Which of the following terms is the series of equal periodic payments or deposits where interest on each one is compounded interest?

Compound Interest: The future value (FV) of an investment of present value (PV) dollars earning interest at an annual rate of r compounded m times per year for a period of t years is:

FV = PV(1 + r/m)mtor

FV = PV(1 + i)n

where i = r/m is the interest per compounding period and n = mt is the number of compounding periods.

One may solve for the present value PV to obtain:

PV = FV/(1 + r/m)mt

Numerical Example: For 4-year investment of $20,000 earning 8.5% per year, with interest re-invested each month, the future value is

FV = PV(1 + r/m)mt   = 20,000(1 + 0.085/12)(12)(4)   = $28,065.30

Notice that the interest earned is $28,065.30 - $20,000 = $8,065.30 -- considerably more than the corresponding simple interest.

Effective Interest Rate: If money is invested at an annual rate r, compounded m times per year, the effective interest rate is:

reff = (1 + r/m)m - 1.

This is the interest rate that would give the same yield if compounded only once per year. In this context r is also called the nominal rate, and is often denoted as rnom.

Numerical Example: A CD paying 9.8% compounded monthly has a nominal rate of rnom = 0.098, and an effective rate of:

r eff =(1 + rnom /m)m   =   (1 + 0.098/12)12 - 1   =  0.1025.

Thus, we get an effective interest rate of 10.25%, since the compounding makes the CD paying 9.8% compounded monthly really pay 10.25% interest over the course of the year.

Mortgage Payments Components: Let where P = principal, r = interest rate per period, n = number of periods, k = number of payments, R = monthly payment, and D = debt balance after K payments, then

R = P � r / [1 - (1 + r)-n]

D = P � (1 + r)k - R � [(1 + r)k - 1)/r]

Accelerating Mortgage Payments Components: Suppose one decides to pay more than the monthly payment, the question is how many months will it take until the mortgage is paid off? The answer is, the rounded-up, where:

n = log[x / (x � P � r)] / log (1 + r)

Future Value (FV) of an Annuity Components: Ler where R = payment, r = rate of interest, and n = number of payments, then

FV = [ R(1 + r)n - 1 ] / r

Future Value for an Increasing Annuity: It is an increasing annuity is an investment that is earning interest, and into which regular payments of a fixed amount are made. Suppose one makes a payment of R at the end of each compounding period into an investment with a present value of PV, paying interest at an annual rate of r compounded m times per year, then the future value after t years will be

FV = PV(1 + i)n + [ R ( (1 + i)n - 1 ) ] / i where i = r/m is the interest paid each period and n = m � t is the total number of periods.

Numerical Example: You deposit $100 per month into an account that now contains $5,000 and earns 5% interest per year compounded monthly. After 10 years, the amount of money in the account is:

FV = PV(1 + i)n + [ R(1 + i)n - 1 ] / i =
5,000(1+0.05/12)120 + [100(1+0.05/12)120 - 1 ] / (0.05/12) = $23,763.28

Value of a Bond:

V is the sum of the value of the dividends and the final payment.

You may like to perform some sensitivity analysis for the "what-if" scenarios by entering different numerical value(s), to make your "good" strategic decision.

Replace the existing numerical example, with your own case-information, and then click one the Calculate.

Lesson 5: Sinking Fund Factor(Assessors’ Handbook 505, Column 3)

This lesson discusses the Sinking Fund Factor (SFF); one of six compound interest functions presented in Assessors’ Handbook Section 505 (AH 505), Capitalization Formulas and Tables. The lesson:

• Explains the function’s meaning and purpose,
• Provides the formula for the calculation of the SFFs, and
• Shows practical examples of how to apply the SFF.

SFF: Meaning and Purpose

The SFF is the equal periodic payment that must be made at the end of each of n periods at periodic interest rate i, such that the payments compound to $1 at the end of the last period.

The SFF is typically used to determine how much must be set aside each period in order to meet a future monetary obligation. The factors for the sinking fund are in column 3 of AH 505.

The SFF can be thought of as the “opposite” of the FW$1/P factor; mathematically, the SFF and the FW$1/P factor are reciprocals:

Conceptually, the FW$1/P factor provides the future amount to which periodic payments of $1 will compound, while the SFF provides the equal periodic payments that will compound to a future value of $1.

Formula for CalculatingSFF

The formula for the calculation of the SFF is

Where:

• SFF = Sinking Fund Factor
• i = Periodic Interest Rate, often expressed as an annual percentage rate
• n = Number of Periods, often expressed in years

In order to calculate the SFF for 4 years at an annual interest rate of 6%, use the formula below:

The table below shows how the sinking fund payments of 0.228591 per year compound to $1 at the end of 4 years. The payments are at the end of each year, so the beginning balance in year 1 is 0.

Viewed on a timeline:

To locate the SFF, go to page 33 of AH 505, go down 4 years and across to column 3. The correct factor is 0.228591.

Link to AH 505, page 33

Practical Applications of SFF

Example 1:
A company has just issued bonds with a face value of $150 million that are due and payable in 10 years. How much should the company deposit at the end of each year in order to retire the bond issue at the end of year 10, assuming the company can earn an annual interest rate of 7% on its deposits?

Solution:

• PMT = FV × SFF (7%, 10 yrs, annual)
• PMT = $150,000,000 × 0.072378
• PMT = $10,856,700 (annual deposit required)

• Find the annual SFF (annual compounding) for 7% and a term of 10 years. In AH 505, page 37, go down 10 years and across to column 3 to find the correct SFF of 0.072378.
• The required annual deposit, $10,856,700, is equal to the future value (the required amount at the end of year 10 multiplied by the SFF).

Link to AH 505, page 37

Example 2:
When you retire in 25 years, you would like to have $500,000 in your 401k retirement account. If you can earn an annual rate of 8%, how much should you deposit at the end of each month in order to reach your goal?

Solution:

• PMT = FV × SFF (8%, 25 yrs, monthly)
• PMT = $500,000 × 0.001051
• PMT = $525.50 (monthly deposit required)

• Find the monthly SFF (monthly compounding) for 8% and a term of 25 years. In AH 505, page 40, go down 25 years and across to column 3 to find the correct SFF of 0.001051.
• The required monthly deposit, $525.50, is equal to the future value (amount desired upon retirement) multiplied by the SFF.

Link to AH 505, page 40

Example 3:
In a balloon payment loan, only interest payments are made during the term of the loan; all of the principal is repaid at the end of the term. Suppose that you must repay a balloon loan in the amount of $1,000,000 that will be due 10 years from today. At an annual interest rate of 8%, how much should you deposit at the end of each year to fund the balloon payment?

Solution:

• PMT = FV × SFF (8%, 10 yrs, annual)
• PMT = $1,000,000 × 0.069029
• PMT = $69,029 (annual deposit required)

• Find the annual SFF (annual compounding) for 8% and a term of 10 years. In AH 505, page 41, go down 10 years and across to column 3 to find the correct SFF of 0.069029.
• The required annual deposit, $69,029, is equal to the future value (in this case the amount of the balloon payment) multiplied by the SFF.

Link to AH 505, page 41

Example 4:
You own a small apartment building and five years from now, you expect to replace the roof at an estimated cost of $50,000. How much should you set aside at the end of each year to fund the future roof replacement, given an annual interest rate of 6%?

Solution:

• PMT = FV × SFF (6%, 5 yrs, annual)
• PMT = $50,000 × 0.177396
• PMT = $8,869.80 (annual deposit required)

• Find the annual SFF (annual compounding) for 6% and a term of 5 years. In AH 505, page 33, go down 5 years and across to column 3 to find the correct SFF of 0.177396.
• The required annual deposit, $8,869.80, is equal to the future value (in this case the future cost of the new roof) multiplied by the SFF.

Link to AH 505, page 33

Example 5:
A borrower has a $200,000 balloon payment due in 10 years. To ensure that he can make the future payment, he plans to make equal annual deposits at the end of each year into an account that earns an annual interest rate of 4%. How much should he deposit at the end of each year?

Solution:

• PMT = FV × SFF (4%, 10 yrs, annual)
• PMT = $200,000 × 0.083291
• PMT = $16,658.20 (annual deposit required)

• Find the annual SFF (annual compounding) for 4% and a term of 10 years. In AH 505, page 25, go down 10 years and across to column 3 to find the correct SFF of 0.083291.
• The required annual deposit, $16,658.20, is equal to the future value (in this case the amount of the balloon payment) multiplied by the SFF.

Link to AH 505, page 25

Which of the following refers to the equal periodic payments or deposits where the interest on each one is compounded?

Which of the following terms refers to a sequence of payments at fixed intervals at compound interest?

What is series of periodic payments?

What is a payment of a debt made by a series of equal periodic payments?