Chapter No 5 Financial mathematics.

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Financial Mathematics

1. The Time Value of Money: The Core Principle of Financial Mathematics

The Time Value of Money, or TVM, is one of the most fundamental concepts in financial mathematics. It is based on the idea that money available today is worth more than the same amount in the future, as money has the potential to earn returns over time. This concept is central in finance, helping to determine how much a future sum is worth in today’s terms, called the present value, or how much a current sum will grow to in the future, known as the future value.

Present value, for instance, allows us to calculate how much a future sum of money is worth in today’s terms, accounting for a specified rate of return. If you expect to receive 1,000 dollars five years from now and the discount rate is 10 percent, the present value of that future amount is around 620 dollars.

Future value, on the other hand, is used to calculate what a present amount of money will grow to over a specified period, given a particular interest rate. For example, if you invest 500 dollars at an annual rate of 8 percent, it will grow to approximately 735 dollars in five years.

2. Compound Interest: The Impact of Exponential Growth

Compound interest is one of the most powerful financial concepts. Unlike simple interest, which is calculated only on the initial principal, compound interest calculates interest on both the initial principal and any interest that has already been added. This results in exponential growth, allowing money to increase significantly over time.

If you invest 1,000 dollars at an annual interest rate of 6 percent, compounded quarterly for three years, the investment will grow to roughly 1,194 dollars. Compound interest is frequently used in savings accounts, retirement funds, and investments, illustrating how money can grow faster over time with consistent compounding.

3. Annuities: Structured Payments Over Time

An annuity is a series of equal payments made at regular intervals, often used for retirement savings, loans, and investments. Financial mathematics allows us to calculate both the present and future values of annuities, providing insight into expected cash flows over time.

An ordinary annuity is when payments are made at the end of each period, which is typical for situations like mortgage payments or regular investments. An annuity due, however, is structured so payments occur at the beginning of each period, as with rental payments. By calculating the future value of an annuity, individuals can see how much to invest periodically to reach financial goals like retirement savings.

4. Loan Amortization: Paying Off Debt Gradually

Loan amortization refers to repaying a loan through regular installments over time. Each installment covers part of the principal, which is the original loan amount, and part of the interest. Loan amortization is commonly used for mortgages, car loans, and personal loans, making it a key concept in financial mathematics.

For example, if you take out a 10,000-dollar loan with an interest rate of 5 percent to be repaid over five years, your monthly payment will be approximately 189 dollars. Understanding loan amortization allows borrowers to see how much of each payment reduces the loan balance and how much goes toward interest.

5. Risk and Return Analysis: Making Informed Investment Decisions

In finance, there is a fundamental relationship between risk and return. Generally, higher returns are associated with higher risks. Financial mathematics provides tools to assess this relationship, helping investors make informed decisions. One way to measure risk is through standard deviation, which calculates how much returns vary from an average. A high standard deviation indicates greater variability, suggesting a riskier investment.

Expected return is another important metric, as it estimates the probable return of an investment based on past performance or set probabilities. By calculating expected return, investors can compare options and build a balanced portfolio that matches their risk tolerance.

6. Applications of Financial Mathematics in Everyday Life

Financial mathematics applies to everyday situations, not just to corporate finance. Here are some examples:

Budgeting and saving: Determining how much to save each month to reach a financial goal.
Retirement planning: Estimating the amount needed for retirement, taking into account inflation and expected returns.
Mortgage payments: Calculating affordable monthly payments for home buyers.
Investment growth: Forecasting how investments will grow over time.

These applications show how financial mathematics empowers people to make smart financial choices in different life scenarios.

7. Conclusion: The Value of Financial Mathematics

Financial mathematics is essential for managing personal and corporate finances effectively. By understanding concepts like time value of money, compound interest, annuities, and risk analysis, individuals and businesses can make better financial decisions. Whether calculating loan payments, planning for retirement, or assessing investments, financial mathematics builds a solid foundation for a financially secure future. As you continue exploring these principles, you’ll see that financial mathematics equips you to handle complex financial situations with confidence.

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1. In a business, A and B invested the same amount at 8% and 10% interest rates, respectively. After 4 years, B withdrew his investment, while A kept his money invested for an additional 4 years. What is the extra amount that A gets compared to B?

a) 18% of the principal amount 𝑃

b) 20% of the principal amount 𝑃

c) 24% of the principal amount 𝑃

d) 26% of the principal amount 𝑃

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2. Hashim invested Rs. 20000 today and will withdraw amount after two years. What Will be
the amount if he withdraws after 3 years at 12% compounded annually?

a) Rs. 25,088

b) Rs. 28,098

c) Rs. 47,488

d) Rs. 67,488

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3. If Ali invested Rs. 30,000 today compounded annually for five years. There is no penalty if
he withdraws money after 2 years. If he will receive Rs. 36,300 after 2 years
1) What is the interest rate?
2) What money he gets, if he withdraws after 3 years

a) 10%, Rs.36300

b) 10%, Rs. 39,930

c) 10.5%, Rs.39450

d) none

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4. A person invested Rs. 100,000 at the start of each year for 6 years @8% simple interest for
first 3 years and 10% simple interest for last 3 years. Find the total amount he received at the
end of 6 years

a) Rs 698,000

b) Rs 654,000

c) Rs 798,000

d) None of these

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5. Basit wants to took a loan for 5 years Which would be the least beneficial for him?

a) 11 % simple interest

b) 10% compounded annually

c) 9% compounded semi annually

d) 8% compounded quarterly

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6. A person invested some amount today @1.8% per quarter for 10 years find his investment if
he receives Rs. 10 million.

a) 8.356 million

b) 4.13 million

c) 4.89 million

d) none of the above

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7. Basit took Rs. 200,000 from bank at 13.5% rate per year. What is the total amount after 5
years?

a) 356,700

b) 376,712

c) 335,000

d) 337,500

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8. Mr. Ikram invests Rs. 400,000 at the start of each quarter@12% for 8 years. What will be the
total amount after 8 years?

a) Rs. 5,510,263

b) Rs. 4,919,877

c) Rs. 21,631,137

d) Rs. 21,001,103

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9. A person has enough amount to invest for 9 years, if Rs. 1 million is received after 9 years
(end of year) at the rate of 7%, What is amount he has to invest each year (at the start)?

a) Rs.83,486

b) Rs.78,025

c) Rs.543,933

d) Rs.613,497

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10. If the fees earned by school in 10th year is 5 Million and fee is increased each year by 10%
and number of students increase by 5%. What was the fee earned by school in 1st year?

a) Rs.3.07Million

b) 1.93 Million

c) 1.42 million

d) 0.81 million

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11. If 5 Million is invested for 3 years and yield 6 Million at compounded quarterly interest rate.
If this amount is not withdrawn and further invested at the same interest rate, what will be the
fee earned by school in further 3 years?

a) Rs.7.2M

b) Rs.6M

c) Rs.7.5M

d) note of the above

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12. If Rs. 500,000 is invested in a scheme @ 1.5% per month simple interest for 5 years then:

a) He will get gain of Rs. 150,000 per annum

b) He will get gain of Rs. 75,000 per annum

c) He will get gain of Rs. 450,000 after 5 years

d) all above

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13. Sohail is receiving interest from Fast Bank Limited (FBL) at 14% compounded semi-
annually. Slow bank limited (SBL) has introduced a scheme whereby interest would be
compounded on a quarterly basis.
SBL should offer a minimum interest rate of 13.16% to motivate Sohail to shift his
investment from FBL to SBL is:

a) True

b) False

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14. Nasir had taken Rs. 900,000 from his office at 13.5% simple interest for a period of 5 years
and 5 months. The principal amount was paid at the expiry of loan period.
He paid interest of Rs. 595,000 during the period of loan.

a) True

b) False

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15. government has issued a five years bond of Rs. 200,000. On maturity the buyer will get Rs
300,000. If the current interest rate is 8% per annum, is purchasing the bond worth?

a) Yes, as present value of Rs 300,000 is more than Rs 200,000

b) No, as present value of Rs 300,000 is more than Rs 200,000

c) Yes, as present value of Rs 200,000 is less than Rs 300,000

d) No, as future value of Rs 200,000 is more than Rs 300,000

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16. Suman borrowed Rs. 900,000 at simple interest of 7.5% per annum. At the end of the loan
period she repaid a total of 1,372,500. Period of the loan was:

a) 7 years

b) 7 years 3 months

c) 7 years 7 months

d) 8 year

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17. Saad borrowed Rs 600,000 from Fahad for a period of 3 years and 9 months at r % simple
interest. He paid Rs 400,000 in excess of the borrowed amount at the end of loan period. The
value of “r” is:

a) 22.22%

b) 17.09%

c) 17.78%

d) 16.67%

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18. Ali invested Rs 500,000 for5 years after which he received a lump sum amount of Rs
762,150. If he earned 10% interest compounded annually during last 2 years, what rate of
interest compounded annually did he earn during the first three years?

a) 6%

b) 8%

c) 7%

d) 9%

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19. if Naina and Maina both invested same amount for 6 years and the rate of interest are 9%
and 10% compounded annually respectively. Then how much more percent amount Maina
will have after 6 years than Naina?

a) 7%

b) 8.5%

c) 9.24%

d) none

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20. What is the present value of $1,000 due in 5 years if the discount rate is 10%

a) $500

b) $620.92

c) $1,000

d) $750

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21. If you invest $200 at an annual interest rate of 8% compounded quarterly, what will be the amount after 3 years

a) $253

b) $262

c) $275

d) $260

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22. You take a loan of $10,000 at an annual interest rate of 12% to be repaid in equal monthly installments over a year. What will be the monthly installment

a) $833

b) $888

c) $1,100

d) $950

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23. Which of the following best describes the concept of ‘Net Present Value’ (NPV)?

a) The interest earned on an investment

b) The difference between present cash inflows and outflows

c) The future value of an investment

d) The total interest paid on a loan

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24. What is the annual percentage yield (APY) for an account with a 6% nominal interest rate compounded monthly?

a) 6.00%

b) 6.17%

c) 6.27%

d) 6.72%

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25. A bond with a face value of $1,000 pays a 5% annual coupon rate. If the current market price of the bond is $950, what is the bond's current yield?

a) 5.26%

b) 5.00%

c) 4.75%

d) 4.50%

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26. If inflation is 3% annually, how long will it take for the purchasing power of money to halve?

a) 23.3 years

b) 24.7 years

c) 18.5 years

d) 14.2 years

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27. In compound interest, the interest earned each period is calculated only on the initial principal amount.

a) True

b) False

Your score is

The average score is 42%

1. The Time Value of Money: The Core Principle of Financial Mathematics

2. Compound Interest: The Impact of Exponential Growth

3. Annuities: Structured Payments Over Time

4. Loan Amortization: Paying Off Debt Gradually

5. Risk and Return Analysis: Making Informed Investment Decisions

6. Applications of Financial Mathematics in Everyday Life

7. Conclusion: The Value of Financial Mathematics

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