Annuities: Ordinary vs. Due

• Total cash available after multiple deposits is the sum of the future values of all deposits.
• Example: Placing aside $1,200 today, $1,400 next year, and $1,000 a year later, earning 8% annual interest, determines the total cash in three years.
• Comparison: Paying $15,500 for a used car today versus $8,000 down and $4,000 payments at the end of the next 2 years with an 8% interest rate helps determine the better deal.

Annuities

• Annuities provide a stream of equal cash flows at fixed intervals for a set period.
• Home mortgages with equal monthly payments over the loan's life are an example.

Ordinary Annuity

• Payments are made at the end of each period.
• To value, calculate the present value of each cash flow and sum them.
• With annual payments 𝐶 and annual interest rate 𝑖, or monthly payments 𝐶 and monthly interest rate 𝑖.

Annuity Due

• Payments are made at the beginning of each period.
• Present value of an annuity due is the present value of an ordinary annuity multiplied by (1 + 𝑖): 𝑃𝑉0 𝑜𝑓 𝑎𝑛𝑛𝑢𝑖𝑡𝑦 𝑑𝑢𝑒 = 𝑃𝑉0 𝑜𝑓 𝑜𝑟𝑑𝑖𝑛𝑎𝑟𝑦 𝑎𝑛𝑛𝑢𝑖𝑡𝑦 × (1 + 𝑖)
• Example: An annuity due of $1,000 annually for 10 years at a 3% interest rate has a specific present value.

Growing Ordinary Annuity

• Payments grow constantly at fixed intervals for a specified number of periods.
• The present value of a growing annuity formula is: 𝑃𝑉0 =( 𝐶/𝑖 – 𝑔) (1 − (1 + 𝑔)^𝑛/(1 + 𝑖)^𝑛)
• Example: A growing ordinary annuity of 4 years with annual payments growing at 2%, a first payment of $100, and a 5% annual interest rate.

Growing Annuity Due

• Payments are made at the beginning of the period and grow constantly.
• Example: A growing annuity due of 30 years with annual payments growing at 2%, a first payment of $20, and a 4% annual interest rate.

Future Value of an Annuity

• Setting aside $3,000 at the end of every year with an 8% annual interest rate accumulates to a specific value after 4 years.
• The sum of the future values of the four payments: 𝐹𝑉4 = 3,000(1 + 0.08)3+ 3,000(1 + 0.08)2+3,000 (1 + 0.08) + 3,000 = 13,518
• Formula: 𝐹𝑉𝑛 = 𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎𝑛𝑛𝑢𝑖𝑡𝑦 × (1 + 𝑖)^𝑛
• Accumulating $500,000 in 50 years requires saving a specific amount annually, depending on the interest rate (e.g., 10% annually).

Effective Interest Rates

• The interest rate must align with the frequency of payments when valuing cash flows.
• Example: Consider a 4-year ordinary annuity with constant monthly payments of 100. If the EAR is 5%, how much is the annuity worth today?
• Example: Consider a 5-year ordinary annuity with quarterly constant payments of 1,200. If the APR is 5%, compounded monthly, how much is it worth today?

Perpetuity

• Perpetuities are streams of equal payments at regular intervals that continue forever.

Ordinary Perpetuity

• Payments occur at the end of each period, with the first cash flow at the end of period 1.
• Formula: PV=C/i
• Example: An ordinary perpetuity of $100 paid annually with a 5% annual interest rate.

Perpetuity Due

• Payments occur at the beginning of each period, starting immediately.
• Formula: PV=(1+i)*C/i
• Example: A perpetuity due of $25 paid annually with a 10% annual interest rate.

Growing Ordinary Perpetuity

• Payments grow at a constant rate and occur at regular intervals, continuing infinitely.
• Example: An ordinary perpetuity with an initial payment of $100 growing at 3% annually, with a 6% interest rate.

Growing Perpetuity Due

• Payments grow at a constant rate, occur at the beginning of each period, and continue forever.
• Example: A perpetuity due with an initial payment of $40 growing at 2% annually, with a 9% interest rate.