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Mathematics Test - 31

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Weekly Quiz Competition

Question 1
5 / -1

Which of the following is correct regarding perpetuaty?

A
Perpetuity is a situation where a stream of cash flow payments continues and ends after a definite period of time.

B
Perpetuity in the financial system is a situation where a stream of cash flow payments continues indefinitely

C
Perpetuity s a situation where a stream of cash flow payments may continue forever or for a definite period of time

D
None of the above.
Solution
CONCEPT:
Perpetuity: A perpetuity is an annuity where payments continue forever.
- We can not calculate the future value of the Perpetuity where payments continue forever but we can calculate the present value of the perpetuity.
- Amount of a Perpetuity: The amount of perpetuity is undefined since it increases beyond all bounds as time goes on.
Present value of Perpetuity: We consider two types of perpetuity which are as follows:
The present value of a perpetuity of Rs. R payable at the end of each period, the first payment due one period hence is the sum of money which is invested now at the rate i per period will yield Rs. R at the end of each period forever. It is given by,
⇒ P = R (I + i)-1 + R (I + i)-2 + ..... ∞
The above is the total present value invested. Here we can not calculate the future value of the Perpetuity where payments continue forever
It is an infinite geometric series with first term R (1 + i)-1 and whose common ratio is (1 + i)-1 Its sum is given by
\(⇒ P = \frac{R(1 + i)^{-1}}{1 - ( 1 + i)^{-1}}\)
Present value of a perpetuity of Rs. R payable a the end of each period, the first being due one period hence is
\(⇒ \rm P = \frac{R}{i}\)
Where R = size of each payment at the end of each period, i = rate per period
Perpetuity of Rs. R payable at the beginning of each period, the first payment due on present value. This annuity can be considered as an initial payment of Rs. R followed by the perpetuity of Rs. R of the above type.
Thus, the present value is given by R + \(\frac{R}{i}\)
Where, R = size of each payment, i = rate per period
EXPLANATION:
From the basic concept of the perpetuity, we can say that perpetuity in the financial system is a situation where a stream of cash flow payments continues indefinitely
So the correct answer is option 2.
Question 2
5 / -1

Which of the following is repressing the formula to calculate the future value of the perpetuity at the rate i per period which yields Rs. R at the beginning of each period?

A
R + \(\frac{R}{i}\)

B
\(\frac{R}{i}\)

C
Can not be defined

D
None
Solution
CONCEPT:
Perpetuity:
- A perpetuity is an annuity where payments continue forever.
- We can not calculate the future value of the Perpetuity where payments continue forever (Time is not defined and it goes forever) but we can calculate the present value of the perpetuity.
- Amount of a Perpetuity: The amount of perpetuity is undefined since it increases beyond all bounds as time goes on.
Present value of Perpetuity: We consider two types of perpetuity which are as follows:
The present value of a perpetuity of Rs. R payable at the end of each period, the first payment due one period hence is the sum of money which is invested now at the rate i per period will yield Rs. R at the end of each period forever. It is given by,
⇒ P = R (I + i)-1 + R (I + i)-2 + ..... ∞
The above is the total present value invested. Here we can not calculate the future value of the Perpetuity where payments continue forever
It is an infinite geometric series with first term R (1 + i)-1 and whose common ratio is (1 + i)-1 Its sum is given by
\(⇒ P = \frac{R(1 + i)^{-1}}{1 - ( 1 + i)^{-1}}\)
Present value of a perpetuity of Rs. R payable a the end of each period, the first being due one period hence is
\(⇒ \rm P = \frac{R}{i}\)
Where R = size of each payment at the end of each period, i = rate per period
Perpetuity of Rs. R payable at the beginning of each period, the first payment due on present value. This annuity can be considered as an initial payment of Rs. R followed by the perpetuity of Rs. R of the above type.
Thus, the present value is given by R + \(\frac{R}{i}\)
Where, R = size of each payment, i = rate per period
EXPLANATION:
From the basic concept of the perpetuity, we can say that perpetuity in the financial system is a situation where a stream of cash flow payments continues indefinitely
A perpetuity is an annuity where payments continue forever.
We can not calculate the future value of the Perpetuity where payments continue forever (Time is not defined and it goes forever) but we can calculate the present value of the perpetuity.
Amount of a Perpetuity: The amount of perpetuity is undefined since it increases beyond all bounds as time goes on.
So the correct answer is option 3.
Question 3
5 / -1

Consider the following statements:
1. Bond is a written contract between borrower and lender.
2. When a bond sells at face value then YTM > current yield > coupon yield
3. The valuation of a bond is the determination of the fair price of a bond.
4. if the market price of the bond is greater than its face value, the bond is sold at the discount.
The correct statements is/are:

A
(1) and (2)

B
(2) and (3)

C
(1) and (3)

D
(3) and (4)
Solution
Concept:
Bond:
- It is a written contract between a borrower and a lender.
- Through this contract, the borrower promises to pay a specified sum at a specific future date and to pay interest payments at a specific rate at equal intervals of time until the bond is redeemed.
Face Value:
- The face value (also known as par value) of a bond is the price at which the bond is sold to buyers (investors) at the time of issue.
- It is also the price at which the bond is redeemed at maturity.
- It is also known as the par value of the bond
Redemption Price: It is the amount the bond issuer pays at maturity. It is usually equal to the face value in case the bond is redeemed at par.
Discount: Where the market price of the bond is less than its face value (par value), the bond is selling at a discount.
Premium: if the market price of the bond is greater than its face value, the bond is selling at a premium.
Bond valuation:
- It is the determination of the fair price of a bond.
- As with any security or capital investment, the theoretical fair value of a bond is the present value of the stream of cash flows it is expected to generate.
The nominal rate of interest: It is the rate at which a bond yields interest. It’s also known as the coupon rate.
Coupon Rate: A bond’s coupon rate denotes the annual interest rate paid by the bond issuer to the bond holder.
Coupon rate \(=\frac {C}{F}\)
Where, C - Coupon payment and F - Face value
Current Yield: The current yield is simply the coupon payment C as a percentage of the current bond price Po.
Coupon yield \(=\frac {C}{P_{0}}\)
Yield to Maturity (YTM): The yield to maturity (YTM) is the discount rate that returns the market price of a bond without embedded optionality; it is identical to the required return.
Relation between Yield to Maturity, current yield, and coupon yield:
1. When a bond sells at a discount, YTM > current yield > coupon yield.
2. When a bond sells at a premium, YTM < current yield < coupon yield.
3. When a bond sells at a face value, YTM = current yield = coupon yield.
Explanation:
- Thus the statements (1) and (3) are true.
- Therefore option (3) is correct.
Question 4
5 / -1

At 6% converted quarterly, find the present value of a perpetuity of Rs. 600 payable at the end of each quarter.

A
48000

B
40000

C
42000

D
None
Solution
CONCEPT:
Perpetuity: A perpetuity is an annuity where payments continue forever.
- We can not calculate the future value of the Perpetuity where payments continue forever but we can calculate the present value of the perpetuity.
- Amount of a Perpetuity: The amount of perpetuity is undefined since it increases beyond all bounds as time goes on.
Present value of Perpetuity: We consider two types of perpetuity which are as follows:

(i) The present value of a perpetuity of Rs. R payable at the end of each period, the first payment due one period hence is the sum of money which is invested now at the rate i per period will yield Rs. R at the end of each period forever. It is given by,
⇒ P = R (I + i)-1 + R (I + i)-2 + ..... ∞
The above is the total present value invested. Here we can not calculate the future value of the Perpetuity where payments continue forever
It is an infinite geometric series with first term R (1 + i)-1 and whose common ratio is (1 + i)-1 Its sum is given by
\(⇒ P = \frac{R(1 + i)^{-1}}{1 - ( 1 + i)^{-1}}\)
Present value of a perpetuity of Rs. R payable a the end of each period, the first being due one period hence is
\(⇒ \rm P = \frac{R}{i}\)
Where R = size of each payment at the end of each period, i = rate per period
(ii ) Perpetuity of Rs. R payable at the beginning of each period, the first payment due on present value. This annuity can be considered as an initial payment of Rs. R followed by the perpetuity of Rs. R of the above type.
Thus, the present value is given by R + \(\frac{R}{i}\)
Where, R = size of each payment, i = rate per period
CALCULATION:
Given: R = 600, \(\rm i = \frac{0.06}{4} = 0.015\)
Then the present value of a perpetuity
\(⇒ P = \frac{R}{i} = \frac{600}{0.015} \)
⇒ P = Rs. 40,000
So the correct answer is option 2.
Question 5
5 / -1

What amount of money will be the present value of the perpetuity of Rs 750 payable at the end of each quarter if the money is worth 12% compounded annually?

A
Rs.30000

B
Rs.25000

C
Rs.20000

D
Insufficient data
Solution
CONCEPT:
Perpetuity:
- A perpetuity is an annuity where payments continue forever.
- We can not calculate the future value of the Perpetuity where payments continue forever but we can calculate the present value of the perpetuity.
- Amount of a Perpetuity: The amount of perpetuity is undefined since it increases beyond all bounds as time goes on.
- Present value of Perpetuity: We consider two types of perpetuity which are as follows:
The present value of a perpetuity of Rs. R payable at the end of each period, the first payment due one period hence is the sum of money which is invested now at the rate i per period will yield Rs. R at the end of each period forever. It is given by,
⇒ P = R (I + i)-1 + R (I + i)-2 + ..... ∞
The above is the total present value invested. Here we can not calculate the future value of the Perpetuity where payments continue forever
It is an infinite geometric series with first term R (1 + i)-1 and whose common ratio is (1 + i)-1 Its sum is given by
\(⇒ P = \frac{R(1 + i)^{-1}}{1 - ( 1 + i)^{-1}}\)
Present value of a perpetuity of Rs. R payable a the end of each period, the first being due one period hence is
\(⇒ \rm P = \frac{R}{i}\)
Where R = size of each payment at the end of each period, i = rate per period
Perpetuity of Rs. R payable at the beginning of each period, the first payment due on present value. This annuity can be considered as an initial payment of Rs. R followed by the perpetuity of Rs. R of the above type.
Thus, the present value is given by R + \(\frac{R}{i}\)
Where, R = size of each payment, i = rate per period
CALCULATION:
Given: R = 750 Rs
- if money is worth 8% compounded annually
- But the payment is done quarter so,
\(\Rightarrow \rm i = \frac{0.12}{4} = 0.03\)
- This is a perpetuity of type (ii) since payments are made at the beginning of each period.
- Then the present value of a perpetuity
\(⇒ P = \frac{R}{i} = \frac{750}{0.03}\)
⇒ P = Rs.25000
So the correct answer is option 2.
Question 6
5 / -1

Mr. and Mrs. want to purchase a house for ₹30,00,000 with a down payment of ₹5,00,000. If they can amortize the balance at 12% per annum compounded monthly for 30 years with the EMI as 25,000 Rs then the total interest paid by them will be?

A
35,00,000 Rs

B
65,00,000 Rs

C
25,00,000 Rs

D
None
Solution
Concept:
- EMI stands for equated monthly installment. It is a monthly payment that we make towards a loan we opted for at a fixed date of every month.
- A loan is amortized when part of each periodic payment is used to pay interest and the remaining part is used to reduce the principal.
Reducing-Balance Method or Amortization Formulas:
- When one is amortizing a loan, at the beginning of any period, the principal outstanding is the present value of the remaining payments.
- Using this fact, we obtain the formulas that describe the amortization of an interest-bearing loan of Rs P, at a rate i per period by n equal payments of Rs R each and each payment is made at the end of each period.
Amortization Formulas:
(i) Periodic payment or installment,
\(⇒ R=P \left(\frac{i}{1-(1+i)^{-n}}\right)\)
(ii ) Principal Outstanding at the beginning of k^th period,
\(=R\left(\frac{1-(1+i)^{-(n-(k-1))}}{i}\right)\)
\(\)Total Interest Paid \(=nR-P\)
Calculations:
Given: DP = 5,00,000 Rs (Amount paid already at the beginning), EMI = 25, 000 Rs
n = 30 × 12 = 360
So, the principal amount at the beginning,
⇒ P = 30,00,000 - 5,00,000 = 25,00,000 Rs
So, total Interest Paid (TI) = n R - P
⇒ TI = 360 × 25,000 - 25,00,000 = 65,00,000 Rs
Hence the option 2 is correct.
Question 7
5 / -1

On 1st April 2021, Balesh purchased a refrigerator costing 20,000 Rs and spent 2000 Rs on its repair. The estimated effective life of it is 10 years with a scrap value of 5000 Rs. The depreciation with the financial year ending on 31st March 2022 is

A
1650 Rs

B
1750 Rs

C
1600 Rs

D
1700 Rs
Solution
Concept:
- The decrease in the value of the assets such as building machinery and equipment of all kinds is called depreciation.
Linear method of calculating depreciation:
- The linear method of depreciation is the simplest and the most widely used method to calculate the depreciation for fixed assets.
According to this method the annual depreciation D is given by,
\(D=\frac{C-S}{n}\)
Where C is the original cost of the asset (Includes the cost of the asset and its repair const in useful life), S is the estimated scrap value of residual value (the value of a depreciable asset at the end of its useful life ) and n is the useful life in years.
Calculation:
Given: S = 5000 Rs, n = 10
- Total original cost of the asset will be the sum of cost of the asset and its repair cost in useful life.
⇒ C = 20,000 + 2,000 = 22,000 Rs
- Therefore the annual depreciation is given by,
\(D=\frac{C-S}{n}\)
\(⇒ D=\frac{22000-5000}{10}\)
\(⇒ D=\frac{17000}{10}=1700 \ Rs\)
Hence option 4 is correct.
Question 8
5 / -1

Consider a bond with its present value of all the periodic payments is ₹1200. The face value is ₹ 1000 and the bond has 5 years to maturity. The yield to maturity is 2% compounded semi-annually. The value of the bond is?
(Given 1.01^-10 ≈ 0.905)

A
₹2100

B
₹2105

C
₹2110

D
₹2115
Solution
Concept:
- The value of the bond can be calculated using the present value approach. In this approach, we first calculate the present value of each expected cash flow and then we add all the individual present values to obtain the value of the purchase price of a bond.
- Bond Value = Present value of first periodic payment + Present value of second periodic payment+ . . . + Present value of nth periodic payment + Present value of Redemption price/Maturity value
The value of the bond can be calculated using the formula below:
Bond Value,
\(V=R\left [ \frac {1-(1+i)^{-n}}{i} \right ]+C(1+i)^{-n}\)
C - Maturity value, n - number of cashflow or periodic payments, i - yield rate
R = C × i_d
Where, i_d - The rate of interest per period
Calculations:
Given: Face Value, C=1000 Rs
- Sum of present values of all the periodic payments P₁ = ₹1200
- Number of periods before redemption \(n=10\)
- Yield rate i = 2 % semi - annually = \(\frac{2}{2×100}=0.01\)
- So, the present value of the Redemption price
\(P_{2}=C(1+i)^{-n}\)
\(\Rightarrow P_{2}=1000(1+0.01)^{-10}\)
\(\Rightarrow P_{2}= 905 \ Rs\)
Therefore the value of a bond,
\(V=P_{1}+P_{2}=1200+905=2105\)
V = P₁ + P₂= 1200 + 905 = 2105 Rs
- Therefore option 2 is correct.
Question 9
5 / -1

Rohan establishes a sinking fund to provide for payment of rupees 237500 for the education of his son maturing in 3 years. The contributions are to be made at the end of each year. Find the amount of each annual deposit if the interest is 15% per annum.(Take 1.153 = 3.375)

A
17000 Rs

B
12000 Rs

C
15000 Rs

D
None
Solution
Concept:
Sinking Fund:
- A sinking fund is a fund established by a company or business entity by setting aside revenue over a period of time to fund a future capital expense or repayment of long-term debt.
- It is a fund that is accumulated for the purpose of paying off a financial obligation at some future designated date.
The periodic payments of Rs R made at the end of each period required to accumulate a sum of Rs A over n periods with interest charged at the rate i per period is,
\(A= R[\frac{(1+i)^n -1}{i}]\)
Where
R = Size of each installment or payment, i = rate per period, n = number of installments, A = lumpsum amount to be accumulated
Remark: The problems relating to Sinking Fund are solved by using known formulas for the amount of an ordinary annuity or annuity due as the case may be depending on whether the payments are set aside at the end or beginning of each payment interval.
Explanation:
Given:
A = 237500 Rs, n = 3, R = ? (Per annum), i = 15 % pa = 0.15
\(⇒237500 = R[\frac{(1+0.15)^3 -1}{0.15}]\)
⇒ R = 15000 Rs
So, the correct answer will be option 3.
Additional InformationDifference between Sinking Fund and Savings Account:
Sinking funds and savings accounts, both, involve setting aside an amount of money for the future.
The main difference is that the sinking fund is set up for a particular purpose and is to be used at a particular time.
While the savings account is set up for any purpose that it may serve.
Question 10
5 / -1

How much money is needed to endure a series of lectures costing Rs 2500 at the beginning of each year indefinitely, if money is worth 2.5% compounded annually?

A
Rs. 75000

B
Rs. 85000

C
Rs. 84040

D
Rs.102500
Solution
CONCEPT:
Perpetuity: A perpetuity is an annuity where payments continue forever.
We can not calculate the future value of the Perpetuity where payments continue forever but we can calculate the present value of the perpetuity.
Amount of a Perpetuity: The amount of perpetuity is undefined since it increases beyond all bounds as time goes on.
Present value of Perpetuity: We consider two types of perpetuity which are as follows:

The present value of a perpetuity of Rs. R payable at the end of each period, the first payment due one period hence is the sum of money which is invested now at the rate i per period will yield Rs. R at the end of each period forever. It is given by,
⇒ P = R (I + i)-1 + R (I + i)-2 + ..... ∞
The above is the total present value invested. Here we can not calculate the future value of the Perpetuity where payments continue forever
It is an infinite geometric series with first term R (1 + i)-1 and whose common ratio is (1 + i)-1 Its sum is given by
\(⇒ P = \frac{R(1 + i)^{-1}}{1 - ( 1 + i)^{-1}}\)
Present value of a perpetuity of Rs. R payable a the end of each period, the first being due one period hence is
\(⇒ \rm P = \frac{R}{i}\)
Where R = size of each payment at the end of each period, i = rate per period
Perpetuity of Rs. R payable at the beginning of each period, the first payment due on present value. This annuity can be considered as an initial payment of Rs. R followed by the perpetuity of Rs. R of the above type.
Thus, the present value is given by R + \(\frac{R}{i}\)
Where, R = size of each payment, i = rate per period
CALCULATION:
Given: R = 2500 Rs
if money is worth 3% compounded annually
\(\Rightarrow \rm i = \frac{0.025}{1} = 0.025\)
This is a perpetuity of type (ii) since payments are made at the beginning of each period.
Then the present value of a perpetuity
\(⇒ P =R + \frac{R}{i} =2500+ \frac{2500}{0.025} \)
⇒ P = Rs.102500
So the correct answer is option 4.

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Mathematics Test - 31

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Question 1

​Perpetuity is a situation where a stream of cash flow payments continues and ends after a definite period of time.

​Perpetuity in the financial system is a situation where a stream of cash flow payments continues indefinitely

​Perpetuity s a situation where a stream of cash flow payments may continue forever or for a definite period of time

None of the above.

Solution

Question 2

R + \(\frac{R}{i}\)

\(\frac{R}{i}\)

Can not be defined

None

Solution

Question 3

(1) and (2)

(2) and (3)

(1) and (3)

(3) and (4)

Solution

Question 4

48000

40000

42000

None

Solution

Question 5

Rs.30000

Rs.25000

Rs.20000

Insufficient data

Solution

Question 6

35,00,000 Rs

65,00,000 Rs

25,00,000 Rs

None

Solution

Question 7

1650 Rs

1750 Rs

1600 Rs

1700 Rs

Solution

Question 8

₹2100

₹2105

₹2110

₹2115

Solution

Question 9

17000 Rs

12000 Rs

15000 Rs

None

Solution

Question 10

Rs. 75000

Rs. 85000

Rs. 84040

Rs.102500

Solution

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Perpetuity is a situation where a stream of cash flow payments continues and ends after a definite period of time.

Perpetuity in the financial system is a situation where a stream of cash flow payments continues indefinitely

Perpetuity s a situation where a stream of cash flow payments may continue forever or for a definite period of time