 # Accrued Interest

### Accrued Interest

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Here are some insights from the field of Accounting on the topic of Accrued Interest:

Accrued Interest calculations are Time Value of Money (TVM) equations applied in estimation of the rate of return from periodic, compound interest on investment, loan, or savings principle.

Effective Annual Rate (EAR)

The Effective Annual Rate (EAR) or (1 + Periodic interest rate)m – 1 is the formula for estimating the total sum of payments where: m = number of compounding periods per year, and the periodic interest rate.

The Effective Annual Rate (EAR) is the simple formula for calculating compounded monthly interest. The EAR formula expresses “m” = number of periods (N). Example to find the EAR on a compounded monthly agreement with an interest rate of 7%:

EAR = (1 + (0.07/12))^12 - 1 = (1.07229) -1 = 0.07229  or 7..23%

Present Value (PV) and Future Value (FV)

Determination of present value (PV) or future value (FV) of TVM estimations is based on the periodic term defined interest rate accrual. Interest for PV and FV is compounded (i.e. quarterly or monthly), or the calculation of the interest rate in years and the time in months. The Rate of Return (r) and Number of Periods (N) consistent with the compounding estimation is performed prior to the calculation of PV and FV problems.

r (or “I”) = periodic interest rate

PV = present value of a single sum of money

FV = future value of a single sum of money

The Present Value (PV) Formula –>Equation

To calculate today’s value or PV of \$10,000 with interest compounding at a rate of 7% accrued for a five year period.

PV = FV * (1/(1 + r)N) = (\$10,000)*(1/(1.07)^5) = (\$10,000)*(1/(1.402552))

PV = (\$10,000)*(0.712986) = \$7129.86

The Future Value (FV) Formula –>Equation

To calculate the FV of a single sum of \$10.000 with annual interest compounding at a rate of 7% accrued for a five year period:

FV = PV * (1 + r)= (\$10,000)*(1.07)^= (\$10,000)*( 1.402552) = \$ 14025.52

To calculate the FV of a single sum of \$10.000 with quarterly interest compounding at a rate of 7% accrued for a five year period:

r = 7%/4 = 0.0175, and periods N = 4*5 = 20 quarters

FV = PV * (1 + r)N = (\$10,000)*(1.0175))^20 = (\$10,000)*(1.4147782) = \$14,1477.82

To calculate the FV of a single sum of \$10,000 with monthly compounded interest compounding at a rate of 7% accrued for a five (5) year period:

r = 7%/12 = 0.005833, and N = 12*5 = 60 months

FV = PV * (1 + r)N = (\$10,000)*(1.005833)^60 = (\$10,000)*(10356.07) = \$10,356.07

The Discounting of Uneven Cash Flows

In cases where it cannot be assumed FV and PV coincide with level and sequential cash flows, the formula for Discounted Cash Flows (DCF) must be applied. In other words, each cash flow of an uneven cash flows statement must be “discounted back” to the PV or compounded to an FV depending on the problem. After determining if an estimation demands finding the PV or FV, the sum of all cash flows is calculated.

The formula for finding the PV of a sequence of uneven cash flows requires the discount rate to compute the results. In this case, the discount rate is 7%.

 Cash Flow Present Value Formula Results \$1,000 (\$1,000)/(1.07)1 \$934.58 \$1,200 (\$1,200)/(1.07)2 \$1373.88 \$500 (\$500)/(1.07)3 \$612.52 \$2,000 (\$2,000)/(1.07)4 \$2621.59 \$1,700 (\$1,700)/(1.07)5 \$2384.34

Total results of present value (PV) estimation = \$7926.91

Where the future value of the same sequence of cash flows for a period is required, the same approach is used by applying the FV formula rather than PV formula. An example is the result of the periodic interest on an annuity with a sequence of uneven cash flows, at an interest rate of 7%. Note: the inversion of the number of periods (N) beginning with zero reflecting principle prior to interest accrual at the end of the first period.

 Cash Flow Present Value Formula Results \$1,000 (\$1,000)/(1.07)4 \$762.90 \$1,200 (\$1,200)/(1.07)3 \$979.56 \$500 (\$500)/(1.07)2 \$436.72 \$2,000 (\$2,000)/(1.07)1 \$1869.17 \$1,700 (\$1,700)/(1.07)0 \$1700.00

Total results of future value (FV) estimation = \$ 5748.35

Valuation of Perpetuities

The rate of return is applied to interest accrual calculation of annuities. An ordinary annuity (first cash flow is one period from today) yet continues indefinitely or “in perpetuity.” Met with equal sequential payments, estimation of perpetuities requires the use of the PV formula to determine N = infinity assuming interest rates are positive.

For example, a \$1,000 annual payment on a perpetuity at an interest rate of 7% is worth:

PV = A/r = (\$1000)/0.07 = \$14,285

Bibliography

“Accrued Interest.” Accounting Tools. Mar 6, 2020.

Chen, James. “Time Value of Money (TVM).” Investopedia. Apr 21, 2020.

“Rate of Return Formula.” Wallstreet Mojo.

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