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## Chapter 5 The Time Value of Money

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**Chapter 5The Time Value of Money**Laurence Booth, Sean Cleary and Pamela Peterson Drake**Simple interest**Simple interest is interest that is paid only on the principal amount. Interest = rate × principal amount of loan**Simple interest: example**A 2-year loan of $1,000 at 6% simple interest • At the end of the first year, interest = 6% × $1,000 = $60 • At the end of the second year,interest = 6% × $1,000 = $60 and loan repayment of $1,000**Compound interest**Compound interest is interest paid on both the principal and any accumulated interest. Interest =**Compounding**Compounding is translating a present value into a future value, using compound interest. Future value interest factor is also referred to as the compound factor.**Compare: simple versus compound**Suppose you deposit $5,000 in an account that pays 5% interest per year. What is the balance in the account at the end of four years if interest is: Simple interest? Compound interest?**Interest on interest**How much interest on interest? Interest on interest = FVcompound – FVsimple Interest on interest = $6,077.53 – 6,000.00 = $77.53**Try it: Simple v. compound**Suppose you are comparing two accounts: The Bank A account pays 5.5% simple interest. The Bank B account pays 5.4% compound interest. If you were to deposit $10,000 in each, what balance would you have in each bank at the end of five years?**Try it: Answer**Bank A: $12,750.00 Bank B: $13,007.78**A note about interest**Because compound interest is so common, assume that interest is compounded unless otherwise indicated.**Short-cuts**Example: Consider $1,000 deposited for three years at 6% per year.**The long way**FV1 = $1,000.00 × (1.06) = $1,060.00 FV2 = $1,060.00 × (1.06) = $1,123.60 FV3 = $1,123.60 × (1.06) = $1,191.02 or FV3 = $1,000 × (1.06)3 = $1,191.02 or FV3 = $1,000 × 1.191016 = $1,191.02 Future value factor**Short-cut: Calculator**Known values: PV = 1,000 n = 3 i = 6% Solve for: FV**Input three known values, solve for the one unknown**Known: PV, i , n Unknown: FV**Short-cut: spreadsheetMicrosoft Excel or Google Docs**=FV(RATE,NPER,PMT,PV,TYPE) TYPE default is 0, end of period =FV(.06,3,0,-1000) or**Problem 1.1**Suppose you deposit $2,000 in an account that pays 3.5% interest annually. How much will be in the account at the end of three years? How much of the account balance is interest on interest?**Problem 1.2**If you invest $100 today in an account that pays 7% each year for four years and 3% each year for five years, how much will you have in the account at the end of the nine years?**Discounting**Discounting is translating a future value into a present value. The discount factor is the inverse of the compound factor: To translate a future value into a present value, PV=**Example**Suppose you have a goal of saving $100,000 three years from today. If your funds earn 4% per year, what lump-sum would you have to deposit today to meet your goal?**Example, continued**Known values: FV = $100,000 n = 3 i = 4% Unknown: PV**Example, continued**PV = = PV = $100,000 × 0.8889964 PV = $88,899.64 Check: FV3 = $88,899.64 × (1 + 0.04)3 = $100,000**Short-cut: spreadsheetMicrosoft Excel or Google Docs**=PV(RATE,NPER,PMT,PV,TYPE) TYPE default: end of period =PV(.06,3,0,-1000) or**Try it: Present value**What is the today’s value of $10,000 promised ten years from now if the discount rate is 3.5%?**Try it: Answer**Given: FV = $10,000 N = 10 I = 3.5% Solve for PV PV = = $7,089.19**Frequency of compounding**If interest is compounded more than once per year, we need to make an adjustment in our calculation. The stated rate or nominal rate of interest is the annual percentage rate (APR). The rate per period depends on the frequency of compounding.**Discrete compounding:Adjustments**• Adjust the number of periods and the rate per period. • Suppose the nominal rate is 10% and compounding is quarterly: • The rate per period is 10% 4 = 2.5% • The number of periods is number of years × 4**Continuous compounding:Adjustments**• The compound factor is eAPR x n. • The discount factor is . • Suppose the nominal rate is 10%. • For five years, the continuous compounding factor is e0.10 x 5 = 1.6487 • The continuous compounding discount factor for five years is 1 ÷ e0.10 x 5 = 0.60653**Try it: Frequency of compounding**If you invest $1,000 in an investment that pays a nominal 5% per year, with interest compounded semi-annually, how much will you have at the end of 5 years?**Try it: Answer**Given: PV = $1,000 n = 5 × 2 = 10 i = 0.05 2 = 0.25 Solve for FV FV = $1,000 × (1 + 0.025)10= $1,280.08**Problem 2.1**Suppose you set aside an amount today in an account that pays 5% interest per year, for five years. If your goal is to have $1,000 at the end of five years, what would you need to set aside today?**Problem 2.2**Suppose you set aside an amount today in an account that pays 5% interest per year, compounded quarterly, for five years. If your goal is to have $1,000 at the end of five years, what would you need to set aside today?**Problem 2.3**Suppose you set aside an amount today in an account that pays 5% interest per year, compounded continuously, for five years. If your goal is to have $1,000 at the end of five years, what would you need to set aside today?**What is an annuity?**• An annuity is a periodic cash flow. • Same amount each period • Regular intervals of time • The different types depend on the timing of the first cash flow.**Key to valuing annuities**The key to valuing annuities is to get the timing of the cash flows correct. When in doubt, draw a time line.**Example: PV of an annuity**What is the present value of a series of three cash flows of $4,000 each if the discount rate is 6%, with the first cash flow one year from today?**Example: PV of an annuityIn table form**PV = $4,000.00 × 2.67301 = $10,692.05