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**Chapter 5 The Time Value of Money**

Laurence Booth, Sean Cleary and Pamela Peterson Drake

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**5.2 Annuities and perpetuities 5.3 Nominal and effective rates**

Outline of the chapter 5.1 Time is money Simple versus compound interest Future value of an amount Present value of an amount 5.2 Annuities and perpetuities Ordinary annuities Annuity due Deferred annuities Perpetuities 5.3 Nominal and effective rates APR EAR Solving for the rate 5.4 Applications Savings plans Loans and mortgages Saving for retirement

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5.1 Time value of money

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Simple interest Simple interest is interest that is paid only on the principal amount. Interest = rate × principal amount of loan Note: Some credit unions offer simple interest loans Booth Cleary Drake

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**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 Key: in the second year, only the principal earns interest Booth Cleary Drake

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Compound interest Compound interest is interest paid on both the principal and any accumulated interest. Interest = rate × principal + accumulated interest

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Compounding Compounding is translating a present value into a future value, using compound interest. Future value = Present value × Future value interest factor Future value interest factor is also referred to as the compound factor.

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**Terminology and notation**

Meaning Future value FV Value at some specified future point in time Present value PV Value today Interest i Compensation for the use of funds Number of periods n Number of periods between the present value and the future value Compound factor (1 + i)n Translates a present value into a future value We subscript FV and PV at times when we are dealing with complex situations and need to distinguish the timing of these values. Booth Cleary Drake

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**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?

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**Simple interest Year Beginning Add interest Ending 1 $5,000.00**

+ (5% × $5,000) = $5,250.00 2 $5,500.00 3 $5,750.00 4 $6,000.00

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**Compound interest Year Beginning Compounding Ending 1 $5,000.00**

× = $5,250.00 2 $5,512.50 3 $5,788.13 4 $6,077.53

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Interest on interest How much interest on interest? Interest on interest = FVcompound – FVsimple Interest on interest = $6, – 6, = $77.53

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**Comparison Year End of year balance Simple interest Compound interest**

$5,000.00 1 $5,250.00 2 $5,500.00 $5,512.50 3 $5,750.00 $5,788.13 4 $6,000.00 $6,077.53 Bottom line: The value grows faster with compound interest compared to simple interest. Booth Cleary Drake

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**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?

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**Try it: Answer Bank A: $12,750.00 Bank B: $13,007.78**

Booth Cleary Drake

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A note about interest Because compound interest is so common, assume that interest is compounded unless otherwise indicated.

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Short-cuts Example: Consider $1,000 deposited for three years at 6% per year.

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The long way FV1 = $1, × (1.06) = $1,060.00 FV2 = $1, × (1.06) = $1,123.60 FV3 = $1, × (1.06) = $1,191.02 or FV3 = $1,000 × (1.06)3 = $1,191.02 FV3 = $1,000 × = $1,191.02 Future value factor

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**Short-cut: Calculator**

Known values: PV = 1,000 n = 3 i = 6% Solve for: FV

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**Input three known values, solve for the one unknown**

Known: PV, i , n Unknown: FV HP10B BAIIPlus HP12C TI83/84 1000 +/- PV 3 N 6 I/YR FV 1000 CHS PV 3 n 6 i [APPS] [Finance] [TVM Solver] N =3 I%=6 PV = -1000 FV [Alpha] [Solve}

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**Short-cut: spreadsheet Microsoft Excel or Google Docs**

=FV(RATE,NPER,PMT,PV,TYPE) TYPE default is 0, end of period =FV(.06,3,0,-1000) or A 1 6% 2 3 -1000 4 =FV(A1,A2,0,A3) Type: 0 = end of period 1 = beginning of period Basically, this is not relevant until we get to annuities. Booth Cleary Drake

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Problems Set 1

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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? Booth Cleary Drake

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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?

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Discounting

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Discounting Discounting is translating a future value into a present value. The discount factor is the inverse of the compound factor: 𝑖 𝑛 To translate a future value into a present value, PV= FV 1+𝑖 𝑛

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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?

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**Example, continued Known values: Unknown: PV FV = $100,000 n = 3**

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Example, continued PV = $100, =$100,000 × PV = $100,000 × PV = $88, Check: FV3 = $88, × ( )3 = $100,000

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**Short-cut: Calculator**

HP10B BAIIPlus HP12C TI83/84 /- PV 3 N 4 I/YR PV CHS PV 3n 4i [APPS] [Finance] [TVM Solver] N =3 I%=4 FV = PV [Alpha] [Solve]

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**Short-cut: spreadsheet Microsoft Excel or Google Docs**

=PV(RATE,NPER,PMT,PV,TYPE) TYPE default: end of period =PV(.06,3,0,-1000) or A 1 6% 2 3 100000 4 =PV(A1,A2,0,A3)

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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%?

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**Try it: Answer Given: PV = $10,000 1+0.035 10 = $7,089.19 FV = $10,000**

Solve for PV PV = $10, = $7,089.19 Booth Cleary Drake

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**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.

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**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

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**Continuous compounding: Adjustments**

The compound factor is eAPR x n. The discount factor is 1 eAPR x n . Suppose the nominal rate is 10%. For five years, the continuous compounding factor is e0.10 x 5 = The continuous compounding discount factor for five years is 1 ÷ e0.10 x 5 =

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**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?

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**Try it: Answer Given: FV = $1,000 × (1 + 0.025)10 = $1,280.08**

PV = $1,000 n = 5 × 2 = 10 i = 0.05 2 = 0.25 Solve for FV FV = $1,000 × ( )10 = $1,280.08

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Problem Set 2

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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?

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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?

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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?

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**5.2 Annuities and Perpetuities**

1 2 3 4 5 | CF PV? FV? 5.2 Annuities and Perpetuities

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**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.

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**Type of annuities Type First cash flow Examples Ordinary**

One period from today Mortgage Annuity due Immediately Lottery payments Rent Deferred annuity Beyond one period from today Retirement savings

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**Time lines: 4-payment annuity**

1 2 3 4 5 | Ordinary PV CF FV Annuity due Deferred annuity

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**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.

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**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? 1 2 3 4 | $4,000

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**Example: PV of an annuity The long way**

1 2 3 4 | $4,000 $3,773.58 3,559.99 3,358.48 $10,692.05

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**Example: PV of an annuity In table form**

Year Cash flow Discount factor Present value 1 $4,000.00 $3,773.58 2 3,559.99 3 3,358.48 $10,692.05 PV = $4, × = $10,692.05

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**Example: PV of an annuity Formula short-cuts**

PV = $4,000 × 1 − PV = $4,000 × PV = $10,692.05

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**Example: PV of an annuity Calculator short cuts**

Given: PMT = $4,000 i = 6% N = 3 Solve for PV

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**Example: PV of an annuity Spreadsheet short-cuts**

=PV(RATE,NPER,PMT,FV,TYPE) =PV(.06,3,4000,0) Note: Type is important for annuities If Type is left out, it is assumed a 0 0 is for an ordinary annuity 1 is for an annuity due Note: the default type is ordinary annuity, so this last term can be left off. Booth Cleary Drake

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**Example: FV of an annuity**

What is the future 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? 1 2 3 4 | $4,000

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**Example: FV of an annuity The long way**

1 2 3 4 | $4,000.00 4,240.00 4,494.40 $12,734.40

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**Example: FV of an annuity In table form**

Year Cash flow Compound factor Future value 1 $4,000.00 1.1236 $4,494.40 2 1.0600 4,240.00 3 1.0000 4,000.00 3.1836 $12,734.40 PV = $4, × = $12,734.40

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**Example: FV of an annuity Calculator short cuts**

Given: PMT = $4,000 i = 6% N = 3 Solve for FV

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**Example: FV of an annuity Spreadsheet short-cuts**

=FV(RATE,NPER,PMT,PV,type) =FV(.06,3,4000,0)

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Annuity due Consider a series of three cash flows of $4,000 each if the discount rate is 6%, with the first cash flow today. What is the present value of this annuity? What is the future value of this annuity?

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The time line 1 2 3 | $4,000.00 PV? FV? This is an annuity due

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**Present value of cash flow**

Valuing an annuity due Present value Future value Year End of year cash flow Compound factor Present value of cash flow $4,000.00 1 3,773.58 2 3,559.99 $11,333.57 Year End of year cash flow Factor Future value $4,000.00 $4,764.06 1 4,494.40 2 4,240.00 $13,498.46

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**Valuing an annuity due: Using calculators**

Present value Future value PMT = 4000 N = 3 I = 6% BEG mode Solve for PV PMT = 4000 N = 3 I = 6% BEG mode Solve for FV

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**Valuing an annuity due: Using spreadsheets**

Present value =PV(RATE,NPER,PMT,FV,TYPE) =PV(0.06,3,4000,0,1) Future value

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Any other way? There is one period difference between an ordinary annuity and an annuity due. Therefore: PVannuity due = PVordinary annuity × (1 + i) and FVannuity due = FVordinary annuity × (1 + i)

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**Valuing a deferred annuity**

A deferred annuity is an annuity that begins beyond one year from today. That means that it could begin 2, 3, 4, … years from today, so each problem is unique.

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**Valuing a deferred annuity**

1 2 3 4 5 | CF 4-payment ordinary annuity, then discount value one period PV0 ←PV1 4-payment annuity due, then discount value two periods ←PV2

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**Example: Deferred annuity**

What is the value today of a series of five cash flows of $6,000 each, with the first cash flow received four years from today, if the discount rate is 8%? 1 2 3 4 5 6 7 8 9 10 | PV? CF

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Example, cont. Using an ordinary annuity: PV3 = $23, PV0 = $19, Using an annuity due: PV4 = $25,872.76 Discount 3 periods at 8% Discount 4 periods at 8%

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**Example: Deferred annuity Calculator solutions**

HP10B BAIIPlus TI83/84 0 CF 6000 CF 8 i NPV 0 CF ↑ 1 0 CF ↑ 1 F1 0 CF ↑ 1 F2 0 CF ↑ 1 F3 6000 CF ↑ 1 F4 6000 CF ↑ 1 F5 6000 CF ↑ 1 F6 6000 CF ↑ 1 F7 6000 CF ↑ 1 F8 [2nd] { } STO [2nd] L1 [APPS] [Finance] [ENTER] 7 NPV(.08,0,L1) [ENTER]

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**Example: Deferred annuity Spreadsheet solutions**

B Year Cash flow 1 $0 2 3 4 $6000 5 6 7 8 =PV(0.08,3,0,PV(0.08,5,6000,0)) =PV(0.08,4,0,PV(0.08,5,6000,0,1)) =NPV(0.08,A1:A9)

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Perpetuities A perpetuity is an even cash flows that occurs at regular intervals of time, forever. The valuation of a perpetuity is simple: PV= 𝐶𝐹 1+𝑖 1 + 𝐶𝐹 1+𝑖 2 + 𝐶𝐹 1+𝑖 3 +… 𝐶𝐹 1+𝑖 ∞ PV= 𝐶𝐹 𝑖

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Problem Set 3

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Problem 3.1 Which do you prefer if the appropriate discount rate is 6% per year: An annuity of $4,000 for four annual payments starting today. An annuity of $4,100 for four annual payments, starting one year from today. An annuity of $4,200 for four annual payments, starting two years from today.

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**5.3 Nominal and effective rates**

% 𝐴𝑃𝑅=𝑖 ×𝑚 𝑬𝑨𝑹= 𝟏+ 𝑨𝑷𝑹 𝒎 𝒎 𝑬𝑨𝑹= 𝒆 𝑨𝑷𝑹 −𝟏 i 5.3 Nominal and effective rates

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APR & EAR The annual percentage rate (APR) is the nominal or stated annual rate. The APR ignores compounding within a year. The APR understates the true, effective rate. The effective annual rate (EAR) incorporates the effect of compounding within a year. Note to instructors: the post-subprime mortgage rules confuse the APR with the EAR, so the term “APR” is used as both an effective rate and a nominal rate, depending on the part of the paperwork. Booth Cleary Drake

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APR EAR EAR = 1+ 𝐴𝑃𝑅 𝑚 𝑚 −1 Suppose interest is stated as 10% per years, compounded quarterly. EAR = −1= −1 EAR = %

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**EAR with continuous compounding**

EAR = 𝑒 𝐴𝑃𝑅 −1 Suppose interest is stated as 10% per years, compounded continuously. EAR = e 0.1 −1= −1 EAR = %

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**Frequency of compounding**

If interest is compounded more frequently than annually, then this is considered in compounding and discounting. There are two approaches Adjust the i and n; or Calculate the EAR and use this

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**Example: EAR & compounding**

Suppose you invest $2,000 in an investment that pays 5% per year, compounded quarterly. How much will you have at the end of 4 years?

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**Example: EAR & compounding**

Method 1: FV = $2,000 ( )16 = $2,439.78 Method 2: EAR = ( )4 – 1 = % FV = $2,000 ( )4 = $2,439.78

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Try it: APR & EAR Suppose a loan has a stated rate of 9%, with interest compounded monthly. What is the effective annual rate of interest on this loan?

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Try it: Answer EAR = −1 EAR = −1 EAR = %

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Problem Set 4

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Problem 4.1 What is the effective interest rate that corresponds to a 6% APR when interest is compounded monthly?

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Problem 4.2 What is the effective interest rate that corresponds to a 6% APR when interest is compounded continuously?

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5.4 Applications

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Saving for retirement Suppose you estimate that you will need $60,000 per year in retirement. You plan to make your first retirement withdrawal in 40 years, and figure that you will need 30 years of cash flow in retirement. You plan to deposit funds for your retirement starting next year, depositing until the year before retirement. You estimate that you will earn 3% on your funds. How much do you need to deposit each year to satisfy your plans?

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**Deferred annuity time line**

1 2 3 4 5 6 7 8 … 39 40 41 42 43 79 | D W D = Deposit (39 in total) W = Withdrawal (30 in total)

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**Deferred annuity time line**

1 2 3 4 5 6 7 8 … 39 40 41 42 43 79 | W PV ← Ordinary annuity ↓ Ordinary annuity → FV D

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**Two steps Step 1: Present value of ordinary annuity**

N = 30; i = 3%; PMT = $60,000 PV39 = $1,176,026.48 Step 2: Solve for payment in an ordinary annuity N = 39; i = 3%; FV = $1,176,026.48 PMT = $16,280.74

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What does this mean? If there are 39 annual deposits of $16, each and the account earns 3%, there will be enough to allow for 30 withdrawals of $60,000 each, starting 40 years from today.

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**Balance in retirement account**

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+ = € $ Practice problems

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Problem 1 What is the future value of $2,000 invested for five years at 7% per year, with interest compounded annually? Booth Cleary Drake

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Problem 2 What is the value today of €10,000 promised in four years if the discount rate is 4%? Booth Cleary Drake

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Problem 3 What is the present value of a series of five end-of-year cash flows of $1,000 each if the discount rate is 4%? Booth Cleary Drake

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Problem 4 Suppose you plan to save $3,000 each year for ten years. If you earn 5% annual interest on your savings, how much more will you have at the end of ten years if you make your payments at the beginning of the year instead of the end of the year? Booth Cleary Drake

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Problem 5 Sue plans to deposit $5,000 in a savings account each year for thirty years, starting ten years from today. Yan plans to deposit $3,500 in a savings account each year for forty years, starting at the end of this year. If both Sue and Yan earn 3% on their savings, who will have the most saved at the end of forty years? Booth Cleary Drake

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Problem 6 Suppose you have two investment opportunities: Opportunity 1: APR of 12%, compounded monthly Opportunity 2: APR of 11.9%, compounded continuously Which opportunity provides the better return? Booth Cleary Drake

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Problem 7 If you can earn 5% per year, what would you have to deposit in an account today so that you have enough saved to allow withdrawals of $40,000 each year for twenty years, beginning thirty years from today? Booth Cleary Drake

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Problem 8 Suppose you deposit ¥50000 in an account that pays 4% interest, compounded continuously. How much will you have in the account at the end of ten years if you make no withdrawals? Booth Cleary Drake

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Chapter 9: Mathematics of Finance

Chapter 9: Mathematics of Finance

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