Unit 4: Time Value of Money Concept - Subjective Questions

Time value of money (TVM) is the principle that money available today is worth more than the same amount available in the future because present money can be invested to earn a return.

The main reasons are:

• Earning capacity: Present money can earn interest or investment returns.
• Inflation: Rising prices reduce the future purchasing power of money.
• Risk and uncertainty: Future cash flows may not be received as expected.
• Liquidity preference: Individuals generally prefer immediate access to money.
• Opportunity cost: Receiving money later means giving up current investment opportunities.

For example, if is invested at for one year, its future value is:

Therefore, today is equivalent to after one year at a return.

Simple interest is calculated only on the original principal, whereas compound interest is calculated on the principal and accumulated interest.

Simple interest:

Compound interest:

where is the principal, is the interest rate per period, and is the number of periods.

Key differences include:

• Under simple interest, the interest amount is constant each period.
• Under compound interest, interest increases because previous interest also earns a return.
• Compound interest produces a higher future value when the rate and time period are positive.

For invested at for two years:

• Simple-interest future value: .
• Compound-interest future value: .

Let the initial investment be and the annual interest rate be .

• At the end of year 1: .
• At the end of year 2: .
• Continuing this process for years gives:

For the given investment:

Thus, the investment will grow to approximately after five years. The total compound interest earned is .

Discounting is the process of converting a future cash flow into its equivalent value today. It is the reverse of compounding.

Starting with the future-value relationship:

Rearranging it gives:

For the given cash flow:

Therefore, today is financially equivalent to received after four years when the required return is .

When interest is compounded more frequently, interest is credited earlier and begins earning additional interest. Therefore, for a positive nominal rate, future value generally rises as compounding frequency increases.

If the nominal annual rate is , interest is compounded times per year, and the investment period is years, then:

Common frequencies are:

• Annual:
• Semiannual:
• Quarterly:
• Monthly:
• Daily: commonly

For the same nominal rate and investment period:

The increase becomes progressively smaller as compounding becomes more frequent.

A nominal annual rate is the stated annual interest rate that does not itself account for the effect of intra-year compounding. The periodic rate is obtained by dividing it by the number of compounding periods.

An effective annual rate (EAR) is the actual annual rate earned or paid after considering compounding during the year.

If is the nominal annual rate and compounding occurs times per year, then:

Key differences are:

• The nominal rate is a quoted rate; the EAR measures the true annual return or cost.
• The EAR is greater than the nominal rate when compounding occurs more than once per year and the rate is positive.
• The EAR allows meaningful comparison of financial products with different compounding frequencies.
• With annual compounding, the nominal rate and EAR are equal.

The effective annual rate is calculated as:

Here, and :

Therefore:

The effective annual rate is approximately . This means that invested for one year would grow to approximately , assuming monthly compounding. The EAR is higher than the quoted nominal rate because interest is compounded each month.

For the first investment:

Thus, .

For the second investment:

Thus, .

Comparison:

• First investment: approximately effective annually.
• Second investment: approximately effective annually.

Therefore, the nominal rate compounded quarterly provides the slightly higher effective annual return. This demonstrates why nominal rates should not be compared without adjusting for compounding frequency.

An annuity is a series of equal cash payments or receipts occurring at regular intervals for a specified period.

An ordinary annuity has payments at the end of each period. Examples include:

• Loan instalments paid at each month-end
• Year-end investment deposits
• Bond interest paid at the end of each interest period

An annuity due has payments at the beginning of each period. Examples include:

• Rent paid at the beginning of each month
• Insurance premiums paid in advance
• Lease payments made at the beginning of each period

Because each annuity-due payment earns interest for one additional period, its present and future values are greater than those of an otherwise identical ordinary annuity:

For an ordinary annuity, each payment is made at the end of a period. At the end of year , the payments accumulate as follows:

This is a geometric series. Applying the geometric-series formula gives:

For , , and :

Thus, the accumulated value immediately after the fifth deposit is approximately . Total deposits are , so the interest earned is approximately .

The present value of an ordinary annuity equals the sum of the discounted values of all end-of-period payments:

This finite geometric series simplifies to:

Using , , and :

Therefore, the six receipts have a present value of approximately at a discount rate.

An annuity due makes every payment one period earlier than an ordinary annuity. Consequently, every payment earns interest for one extra period when future value is calculated and is discounted for one fewer period when present value is calculated.

The relationships are:

For example, if the present value of an ordinary annuity is and the periodic rate is :

The annuity due has a higher value because payments are received earlier. When applying the formulas, the interest rate and payment period must use the same time interval.

A deferred annuity is an annuity whose first payment begins after a specified delay. Its value is first calculated one period before the first payment and then discounted back to the valuation date.

There are five payments from year 4 through year 8. Their value at the end of year 3 is:

Discounting this amount for three years:

Therefore, the present value of the deferred annuity is approximately .

A perpetuity is a stream of equal periodic cash flows that continues indefinitely. If the first payment of occurs one period from today and the discount rate is , then:

For example, a perpetuity paying annually at a discount rate of is worth:

A growing perpetuity is an infinite stream in which payments grow at a constant rate . If the first payment next period is , then:

The formula requires:

• The first cash flow to occur one period after valuation.
• Cash flows to continue indefinitely.
• The discount rate to exceed the growth rate, so .
• The rate and growth period to be expressed consistently.

Let the initial value be . If it doubles in six years, the future value is .

Using the compound-value equation:

Dividing both sides by :

Taking the sixth root:

Therefore:

Thus, the investment earns a compound annual return of approximately . This is the annual rate that would produce exactly the same doubling over six years.

Cash flows occurring at different dates must be moved to a common date before comparison. Discount the future payment to the present:

Comparison at the present date:

• Immediate receipt: .
• Present value of future receipt: approximately .
• Difference: approximately .

Therefore, receiving after four years is financially preferable, assuming an required return and no material difference in risk. Its present value exceeds the immediate offer by approximately .

Loan amortization is the repayment of a loan through periodic instalments containing both interest and principal. Interest is calculated on the outstanding balance, so the interest component declines and the principal component generally rises over time.

The monthly rate and number of payments are:

The present value of the instalments must equal the loan amount:

Therefore:

The first month's interest is . The remainder of the first instalment reduces principal. Minor differences in the instalment may arise from rounding.

A sinking fund is a fund created through regular deposits to accumulate a specified amount for a future obligation, such as debt repayment or asset replacement.

Using the future value of an ordinary annuity:

Solving for the periodic deposit:

Substituting the values:

Therefore, approximately must be deposited at the end of each year to accumulate in five years.

Uneven cash flows cannot be valued using a standard annuity formula. Each cash flow must be discounted separately and the resulting present values added.

The project's net present value is:

The individual present values are approximately:

• Year 1:
• Year 2:
• Year 3:

Thus:

Because the NPV is negative, the project's discounted inflows are insufficient to recover the investment at a required return.

Under continuous compounding, interest is compounded an infinitely large number of times per year. The future-value formula is:

where is the mathematical constant approximately equal to .

The effective annual rate is:

For :

Therefore, the effective annual rate is approximately .

The future value after three years is:

Thus, grows to approximately after three years.