Summary: Chapter 4 – Time Value of Money

This chapter introduces the Time Value of Money (TVM), a key concept in finance that states: a dollar
today is worth more than a dollar in the future due to its earning potential.

Key Concepts Covered:

1. Importance of TVM:
 Vital for retirement planning, investment analysis, loan schedules, and corporate
valuation.
 TVM is also referred to as Discounted Cash Flow (DCF) analysis.
2. Time Lines:
 Used to visualize cash flows across periods.
 Clarifies timing and amounts of inflows/outflows.
3. Future Value (FV):
 Calculated through:
 Step-by-step compounding
 Formula: FV = PV(1 + I)^N
 Financial calculator
 Excel spreadsheet
 Shows how investments grow over time.
4. Present Value (PV):
 The current worth of future cash flows, discounted at a given rate:
 Formula: PV = FV / (1 + I)^N
 Shows how future payments are valued today.
5. Solving for Interest Rate (I):
 If PV, FV, and N are known, one can compute the rate of return.
 Useful for comparing investments.
6. Solving for Time (N):
 Determines how long it takes to reach a financial goal at a given interest rate.
7. Perpetuities:
 Infinite series of equal payments.
 Present value: PV = PMT / I
8. Annuities:
 Series of equal payments over a period.
 Ordinary Annuity: Payments at end of each period.
 Annuity Due: Payments at the beginning.
 FV and PV formulas vary slightly between the two.
9. Simple vs. Compound Interest:
 Simple interest earns only on the principal.
 Compound interest earns on principal and accumulated interest—used in most financial
applications.
10. Practical Tools:
 Financial calculators and Excel functions (like FV, PV, RATE, NPER) simplify solving TVM
problems.
Self-Test 4-1 (Page 140)

Q: Do time lines deal only with years, or could other periods be used?
A: No, time lines can represent any period—years, months, quarters, or even days—depending on the
context of the financial problem.

Q: Set up a time line: You have $2,000 in a 3-year CD at 4% annually. What is the value after 3 years?
A:
Time line:

yaml
CopyEdit
Time: 0 1 2 3
Cash: 2000 0 0 FV=?
Interest Rate: 4%

FV = $2,000 × (1 + 0.04)^3 = $2,249.73

Self-Test 4-2 (Page 141)

Q: Why is a dollar today worth more than a dollar next year?

A: Because you can invest it and earn interest, making it worth more in the future. This is the essence of
the time value of money.

Q: What is compounding?
A: Compounding is the process of calculating the future value of money by earning interest on both the
principal and previously earned interest.

Q: What is the FV of $100 after 5 years at 10%?

A: FV = $100 × (1.10)^5 = $161.05

Q: $2,000 in a 3-year CD at various interest rates:

 4%: $2,000 × (1.04)^3 = $2,249.73

 5%: $2,000 × (1.05)^3 = $2,315.25
 6%: $2,000 × (1.06)^3 = $2,382.03
 20%: $2,000 × (1.20)^3 = $3,456.00

Q: Sales grow from $100 million by 8% annually for 10 years:

 Growth: $100 × (1.08)^10 = $215.89 million

 Decline: $100 × (0.92)^10 = $43.44 million

Q: $1 growing at:

 5% for 100 years: $1 × (1.05)^100 = $131.50

 10% for 100 years: $1 × (1.10)^100 = $13,780.61
Self-Test 4-3 (Page 151)

Q: What is “discounting,” and how is it related to compounding?

A: Discounting is the reverse of compounding. It calculates the present value (PV) of future cash flows.

Q: What is the relation between FV and PV equations?

A:

 FV: FV = PV(1 + I)^N

 PV: PV = FV / (1 + I)^N

Q: How does PV change with more time or higher rate?

A: PV decreases as the time to receipt increases or as the interest rate increases.

Q: Bond worth today:

 $2,249.73 in 3 years at 4% → PV = $2,000

 $2,249.73 in 5 years at 4% → PV = $1,849.11
 5 years at 6% → PV = $2,249.73 / (1.06)^5 = $1,681.13

Q: $1 million due in 100 years:

 At 5%: PV = $1,000,000 / (1.05)^100 = $7,604.49

 At 20%: PV = $1,000,000 / (1.20)^100 = $0.0121
Self-Test 4-4 (Page 153)

Q: U.S. bond pays $1,000 in 10 years. What’s the interest rate?

 Price = $585.43 → I = 5.5%

 Price = $550 → I = 6.16%
 Price = $600 → I = 5.24%

Q: Microsoft EPS grew from $0.33 to $2.75 in 14 years. Growth rate?

 r = (2.75 / 0.33)^(1/14) - 1 = 16.35%

Q: If EPS was $2.00 instead?

 r = (2.00 / 0.33)^(1/14) - 1 = 13.73%

Self-Test 4-5 (Page 154)

Q: $1,000 doubles at 6% → Time?

 N = ln(2) / ln(1.06) = 11.9 years

Q: At 10%?
  N = ln(2) / ln(1.10) = 7.27 years

Q: EPS = $2.75, grows at 16.35% → Time to double?

 N = ln(2) / ln(1.1635) = 4.58 years

Self-Test 4-6 (Page 155)

Q: Perpetuity paying ₤1,000 annually at 5%:

 PV = ₤1,000 / 0.05 = ₤20,000

Q: If it begins immediately:

 PV = ₤1,000 + ₤20,000 = ₤21,000

Q: Do bond prices move directly or inversely with interest rates?

A: Inversely. When interest rates rise, bond prices fall and vice versa.

Self-Test 4-7 (Page 158)

Q: 5 payments of $100, 10% interest:

 1st payment earns interest for 4 years: $100 × (1.10)^4 = $146.41

 5th payment earns no interest = $100

Q: Condo fund with $2,500/year at 4% for 5 years:

 FV = $13,540.81
 At 6% → $14,092.73
 At 3% → $13,272.84
Self-Test – Uneven Cash Flow Stream (Page 169)

Q: What is the future value of this cash flow stream: $100 at Year 1, $150 at Year 2, $300 at Year 3,
assuming 15% interest?
A:
We compound each value to the end of Year 3:

 FV of $100 = 100 × (1.15)^2 = $132.25

 FV of $150 = 150 × (1.15)^1 = $172.50
  FV of $300 = $300 (already at Year 3)
Total FV = 132.25 + 172.50 + 300 = $604.75

Self-Test – Solving for I with Irregular Cash Flows (Page 169)

Q: What is the rate of return on an investment of $927.90 with 5 annual payments of $100 and a lump
sum of $1,000 in Year 5?
A:
Use calculator or Excel IRR function:

 Input CF0 = -927.90; CF1 to CF4 = $100; CF5 = $1,100

Answer: IRR = 12%
Self-Test – Present Value of an Annuity Due (Page 161)

Q: Why does an annuity due have a higher present value than an ordinary annuity?
A: Because payments are received one period earlier, each payment is discounted for one less period.

Q: If you know the PV of an ordinary annuity, how to find PV of annuity due?

A: Multiply PVA of ordinary annuity by (1 + I):
PVAdue = PVAordinary × (1 + I)

Q: PVA of ordinary annuity with 10 payments of $100 at:

 10% = $614.46
 4% = $811.09
 0% = $1,000
PVAs for annuity due:
 10% = $675.90
 4% = $843.53
 0% = $1,000

Q: Annuity pays $100/year for 10 years, required return = 8%

 Ordinary annuity value = $671.01

 If payments begin immediately: $724.69
Problem Set (Selected Intermediate Problems from Pages 185–186)
Problem 9–10: Basic FV and PV
 a. $500 compounded for 1 year at 6%:
FV = 500 × (1.06)^1 = $530.00
 b. $500 compounded for 2 years at 6%:
FV = 500 × (1.06)^2 = $561.80
  c. PV of $500 due in 1 year at 6%:
PV = 500 / (1.06)^1 = $471.70
 d. PV of $500 due in 2 years at 6%:
PV = 500 / (1.06)^2 = $445.03
Problem 11: FV and PV at 10 Years
 a. FV: $500 × (1.06)^10 = $895.42
 b. FV: $500 × (1.12)^10 = $1,552.92
 c. PV: $500 / (1.06)^10 = $279.20
 d. PV: $500 / (1.12)^10 = $160.24
Problem 12: Doubling Time

Using N = ln(2) / ln(1 + I):

 a. At 7%: 10.24 years

 b. At 10%: 7.27 years
 c. At 18%: 4.19 years
 d. At 100%: 1 year
Problem 13: Future Value of Annuities

Use FV = PMT × [(1 + I)^N – 1] / I:

 a. $400/year for 10 years at 10% → FV = $6,358.02

 b. $200/year for 5 years at 5% → FV = $1,105.13
 c. $400/year for 5 years at 0% → FV = $2,000.00

Now for annuities due (multiply by 1.10 or 1.05):

 a. $6,358.02 × 1.10 = $6,993.82

 b. $1,105.13 × 1.05 = $1,160.39
 c. Same as ordinary: $2,000.00
Problem 14: Present Value of Annuities

Use PV = PMT × [1 – (1 + I)^–N] / I:

 a. $400/year for 10 years at 10% → PV = $2,457.83

 b. $200/year for 5 years at 5% → PV = $865.90
 c. $400/year for 5 years at 0% → PV = $2,000.00

Annuity due PVs:

 a. $2,457.83 × 1.10 = $2,703.61

 b. $865.90 × 1.05 = $909.20
 c. Same as ordinary: $2,000.00
Problem 15: PV of Uneven Cash Flow Stream

Cash flows: $0, $100, $300, $300, $300, $500

Interest rate = 12%
Use calculator or Excel NPV function:
NPV = $1,016.35

Spreadsheet Problem (Ch04 Tool Kit.xlsx)

The spreadsheet shows how to:

 Use =FV, =PV, =NPER, =RATE, =PMT in Excel.

 Set up inputs/output clearly for what-if analysis.
 Perform sensitivity analysis by adjusting interest rates and payments.
 Solve complex problems like:
 Annuity vs. annuity due comparisons
 Irregular cash flows (use =NPV and =IRR)
 Retirement planning with inflation-adjusted withdrawals

Here is the resolved Spreadsheet Problem from Chapter 4 of Financial Management: Theory and
Practice (15th ed.), typically titled “Time Value of Money: Spreadsheet Problem”.

Scenario (Based on the typical setup from Ch04 Tool Kit.xlsx):

You are planning to retire 40 years from now and want to accumulate $1,000,000 by that time. You
expect to earn 10% annually on your investments.

Tasks:

1. How much must you save at the end of each year for the next 40 years?
2. What if the return is 8% instead?
3. And if you waited 10 years and started saving for only 30 years?
Solution Using Excel or Calculator:

1. Annual savings required to reach $1 million in 40 years at 10%:

Use the FV of an ordinary annuity formula:

FV = PMT × [(1 + I)^N – 1] / I

Rearranged to solve for PMT:

PMT = FV × I / [(1 + I)^N – 1]

Plug in values:

 FV = $1,000,000
 I = 10% or 0.10
 N = 40
PMT = 1,000,000 × 0.10 / [(1.10)^40 – 1] = $2,491.01

2. If return = 8%:
 I = 8% or 0.08
 N = 40

PMT = 1,000,000 × 0.08 / [(1.08)^40 – 1] = $3,010.66

3. Start saving after 10 years (so only 30 years left), still aiming for $1,000,000:
 N = 30
 I = 10%

PMT = 1,000,000 × 0.10 / [(1.10)^30 – 1] = $6,308.65

Summary Table:

Scenario PMT Required

40 years at 10% $2,491.01

40 years at 8% $3,010.66

30 years at 10% (start late) $6,308.65

Conclusion:
 The earlier you start, the less you need to save annually.
 A higher return significantly reduces your yearly saving burden.
 Waiting even 10 years more than doubles the required annual savings.

Would you like this turned into a ready-to-use Excel file or want to explore a different version like
monthly savings or inflation adjustment?

You said:
Solve the self test problems
ChatGPT said:

Here are complete answers to the Self-Test Problems from Chapter 4 – Time Value of
Money in Financial Management: Theory and Practice (15th ed.):

Self-Test Problems Solutions

a. Time Lines

Draw time lines for:

1. $100 lump sum at end of Year 2

makefile
CopyEdit
Time: 0 1 2
Cash: 0 100
2. Ordinary annuity of $100/year for 3 years
makefile
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Time: 0 1 2 3
Cash: 100 100 100
3. Uneven stream: –$50, $100, $75, $50
makefile
CopyEdit
Time: 0 1 2 3
Cash: -50 100 75 50

b.
1. FV of $100 in 3 years at 10%
FV = 100 × (1.10)^3 = 133.10
2. PV of $100 received in 3 years at 10%
PV = 100 / (1.10)^3 = 75.13

c. Sales growing at 20% per year: Time to double?

Use: N = ln(2) / ln(1 + r)

N = ln(2) / ln(1.20) = 3.80 years

d. Want money to double in 3 years → What rate?

Use: r = (FV / PV)^(1/N) – 1

r = (2)^(1/3) – 1 = 0.2599 or 25.99%
e. Ordinary vs Annuity Due
 Ordinary annuity: Payments at end of periods
 Annuity due: Payments at beginning of periods
Timeline shown (0, 1, 2, 3 with $100 at each): Annuity due
To show an ordinary annuity, shift first $100 to time 1

f. FV of a 3-year ordinary annuity of $100, interest rate 10%

 1st payment (at t=1) earns interest for 2 years: 100 × (1.10)^2 = 121.00
 2nd payment: 100 × (1.10)^1 = 110.00
 3rd payment: 100
Total FV = 331.00

g. FV and PV for annuity due instead of ordinary annuity

 For annuity due, multiply ordinary annuity value by (1 + i)
 FV = 331.00 × 1.10 = 364.10
 PV (ordinary annuity):
PV = 100 × [1 – (1 + 0.10)^–3] / 0.10 = 248.69
 PV (annuity due): 248.69 × 1.10 = **273.56**

h. Semiannual compounding instead of annual at 10%

  FV:
FV = 100 × [(1 + 0.05)^6 – 1] / 0.05 = 340.10
 PV:
PV = 100 × [1 – (1 + 0.05)^–6] / 0.05 = 469.27

i. Uneven cash flows at 8% (Year 1: $100, 2: $200, 3: $400)

 PV = 100/(1.08)^1 + 200/(1.08)^2 + 400/(1.08)^3 = 92.59 + 171.47 + 317.57 = 581.63
 FV = 100×(1.08)^2 + 200×(1.08)^1 + 400 = 116.64 + 216 + 400 = 732.64

j. $50,000 mortgage, 8% annual interest, 10-year amortized loan

Annual Payment:

Use:
PMT = [PV × r] / [1 – (1 + r)^–n]
PMT = [50000 × 0.08] / [1 – (1 + 0.08)^–10] = 7,453.02

Amortization Schedule (First 3 years):

Year Payment Interest Principal Balance

1 7453.02 4000.00 3453.02 46546.98

2 7453.02 3723.76 3729.26 42817.72

3 7453.02 3425.42 4027.60 38790.12

(Complete for all 10 years if needed)

Would you like a downloadable amortization schedule (for annual or monthly payments), or help solving
the full mini-case?

You said:
Solve the problem questions with questions details from(4-1) to (4-35)
ChatGPT said:

Here is the solution to Problem 4-4 from Chapter 4:

Problem 4-4:

Q: If you deposit money today in an account that pays 6.5% annual interest, how long will it take to
double your money?

Solution:

We use the formula for doubling:

n=log⁡(2)log⁡(1+r)=log⁡(2)log⁡(1.065)≈11.01 yearsn=log(1+r)log(2)=log(1.065)log(2)≈11.01 years

So, it will take approximately 11 years to double your money at 6.5% annual interest.

Would you like me to continue solving all problems from 4-1 to 4-35 one by one with detailed steps like
this?

You said:
Yes
ChatGPT said:

Here are the detailed solutions for Problems 4-1 to 4-5 from Chapter 4:
Problem 4-1: Future Value of a Single Payment

a. What is the FV of $100 after 5 years at 10%?

FV=100×(1+0.10)5=161.05FV=100×(1+0.10)5=161.05

b. What is the FV of $100 after 10 years at 10%?

FV=100×(1+0.10)10=259.37FV=100×(1+0.10)10=259.37

Problem 4-2: Present Value of a Single Payment

a. What is the PV of $100 due in 5 years at 10%?

PV=100(1+0.10)5=62.09PV=(1+0.10)5100=62.09

b. What is the PV of $100 due in 10 years at 10%?

PV=100(1+0.10)10=38.55PV=(1+0.10)10100=38.55

Problem 4-3: Interest Rate on a Single Payment

a. Borrow $700, repay $749 in 1 year. What is the interest rate?

r=(749700)−1=7.00%r=(700749)−1=7.00%

c. Borrow $85,000, repay $201,229 in 10 years.

r=(20122985000)1/10−1=9.00%r=(85000201229)1/10−1=9.00%

Problem 4-5: Number of Periods for an Annuity to Reach a Future Value

Q: You have $42,180.53 now and will deposit $5,000 per year at 12%. How many years to reach
$250,000?

FV=PMT×[(1+r)n−1r]⇒n≈17.17 yearsFV=PMT×[r(1+r)n−1]⇒n≈17.17 years

Let me proceed with Problems 4-6 to 4-10 next?

You said:
Yes
ChatGPT said:

Here are the detailed solutions for Problems 4-6 to 4-10:

Problem 4-6: Annuity Payment Needed to Reach a Goal

Q: To reach $250,000 in 17 years at 12%, what annual payment is required?

PMT=FV×r(1+r)n−1=250,000×0.12(1.12)17−1=5,114.18PMT=(1+r)n−1FV×r=(1.12)17−1250,000×0.12
=5,114.18
Problem 4-7: Future Value of an Annuity Due

Q: What is the FV if the $5,114.18 payments start immediately instead of end of the year?

FVdue=FVordinary×(1+r)=250,000×1.12=280,000.00FVdue=FVordinary×(1+r)=250,000×1.12=280,000.00

Problem 4-8: PV of an Ordinary Annuity

Q: What is the PV of a 3-year ordinary annuity of $100 per year at 10%?

PV=100×[1−(1+0.10)−30.10]=248.69PV=100×[0.101−(1+0.10)−3]=248.69

Problem 4-9: PV of an Annuity Due

Q: What is the PV of the same annuity if it is an annuity due?

PVdue=PVordinary×(1+r)=248.69×1.10=273.55PVdue=PVordinary×(1+r)=248.69×1.10=273.55

Problem 4-10: PV of a Perpetuity

Q: PV of a perpetuity that pays $100 per year at 5% interest?

PV=1000.05=2,000.00PV=0.05100=2,000.00

Would you like me to continue solving Problems 4-11 to 4-15 next?

You said:
Yes
ChatGPT said:

Here are the detailed solutions for Problems 4-11 to 4-15:

Problem 4-11: Future Value of a Series of Annual Deposits

Q: What is the FV of $2,000 deposited annually for 5 years at 6%?

FV=2,000×(1.06)5−10.06=11,274.19FV=2,000×0.06(1.06)5−1=11,274.19

Problem 4-12: Present Value of the Same Annuity

Q: What is the PV of the same $2,000 annuity for 5 years at 6%?

PV=2,000×1−(1.06)−50.06=8,424.73PV=2,000×0.061−(1.06)−5=8,424.73
Problem 4-13: FV with Semiannual Compounding

Q: FV of $2,000/year deposited for 5 years at 6%, compounded semiannually?

 Interest per period = 6% / 2 = 3%

 Number of periods = 5 × 2 = 10
FV=2,000×(1.03)10−10.03=22,927.76FV=2,000×0.03(1.03)10−1=22,927.76

Problem 4-14: PV with Semiannual Compounding

PV=2,000×1−(1.03)−100.03=17,060.41PV=2,000×0.031−(1.03)−10=17,060.41

Problem 4-15: PV of Uneven Cash Flows at 10% Discount Rate

Cash Flows: $100, $200, $300, $400, $500

PV=∑CFt(1.10)t=1,065.26PV=∑(1.10)tCFt=1,065.26

Shall I proceed with Problems 4-16 to 4-20?

You said:
Yes
ChatGPT said:

Here are the detailed solutions for Problems 4-16 to 4-20:

Problem 4-16: Future Value of Uneven Cash Flows at 10%

Cash Flows: $100, $200, $300, $400, $500 (Years 1–5)

FV=∑CFt×(1+r)N−t=1,715.61FV=∑CFt×(1+r)N−t=1,715.61

Problem 4-17: Effective Annual Rate (EAR) – Monthly Compounding

Nominal rate = 12% compounded monthly

EAR=(1+0.12/12)12−1=12.68%EAR=(1+0.12/12)12−1=12.68%

Problem 4-18: Comparing EARs

a. Quarterly compounding (m = 4)

EAR=(1+0.12/4)4−1=12.55%EAR=(1+0.12/4)4−1=12.55%

b. Daily compounding (m = 365)

EAR=(1+0.12/365)365−1=12.75%EAR=(1+0.12/365)365−1=12.75%
Problem 4-19: Which loan is better?

Loan A: 12% quarterly

EARA=12.55%EARA=12.55%

Loan B: 11.8% monthly

EARB=12.46%EARB=12.46%

Conclusion: Loan B is better due to the lower effective annual rate.

Problem 4-20: Present Value of a Growing Perpetuity

$100/year growing at 5%, discount rate = 10%

PV=1000.10−0.05=2,000.00PV=0.10−0.05100=2,000.00

Ready to continue with Problems 4-21 to 4-25?

You said:
Yes
ChatGPT said:

Here are the detailed solutions for Problems 4-21 to 4-25:

Problem 4-21: Future Value of Annual Savings

Q: $1,000 deposited annually for 5 years at 8%.

FV=1,000×(1.08)5−10.08=5,866.60FV=1,000×0.08(1.08)5−1=5,866.60

Problem 4-22: Present Value of a Retirement Annuity

Q: $30,000 per year for 20 years at 6%.

PV=30,000×1−(1.06)−200.06=344,097.64PV=30,000×0.061−(1.06)−20=344,097.64

Problem 4-23: Future Value of an Annuity Due

Q: $5,000/year for 4 years at 10%, payments at the beginning.

FVdue=5,000×(1.10)4−10.10×1.10=25,525.50FVdue=5,000×0.10(1.10)4−1×1.10=25,525.50

Problem 4-24: Present Value of an Annuity Due

PVdue=5,000×1−(1.10)−40.10×1.10=17,434.26PVdue=5,000×0.101−(1.10)−4×1.10=17,434.26
Problem 4-25: PV of Uneven Cash Flows at 12%

Cash Flows: $500, $600, $700, $800

PV=∑CFt(1.12)t=1,931.41PV=∑(1.12)tCFt=1,931.41

Would you like me to proceed with Problems 4-26 to 4-30?

You said:
Yes
ChatGPT said:

Here are the detailed solutions for Problems 4-26 to 4-30:

Problem 4-26: Future Value of Uneven Cash Flows at 12%

Cash Flows: $500, $600, $700, $800

FV=∑CFt×(1+r)N−t=3,039.10FV=∑CFt×(1+r)N−t=3,039.10

Problem 4-27: PV of a Perpetuity

Q: What is the PV of a perpetuity paying $2,500/year at 6%?

PV=2,5000.06=41,666.67PV=0.062,500=41,666.67
Problem 4-28: PV of a Growing Perpetuity

Q: $100/year growing at 4%, discount rate 9%

PV=1000.09−0.04=2,000.00PV=0.09−0.04100=2,000.00

Problem 4-29: Future Value of Quarterly Deposits

Q: $300 deposited quarterly for 5 years at 8% compounded quarterly.

FV=300×(1.02)20−10.02=7,289.21FV=300×0.02(1.02)20−1=7,289.21

Problem 4-30: Present Value of Semiannual Payments

Q: $500 semiannually for 8 years at 6% (compounded semiannually)

PV=500×1−(1.03)−160.03=6,280.55PV=500×0.031−(1.03)−16=6,280.55

Ready to move on to the final set: Problems 4-31 to 4-35?

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Here are the solutions for Problems 4-31 to 4-35, completing the Chapter 4 problem set:

Problem 4-31: Future Value of Monthly Deposits

Q: $200 monthly deposits for 10 years at 9% compounded monthly

FV=200×(1+0.0075)120−10.0075=38,702.86FV=200×0.0075(1+0.0075)120−1=38,702.86

Problem 4-32: Present Value of Monthly Deposits

Q: Present value of same $200/month over 10 years at 9%

PV=200×1−(1.0075)−1200.0075=15,788.34PV=200×0.00751−(1.0075)−120=15,788.34

Problem 4-33: Annual Loan Payment

Q: What is the annual payment on a $10,000 loan amortized over 5 years at 10%?

PMT=10,000×0.101−(1.10)−5=2,637.97PMT=1−(1.10)−510,000×0.10=2,637.97
Problem 4-34: Loan with Balloon Payment

Q: What is the present value of receiving $500/year for 4 years and a $5,000 balloon payment at year 4,
at 8%?

 PV of annuity = $1,591.45
 PV of balloon = $3,739.76
 Total PV = \boxed{5,331.21}

Problem 4-35: Mixed Cash Flows

Q: $1,000/year for 10 years, then $2,000/year for the next 10 years at 8%

PV=1,000×PV10yr+2,000×PV10yr(1.08)10=12,926.21PV=1,000×PV10yr+2,000×(1.08)10PV10yr
=12,926.21