Topic 9 Time Value of Money

Topic 9: Time Value of Money Learning Objectives Calculate present value and future value of single amounts, annuities, annuities due, uneven cash flows and serial payments. Calculate NPV and IRR and be able to apply the techniques to financial planning problems.

Topic 9: Present Value The value today of a known amount or series of amounts to be received in the future, given a specified interest rate and number of periods for discounting Formula for single sum PV = FV / (1 + i)n

Topic 9: Present Value Example What is the present value of $20,000 in 5 years assuming an 8% rate of return? It will be worth $13,612 N = 5 I = 8 PMT = 0 FV = 20,000 Solve for PV

Topic 9: Future Value The value that will be available when a present sum or series of periodic sums is invested, with earnings added, most often on a compound basis Formula for a single sum FV = PV(1 + i)n

Topic 9: Future Value Example What is the future value of $5,000 invested today in a 4% CD for 3 years? It will be worth $5,624 N = 3 I = 4 PV = 5,000 PMT = 0 Solve for FV

Topic 9: Rule of 72 A tool for estimating the approximate length of time it will take for a single sum of money to double in value at a given compound annual rate of interest Divide 72 by the interest rate to produce the answer Example: At a 6% rate of interest it will take approximately 12 years (72/6 = 12) for a lump sum of money to double in value

Topic 9: Annuity A specified amount of money, paid or received at a specified uniform interval, for a specified period of time Ordinary annuity Payments or receipts are made at the end of each period Annuity due Payments or receipts are made at the beginning of each period

Topic 9: Annuity Example How much should a client pay for an investment that will pay him $1,000 a year for 10 years given a 6% required rate of return? The client should pay $7,360 N = 10 I = 6 PMT = 1,000 FV = 0 Solve for PV

Topic 9: Net Present Value The present value of all cash outflows and cash inflows It is how much a client should pay for an asset

Topic 9: Net Present Value Example How much should a client be willing to pay for an investment with the following cash flows assuming a 7% interest rate? End of Year 1 = $1,000 End of Year 2 = $2,000 End of Year 3 = ($5,000) End of Year 4 = $10,000 The client should be willing to pay $6,229 Input the cash flows above starting with $0 as the first cash flow Input 7% as the interest rate Solve for NPV

Topic 9: Internal Rate of Return (IRR) It is the rate of return on an investment given the cash inflows and outflows The client wants the IRR to be equal to or greater than the client’s required rate of return

Topic 9: Internal Rate of Return Example If a client pays $6,000 for the investment used in the NPV example, what will be the client’s IRR? The client’s IRR will be 8.1% Input the following cash flows ($6,000) $1,000 $2,000 ($5,000) $10,000 Then solve for IRR which is 8.1% The IRR will be higher than the 7% since the client paid less than the PV of the future cash flows (NPV>0)

Topic 9: Uneven Cash Flows Typically you will be solving for how much a client should pay for the investment or how much they will have saved at the end of a period of time given the different cash flows from an investment

Topic 9: Uneven Cash Flows Example How much should Peter pay for a bond with a 9% coupon that matures in 7 years if comparable bonds are yielding 10%? Peter should pay $951 N = 7 x 2 = 14 Due to semi annual coupon payments I = 10 / 2 = 5 Semi annual interest rate PMT = 90 / 2 = 45 Semi annual coupon payment FV = 1000 Solve for PV

Topic 9: Serial Payments The calculation will involve an inflation adjusted rate of return [(1 + nominal rate)/(1 + inflation rate) – 1] x 100 Example 9% yield in a year with 2.5% inflation [(1 + 9%) / (1 + 2.5%) –1] x 100 6.34%

Topic 9: Annuity Due Serial Payment Example Susan wishes to accumulate $50,000 in today’s dollars in 4 annual deposits, beginning immediately. The deposits will earn 7% after taxes, and the deposits must also increase each year by 3% to keep up with anticipated inflation. What should be the size of the first deposit? Susan should deposit $11,354 Set for beginning-of-period payments N = 4 I = [(1.07 ÷ 1.03) – 1] x 100 = 3.88 PV = 0 FV = 50,000 Solve for PMT

Topic 9: Ordinary Annuity Serial Payment Example Susan wishes to accumulate $50,000 in today’s dollars in 4 annual deposits. The deposits will earn 7% after taxes, and the deposits must also increase each year by 3% to keep up with anticipated inflation. What should be the size of the first deposit? Susan should deposit $12,149 Set for end-of-period payments N = 4 I = [(1.07 ÷ 1.03) – 1] x 100 = 3.88 PV = 0 FV = 50,000 Solve for PMT which is 11,795 Add inflation that occurred between today and when the first payment is made by multiplying 11,795 x 1.03 = 12,149

Topic 9: Growing Annuity Formulas

Topic 9: Loan Amortization Example Ella took out a $500,000 loan fixed at 6% interest for 30 years? How much is her monthly payment? N = 30 x 12 = 360, I = 6/12 = 0.50, PV = 500,000, FV = 0, Solve for PMT which is $2,998 How much interest and principal would Ella have paid after 5 years of monthly payments (60 payments) and what is her remaining loan balance? Without clearing your calculator from the problem above: 2nd, AMORT, 1, set, ↓, 60, set, ↓ $145,137 is the total amount of interest paid $34,728 is the total amount of principal paid $465,272 is the remaining loan balance

End of Topic 9