not begin receiving payments until
a specified event occurs. A contingent annuity may be set up to begin sending payments to a beneficiary upon the death of another individual who wishes to ensure financial stability for the beneficiary or upon retirement or disablement of the beneficiary. ”

2. Ordinary annuity – Payments are required at the end of each period. For example, straight bonds usually pay coupon payments at the end of every six months until the bond’s maturity date.

3. Annuity due – Payments are required at the beginning of each period. Rent is an example of an annuity due. You are usually required to pay rent when you first move in at the beginning of the month, and then on the first of each month thereafter.

4. Deferred Annuity- An annuity in which the annuitant does not begin to receive payments until some future date. A deferred annuity has two phases: a savings phase and an income phase. During the savings phase, the annuitant places money into the annuity, which invests it on behalf of the annuitant. In the income phase, the annuitant receives payments. It is important to note that a deferred annuity is not taxed until the income phase begins. It also pays a death benefit to the survivor(s) of the annuitant. Nearly all retirement plans are deferred annuities

5. Immediate annuity – If at the end of the period the periodic payments are made.

6. Forborne annuity – The annuity that is left unpaid for years is called a forborne annuity.

6. Amount of annuity or future value of the annuity
The value of a group of payments at a specified date in the future. These payments are known as an annuity or a set of cash flows. The future value of annuity measures how much you would have in the future given a specified rate of return or discount rate.
The total amount of Annuity = sum of all periodic payments+ sum of all interest on periodic payments.

1. Ordinary annuity – Here the payments are made when the period ends. It is also called an immediate annuity. Most installment loans can be classified as ordinary annuities. Mortgages with the first payment due a month after the initial loan date are one of the most common examples of an ordinary annuity.

2. Annuity due – The payments are made at the beginning of each period. Any fixed payment for a service or property that occurs before a service period begins is an example of an annuity payment. Common applications include rent payments
Present Value – Present Value (PV) is a formula used that calculates the present-day value of an amount that is received at a future date. The premise of the equation is that there is a “time value of money”.
Time value of money is the concept that receiving something today is worth more than receiving the same item at a future date.

7. Present value of the annuity
The present value of the annuity formula determines the value of a series of future periodic payments at a given time. The present value of the annuity formula relies on the concept of the time value of money, in that one Rupee present day is worth more than that same Rupee at a future date. It is helpful for calculating the series of retirement payments etc.

8. Present value of ordinary annuity
The Present Value of an Ordinary Annuity is the value of a stream of expected or promised future payments that have been discounted to a single equivalent value today. It is extremely useful for comparing two separate cash flows that differ in some way.

9. Present value of an annuity due
The Present Value of an Annuity Due is identical to an ordinary annuity except that each payment occurs at the beginning of a period rather than at the end. Since each payment occurs one period earlier, we can calculate the present value of an ordinary annuity and then multiply the result by (1 + i).
PV _{Ordinary Annuity} = C * \(\left[\frac{1-(1+\mathrm{i})^{-\mathrm{n}}}{\mathrm{i}}\right]\)
C = Cash flow per period
i = Interest rate
n = Number of payments
This calculates the present value of an ordinary annuity.
To calculate the present value of an annuity due, multiply the result by (1+i). (The payments start at time zero instead of one period into the future.)

Mathematics Of Finance MCQ Questions

Question 1.
Interest is
a. Money Loaded
b. Money borrowed
c. Extra money paid on borrowed money
d. Borrowed run and above
Answer:
c. Extra money paid on borrowed money

Question 2.
The conversion period is
a. The time for which loan is given
b. The time for which interest is calculated
c. Period of conversion of simple interest into compound interest
d. None of the above
Answer:
b. The time for which interest is calculated

3. It is correct for annuity
a. It is the run paid
b. Fixed run paid at regular intervals
c. Fixed sum paid at regular interest under stared condition
d. All of the above
Answer:
c. Fixed sum paid at regular interest under the stared condition

Question 4.
Annuity due is the periodic payment made
a. At the end of each period
b. At the beginning of each period
c. Start only after a specified period
d. After a fixed number of interest
Answer:
b. At the beginning of each period

Question 5.
The time between two successive payment dates of an annuity is called
a. Annuity certain
b. For borne annuity
c. Payment Period
d. Terms of an annuity
Answer:
c. Payment Period

6. Amount of an annuity is
a. Total amount of all provide payment and total interest on the payments
b. Total time from the beginning of the first payment to the last payment
c. The total amount paid from the time loan taken till loan and
d. None of the above
Answer:
a. Total amount of all provide payments and total interest on the payments

Question 7.
If the principal amount is 99 and the amount payable is 199 at the end of the year then the rate of interest is
a. 100%
b. 99%
c. 100.5%
d. 200%
Answer:
c. 100.5%

Question 8.
There are __________types of annuity
a. 7
b. 8
c. 9
d. 6
Answer:
b. 8

Question 9.
An insurance company designed to pay a certain amount at a specified intervals is
a. Annuity
b. Simple interest
c. Compound interest
d. none of the center
Answer:
a. Annuity

Question 10.
The rate of interest is decided by
a. Borrower
b. Lender
c. Bank
d. Both a & b
Answer:
d. Both a & b

Question 11.
Compound interest is
a. The interest on principal amount
b. The interest on previously accumulated interest as well as interest on principal amount
c. The interest on principle for a number of years
d. All of the above
Answer:
b. The interest on previously accumulated interest as well as interest on the principal amount

Question 12.
If an investment yields an interest rate of 10% annually compounded quarterly. What is the effective interest rate?
a. 10.4%
b. 4%
c. 10%
d. 10.2%
Answer:
a. 10.4%

Question 13.
Calculate the total amount received after 5 years if yearly Rs. 200 is being invested and the rate of interest is 6% annually compounded
a. Rs. 1000
b. Rs. 1200
c. Rs. 1500
d. Rs. 1127.42
Answer:
d. Rs. 1127.42

Question 14.
If after 10 years the desired balance is Rs. 5000 then how much should be invested if the rate of interest is 5% compounded quarterly
a. Rs. 3042
b. Rs. 3042.07
c. Rs. 3042.70
d. Rs. 3040
Answer:
b. Rs. 3042.07

Question 15.
Historically which type of annuity did all the companies possessed?
a. Fixed annuity
b. Annuity due
c. Deferred annuity
d. None of the above
Answer:
a. Fixed annuity

Question 16.
Raj borrowed Rs. 8000 for 180 days at 5% interest. On the 90th day he makes a partial payment of Rs. 2500. What will be the adjusted principal on the maturity date?
a. Rs. 2500
b. Rs. 8000
c. Rs. 5600
d. None of the above
Answer:
c. Rs. 5600

Question 17.
Ajay has purchased a second-hand car by taking a loan of Rs. 5000 from the bank at a rate of interest of 8% for 4 years. Calculate the total amount he will be paying to the bank on a loan of Rs. 5000 at the end of 4 years
a. Rs. 6820
b. Rs. 6820.44
c. Rs. 6800.44
d. Rs. 6802.44
Answer:
d. Rs. 6802.44

Question 18.
If it is written, “interest is compounded semi-annually”. It means.
a. In 2 years there is one conversion period
b. The conversion period is 2 per year
c. Every year interest is halved
d. None of the above
Answer:
b. Conversion period is 2 per year

Question 19.
If the final amount received is 1938.84 for a deposit made for 6 years. What will be the principal amount? The annual interest rate is 4.3% compounded quarterly
a. Rs. 1200
b. Rs. 1000
c. Rs. 1600
d. Rs. 1500
Answer:
d. Rs. 1500

Question 20.
Interest rate per conversion period is calculated by.
a. Rate of interest
b. Conversion period/annual interest rate
c. Annual interest rate/conversion period
d. None of the above
Answer:
c. Annual interest rate/conversion period

Question 21.
Suppose you purchase a share of a company at a price of Rs. 77 a share and sold it after a month for Rs. 82 a share. You have received a dividend of Rs. 1 at the end of the month. Calculate your annual rate of return, assuming monthly compounding
a. 146%
b. 140%
c. 100%
d. 200%
Answer:
a. 146%

Question 22.
If the question arises that how much the company should pay lump sum today so that a future pension plan could be purchased then it is.
a. Future value of annuity
b. Present value of annuity
c. Both a & b
d. None of the above
Answer:
b. Present value of the annuity

Question 23.
What will be the total amount of interest (rounded off) received .by Ajay if he invests Rs. 10000 for 5 years at an interest rate of 7.5 % compounded quarterly.
a. Rs. 4499
b. Rs. 4490
c. Rs. 4500
d. Rs. 4994
Answer:
a. Rs. 4499

Question 24.
If a firm’s debt ratio is 45% this means ________ of the firm’s assets are financed by equity financing provided debt ratio = debt/Total Assets
a. 50%
b. 55%
c. 45%
d. Cannot be terminated without more information
Answer:
b. 55%

Question 25.
A firm has paid out Rs. 150,000 as dividends from its net income of Rs. 250,000 what is the retention ratio for the firm? (Retention ratio = net income – dividend/Net income)
a. 12%
b. 25%
c. 40%
d. 60%
Answer:
c. 40%

Question 26.
When the market’s required rate of return for a particular bond is much less than its coupon rate, the bond is selling at:
a. Premium
b. Discount
c. Par
d. Cannot be determined without more information
Answer:
a. Premium

Question 27.
If you plan to save Rs. 5,000 with a bank at an interest rate of 8% what will be the worth of your amount after 4 years if interest is compounded annually?
a. Rs. 5,400
b. Rs. 5,900
c. Rs. 6,600
d. Rs. 6,802
Answer:
d. Rs. 6,802

Question 28.
Which of the following measure reveals how much profit a company generates with the money shareholders have invested ?
a. Profit margin
b. Return on Assets
c. Return on equity
d. Debt-Equity Ratio
Answer:
c. Return on equity

Question 29.
If you have Rs. 850 and you plan to save it for 4 years with an interest rate of 10%, what will be the future value of your savings?
a. Rs. 1,000
b. Rs. 1,244
c. Rs. 1,331
d. Rs. 1,464
Answer:
b. Rs. 1,244

Question 30.
You need Rs. 10,000 to buy a new television. If you have Rs. 6, 000 to invest at 5 percent compounded annually, how long will you have to wait to buy the televisions?
a. 8.42 years
b. 10.51 years
c. 15.75 years
d. 18.78 years
Answer:
b. 10.51 years

Question 31.
How many years will it take to pay a Rs. 11,000 loan with an Rs. 1241.08 annual payment and a 5% interest rate?
a. 6 years
b. 12 years
c. 24 years
d. 48 years
Answer:
b. 12 years

Question 32.
Which one of the following terms refers to the risk that arises for bond owners from fluctuating interest rates?
a. Fluctuations Risk
b. Interest rate Risk
c. Real-Time Risk
d. Inflation Risk
Answer:
b. Interest rate Risk

Question 33.
A sum of Rs.1,200 becomes Rs.1,323 in two years at compound interest compounded annually. Find the rate percent?
a. 5%
b. 6%
c. 7%
d. 8%
Answer:
a. 5%
Hint
Amount=P(1+r)ⁿ
1323 = 1200(1 + r)²
r=0 .05
r = 5%

Question 34.
The difference between simple interest and compound interest compounded annually on X sum of money for 2 years at 4% per annum is Rs.1. The value of ‘X’ is
a. Rs. 100
b. Rs. 300
c. Rs. 500
d. Rs. 625
Answer:
d. Rs. 625
Hint
S.I. = PRT
where I am interested,
P is Principal,
R is the percentage rate expressed as decimal and T is time x is principal
S.I. =x 2.4/100 = ,08x
Total Amount = Principal + Cl (Compound Interest)
Cl = Total amount – Principal
Amount =P(1+r)ⁿ
= X (1+.04)²
Cl = X(1+.04)² – X = x({1 +.04}² – 1)
= 0816x
According to question C.l. – S.l. = 1
X (.0816- .08) = 1
X = 170016 = 625

Question 35.
The time between two successive dates of an annuity is called
a. Payment interval
b. Future value of annuity
c. Contingent annuity
d. Annuity certain
Answer:
a. Payment interval
Hint
The time period between two successive dates of an annuity is called payment interval.

Question 36.
Compound interest for Rs. 1 ,000 for 4 years at 5% per annum when it is compounded quarterly-
a. Rs. 215
b. Rs. 218
c. Rs. 220
d. Rs. 225
Answer:
c. Rs. 220
Hint
Total amount when Interest compounded quarterly = P(1+(R/400))⁴ⁿ
A = 1000(1+(5/400)^4×4
A = 1220
C.l. = A-P = 1220-1000 = 220

Question 37.
Amjad invested a sum of money at 8% per annum at simple interest for’t’ years. At the end of’t’ year, Amjad got back 4 times his original investment.’ The value of ‘t; 3
a. 5 Years
b. 10.5 Years
c. 25.5 Years
d. 37.5 Years
Answer:
d. 37.5 Years
Hint
r = 8% , Let Principal = P
S.I. = Prt
A = S.I. + P
As per question amount = 4 P
Thus putting these value we get
A = S.I. + P
4P = S.I. + P
S.I. = 3P
Prt = 3 P
P x ,08 x t = 3P
t = 3P/Px .08 = 3/.08 =37.5 yrs

Question 38.
An annuity is a fixed sum paid at regular intervals. An annuity that continues for a number of years is called:
a. Deferred annuity
b. Immediate annuity
c. Uniform annuity
d. Perpetual annuity
Answer:
d. Perpetual annuity
Hint
A perpetual annuity is an annuity in which the periodic payments begin on a fixed date and continue indefinitely.

Question 39.
In what period, the compound interest on ~ 30,000 at 7% per annum amounts to Rs.4,347
a. 2 years
b. 1.5 years
c. 3 years
d. 4 years
Answer:
a. 2 years
Hint
C.I.= 4347
r = 7%
P= 30000
A = P + C.I.
A = 34347
A = P(1+(R/100))ⁿ
34347 = 30000 (1+(7/100)ⁿ
34347= 30000 (1+ .07)ⁿ
1.1449 = 1.07ⁿ
n = 2 yrs

Question 40.
What is the present value of Annuity on t1 for 2 years @ 10% p.a.?
a. 0.18
b. 2
c. 3
d. 0.67
Answer:
b. 2
Hint
PV_{Ordinary Annuity} = C * \(\)

C=cash flow per period
i = interest rate
n=number of payments
i= 10%/12 = .1/12 = .00833
n= 1 × 2 = 2
c = 1
PV = 1(1 – \(\frac{.00833)^{-2}}{(.00833)}\)
PV = 1 × 1.975
PV =2

Question 41.
Cl = Rs. 30,000, @7% rate is Rs. 4347. Find the number of years.
a. 4 years
b. 2 years
c. 2.5 years
d. 3 years
Answer:
b. 2 years
Hint
P =A – C.l.
= 30000-4347 = 25653
A =P (1+r/100)ⁿ
30000 = 25653 (1+.07)ⁿ
1.1695 = (1.07)ⁿ
(1.07)² = (1.07)ⁿ
2 = n

Question 42.
When the loan amount is given @ 9% then what will be the outstanding amount after 3 months?
a. 1014
b. 1000
c. 1012.5
d. 1013.6
Answer:
c. 1012.5
Hint
t= 3 months = 1/4 yrs
r =9% p.a.= 2.25 per quarter
A = 1000(1+ 2.25/100)^1/4×4
A = 1000 x 1.0225 = 1022.5

Question 43.
The difference, between 81 & Cl is 1, the time period is 2 years, rate 4% p.a., find the principal amount?
a. 629
b. 625
c. 700
d. 600
Answer:
b. 625
Hint
S.I. = PRT
where I am interested,
P is Principal,
R is the percentage rate expressed as decimal and T is time
x is principal
5.1. = × 2 .4/100 = .08x
Total Amount = Principal + Cl (Compound Interest)
CI = Total amount – Principal
Amount =P(1+r)ⁿ
= x (1+.04)²
CI = x (1+.04)² – X
= X({1+.04}² – 1)
= 0816x
According to question C.I. – S.l. = 1 x(.0816 – .08) = 1 x= 1/.0016 = 625

Question 44.
On what sum will the compound compounded interest at 5% p.a. for 2 yeas annually be Rs. 1,640?
a. 10,000
b. 12,000
c. 15,525
d. 16,000
Answer:
d. 16,000
Hint
Amount = P(1+r)ⁿ
1640 = P{(1 +5/100)² -1}
1640 = P{1.1025-1}
1640 = 1025 P
P = 16000

Question 45.
A man took a loan from a bank at the rate of 12% р.a. on simple interest. After 3 years he had to pay Rs.5,400 interest only for the period. The principal amount borrowed by him was:
a. Rs. 10,000
b. Rs. 20,000
с. Rs. 15,000
d. Rs. 18,000
Answer:
с. Rs. 15,000
Hint
S.l. =PRT
P = S.I./RT = 5400 × 100/12 × 3 =15000

Question 46.
With an interest rate of 5 percent, the present value of ~ 100 received one year from now is approximate:
a. Rs. 95.238
b. Rs. 105
c. Rs. 100
d. Rs. 95
Answer:
b. Rs. 105
Hint
A =P(1+R/100)^t
= 100(1+5/100)^t = 105

Question 47.
Vidhya Signed a contract in which she will receive ~ 3 million immediately and 1 million will be paid to her every year for the next 5 years. If the interest rate is 10%, the present value of the contract is approximate:
a. Rs. 7 million
b. Rs. 6.79 million
c. Rs. 8 million
d. Rs. 8.79 million
Answer:
b. Rs. 6.79 million
Hint
PV = 3 + 1 × \(\frac{1}{(1+0.01)^{1}}\) + 1 × \(\frac{1}{(1+0.01)^{2}}\) + 1 × \(\frac{1}{(1+0.01)^{3}}\) + 1 × \(\frac{1}{(1+0.01)^{4}}\) + 1 × \(\frac{1}{(1+0.01)^{5}}\)
= 3 + 3.79 = 6.79 million

Question 48.
A sum fetched a total Simple Interest of Rs 4016.25 at the rate of 9% p.a. in 5 years. Find the sum?
a. Rs. 8900
b. Rs. 4462.50
c. Rs. 8032.50
d. Rs. 8925
Answer:
d. Rs. 8925
Hint
S.l. = PRT
P = S.I./RT = 4016.25 × 100/5 x 9 = 8925

CS Foundation Business Economics Notes