HEXCO · Polytechnic · NC Accountancy

Financial Mathematics and Statistics 1: Time Value of Money

Learning Outcome 3 of 3

Free HEXCO NC Accountancy notes on Time Value of Money, learning outcome 3 of 3 in Financial Mathematics and Statistics 1. Business computations, financial statistics and the time value of money. Semester 2, module 310/22/M05.

Here we learn why money today is worth more than money in the future, how to calculate simple and compound interest, and how to find the present and future values of lump sums and annuities.

A dollar received today can be invested to earn interest, so it is worth more than a dollar received next year, and this idea, the time value of money, underlies loans, savings, hire purchase and investment decisions.

3.1 Simple Interest

Interest is the price paid for the use of money, earned by a lender or investor and paid by a borrower.

With simple interest, interest is calculated only on the original amount, so it is the same every year.

Calculating Simple Interest

Simple interest is interest calculated on the original amount, called the principal, only. The formula is I = P × r × t, where P is the principal, r the interest rate per year as a decimal, and t the time in years. The amount (future value) at the end is A = P + I = P(1 + rt).

Here you are finding interest that is charged only on the original amount. Write down P (the principal), r (the yearly rate as a decimal) and t (the time in years, so months must be divided by 12). Multiply them to get the interest, and add it to P for the final amount. Below, we show you how it is done, step by step.

Find the simple interest on $5,000 invested at 8% a year for 3 years, and the amount at the end.
  1. I = P × r × t = 5,000 × 0.08 × 3 = $1,200
  2. A = 5,000 + 1,200 = $6,200

Answer: Interest $1,200; amount $6,200

What principal invested at 6% simple interest for 4 years grows to $3,720?
  1. A = P(1 + rt), so 3,720 = P(1 + 0.06 × 4) = 1.24P
  2. P = 3,720 ÷ 1.24 = $3,000

Answer: $3,000

Find the simple interest on $1,500 borrowed for 8 months at 12% a year.
  1. Time in years = 8 ÷ 12
  2. I = 1,500 × 0.12 × 8 ÷ 12
  3. I = 180 × 8 ÷ 12 = $120

Answer: $120

3.2 Compound Interest

With compound interest, each period's interest is added to the principal, so the next period's interest is earned on a larger amount.

This 'interest on interest' makes compound interest grow faster than simple interest.

Simple and Compound Interest Compared

Compound interest is interest calculated on the principal plus all interest already added. The amount after n periods is A = P(1 + r)ⁿ, and the compound interest is A − P. When interest is compounded more often than yearly, divide the yearly rate by the number of periods per year (m) and multiply the years by m: A = P(1 + r/m)ⁿᵐ.

Here you are finding interest that is added to the balance each period, so interest is then earned on interest. If interest is added more than once a year, divide the yearly rate by the number of times and multiply the years by the same number. Use A = P(1 + r)ⁿ and subtract P to get the interest. Below, we show you how it is done, step by step.

Find the compound interest on $5,000 at 8% a year for 3 years, and compare it with simple interest.
  1. A = 5,000 × (1.08)³ = 5,000 × 1.259712 = $6,298.56
  2. Compound interest = 6,298.56 − 5,000 = $1,298.56
  3. Simple interest for the same terms = $1,200 (from the previous example)
  4. Difference = 1,298.56 − 1,200 = $98.56

Answer: Compound interest $1,298.56, which is $98.56 more than simple interest

Find the amount of $2,000 invested for 2 years at 12% a year compounded quarterly.
  1. Rate per quarter = 12% ÷ 4 = 3% = 0.03
  2. Number of quarters = 2 × 4 = 8
  3. A = 2,000 × (1.03)⁸ = 2,000 × 1.266770 = $2,533.54

Answer: $2,533.54

Find the amount of $10,000 invested for one year at 6% a year compounded monthly, and the effective yearly rate.
  1. Rate per month = 6% ÷ 12 = 0.5% = 0.005; periods = 12
  2. A = 10,000 × (1.005)¹² = 10,000 × 1.061678 = $10,616.78
  3. Interest = $616.78, which is 616.78 ÷ 10,000 × 100 = 6.17% a year

Answer: $10,616.78; effective rate 6.17%

Future Value of a Lump Sum

The future value (FV) of a lump sum is what a single amount invested today will grow to after n periods at interest rate r: FV = PV × (1 + r)ⁿ. The factor (1 + r)ⁿ is called the compounding factor and can be found from tables or a calculator.

Here you are finding what a single amount invested today will grow to. Find the rate per period and the number of periods, then work out (1 + r)ⁿ on your calculator. Multiply today's amount by that factor to get the future value. Below, we show you how it is done, step by step.

A business deposits $10,000 today at 10% a year compound interest. Find its value after 5 years.
  1. FV = 10,000 × (1.10)⁵
  2. (1.10)⁵ = 1.61051
  3. FV = 10,000 × 1.61051 = $16,105.10

Answer: $16,105.10

Find the value of $3,000 after 4 years at 7% a year compound interest.
  1. (1.07)⁴ = 1.310796
  2. FV = 3,000 × 1.310796 = $3,932.39

Answer: $3,932.39

Find the value of $8,000 after 3 years at 12% a year compounded half-yearly.
  1. Rate per half-year = 12% ÷ 2 = 6%; periods = 3 × 2 = 6
  2. (1.06)⁶ = 1.418519
  3. FV = 8,000 × 1.418519 = $11,348.15

Answer: $11,348.15

Present Value of a Lump Sum

The present value (PV) is today's value of an amount to be received in the future, found by discounting: PV = FV ÷ (1 + r)ⁿ = FV × (1 + r)⁻ⁿ. The factor 1 ÷ (1 + r)ⁿ is the discount factor. Present value is used to compare amounts received at different times and to decide whether investments are worthwhile.

Here you are working backwards to find what a future amount is worth today. Work out (1 + r)ⁿ for the rate and number of periods. Divide the future amount by it, and use the result to compare money received at different times. Below, we show you how it is done, step by step.

How much must be invested today at 10% a year to have $16,105.10 in 5 years? And what is the present value of $5,000 receivable in 3 years at 8%?
  1. PV = 16,105.10 ÷ (1.10)⁵ = 16,105.10 ÷ 1.61051 = $10,000
  2. PV = 5,000 ÷ (1.08)³ = 5,000 ÷ 1.259712 = $3,969.16

Answer: $10,000 and $3,969.16

A business needs $20,000 in 4 years. How much must it invest today at 9% a year?
  1. (1.09)⁴ = 1.411582
  2. PV = 20,000 ÷ 1.411582 = $14,168.51

Answer: $14,168.51

Would you rather receive $5,000 now or $6,000 in 2 years, if money can earn 10% a year?
  1. (1.10)² = 1.21
  2. PV of $6,000 = 6,000 ÷ 1.21 = $4,958.68
  3. Compare: $5,000 now is worth more than $4,958.68

Answer: Take $5,000 now

Future Value of an Annuity

An annuity is a series of equal payments made at regular intervals, such as monthly savings, rentals or loan repayments. For an ordinary annuity, with payments at the end of each period, the future value is FV = PMT × [(1 + r)ⁿ − 1] ÷ r, where PMT is the payment per period. It answers questions such as how much regular savings will grow to.

Here you are finding what regular equal savings will grow to. Find the payment per period, the rate per period and the number of payments, matching them all to the same period. Work out [(1 + r)ⁿ − 1] ÷ r and multiply it by the payment. Below, we show you how it is done, step by step.

A trader saves $1,000 at the end of every year for 4 years at 10% compound interest. How much will she have after 4 years?
  1. FV = 1,000 × [(1.10)⁴ − 1] ÷ 0.10
  2. (1.10)⁴ = 1.4641
  3. FV = 1,000 × (1.4641 − 1) ÷ 0.10 = 1,000 × 4.641 = $4,641

Answer: $4,641

A worker saves $200 at the end of every month for 2 years at 12% a year compounded monthly. How much will she have?
  1. Rate per month = 1% = 0.01; payments = 24
  2. (1.01)²⁴ = 1.269735
  3. Factor = (1.269735 − 1) ÷ 0.01 = 26.9735
  4. FV = 200 × 26.9735 = $5,394.69

Answer: $5,394.69

How much must be saved at the end of each year for 5 years at 8% to have $10,000?
  1. (1.08)⁵ = 1.469328
  2. Factor = (1.469328 − 1) ÷ 0.08 = 5.866601
  3. Payment = 10,000 ÷ 5.866601 = $1,704.56

Answer: $1,704.56 a year

Present Value of an Annuity

The present value of an ordinary annuity is today's value of all the future equal payments: PV = PMT × [1 − (1 + r)⁻ⁿ] ÷ r. It is used to value a stream of income, and, turned around, to find the equal instalment needed to repay a loan: PMT = PV × r ÷ [1 − (1 + r)⁻ⁿ].

Here you are finding today's value of a series of equal payments, or the equal payment needed to repay a loan. Work out (1 + r)⁻ⁿ, subtract it from 1 and divide by r to get the annuity factor. Multiply the payment by the factor for the present value, or divide the loan by the factor to get the instalment. Below, we show you how it is done, step by step.

Find the present value of $2,000 received at the end of each year for 3 years at 10%.
  1. (1.10)⁻³ = 1 ÷ 1.331 = 0.751315
  2. PV = 2,000 × (1 − 0.751315) ÷ 0.10
  3. PV = 2,000 × 2.48685 = $4,973.70

Answer: $4,973.70

A loan of $4,973.70 at 10% a year is to be repaid in 3 equal yearly instalments at the end of each year. Find the instalment.
  1. PMT = PV × r ÷ [1 − (1 + r)⁻ⁿ]
  2. PMT = 4,973.70 × 0.10 ÷ (1 − 0.751315) = 497.37 ÷ 0.248685 = $2,000

Answer: $2,000 a year

A car loan of $15,000 at 1% a month is repaid in 36 equal monthly instalments. Find the instalment.
  1. (1.01)⁻³⁶ = 0.698925
  2. Factor = (1 − 0.698925) ÷ 0.01 = 30.1075
  3. Instalment = 15,000 ÷ 30.1075 = $498.21

Answer: $498.21 a month

All Financial Mathematics and Statistics 1 learning outcomes

  1. Learning Outcome 1: Business Computations
  2. Learning Outcome 2: Financial Statistics
  3. Learning Outcome 3: Time Value of Money