Introduction to Agribusiness Finance– Annuities and Loans Part II Objectives: Practice more complicated annuity problems In-class work (more loan amortization, complex annuity problems) Advanced Time Value of Money Problems Deferred Annuities When the annuity doesn’t begin in year 0 or 1. Example would be our social security payments, which won’t start until we are 62 Combining Multiple Annuities When an annuity is followed by a different annuity Example would be annual savings followed by mortgage payment or social security income. Advanced Time Value of Money Problems We already know how to solve these problems – they are just cases of unequal cash flows! The annuity equations just give us additional ways to solve them. Tips for Solving Advanced TVM Problems Review the information given in the problem and jot down what is known Draw a timeline Select a focal date Determine whether cash flows are lump-sums or annuities Determine the compounding frequency of the problem and decide whether the cash flows are beginning or end of period Timeout! Let’s review the “timing” of =PV() =PV(rate, nper, payment) 𝑃𝑃𝑃𝑃 = 𝑃𝑃𝑃𝑃𝑃𝑃 × 1− 1 1+𝑟𝑟 𝑛𝑛 𝑟𝑟 These assume that the first payment is received 1 year from today. Today = year 0 Deferred Annuities Example – A contract agreement requires the payment of $15,000 at the end of the 3rd, 4th, and 5th years but nothing in the 1st and 2nd years. Find PV when discount rate = 0.09. 0 Cash Flow PV 1 0 2 0 =0/(1.09)^0 =0/(1.09)^1 0 3 0 =0/(1.09)^2 0 Total PV = $31,958.10 4 15000 =15000/(1.09)^3 0 11,582.75 5 15000 =15000/(1.09)^4 10,626.38 15000 =15000/(1.09)^5 9,748.97 Deferred Annuities We can solve this problem another way using the PV of annuity formula. 1. Find PV of annuity at start of annuity (end of year 2) 𝑃𝑃𝑃𝑃 = 15000 ∗ 1 1.093 1− 0.09 = $37,969.42 2. Discount this amount back to “today” (i.e. end of year zero). $37,969.42 𝑃𝑃𝑃𝑃 = = $31,958.10 2 1.09 Combined Annuities This is just an annuity with other payments mixed in. Two ways to solve: You can either write out on timeline and adjust each cash flow to present value individually and sum them, OR You can use the annuity formulas for the annuities and the PV and FV formulas for individual cash flows and sum them. Imbedded and Combined Annuities Example: You turned 30 years old today and don’t have a penny to your name. You want to retire at age 60 with $2,200,000 in the bank. Realistically, you know that the most that you can save from your 31st birthday until your 50th is $6,700 per year (you only save on your birthdays!). How much do you have to save each year from your 51st to your 60th birthday in order to achieve your retirement goal if you can earn 5.5% on your savings? In-Class Activity Practice solving advanced TVM problems Retirement plan Savings plan Loan amortization