Analyzing Car Buying Options: Rebate vs. 0% Financing at 1% Interest

You can buy a car that is advertised for $24,000 on the following terms: 

• (a) pay $24,000 and receive a $2,000 rebate from the manufacturer; 

• (b) pay $500 a month for four years for total payments of $24,000, implying zero percent financing. Which is the better deal if the interest rate is 1% per month?

Solution:

To determine which option is better, we compare the present value (PV) of the total cost under each scenario, given a monthly interest rate of 1% (0.01 per month).

Option A: Pay $24,000 upfront and receive a $2,000 rebate

• Net cost = $24,000 - $2,000 = $22,000
• Since this is a lump-sum payment today, its present value is simply $22,000.

Option B: Pay $500 per month for 4 years

• This is a 48-month annuity where each payment is $500.

• Using the present value formula for an annuity:

  $$PV = P \times \left( 1 - \frac{1}{(1 + r)^n} \right) \div r$$
  
  

  where:

  • $P=\mathrm{<div\; style="text-align:\; justify;"><span\; style="font-family:\; arial;\; font-size:\; large;">500(monthly\; payment)</span></div><span\; style="font-family:\; arial;\; font-size:\; large;\; text-align:\; justify;"><br></span>}$
  • $r=\mathrm{<div\; style="text-align:\; justify;"><span\; style="font-family:\; arial;\; font-size:\; large;">0.01(</span><span\; style="font-family:\; arial;\; font-size:\; large;">monthly\; interest\; rate)</span></div><span\; style="font-family:\; arial;\; text-align:\; justify;"><br></span>}$
  • $n=\mathrm{48(number\; of\; months) <span\; style="font-family:\; arial;\; font-size:\; large;"><br></span>}$

  Plugging in the values:

  $$PV = 500 \times \left( 1 - \frac{1}{(1.01)^{48}} \right) \div 0.01$$
  
  
  $$PV = 500 \times \left( 1 - \frac{1}{1.6084} \right) \div 0.01$$
  
  
  $$PV = 500 \times \left( 1 - 0.6218 \right) \div 0.01$$
  
  
  $$PV = 500 \times 0.3782 \div 0.01$$
  
  
  $$PV = 500 \times 37.82 = 18,910$$
  
  

  So, the present value of payments under the financing option is $18,910.

Conclusion

• Option A (lump sum payment) has a present value of $22,000.
• Option B (zero-percent financing) has a present value of $18,910.
• Since the present value of payments is lower in Option B, it is the better deal financially.

Thus, taking the zero-percent financing and making $500 monthly payments is the better choice.

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Lifetime Income Analysis: Determining The Value Of Professor's Annuity Corp.

Professor’s Annuity Corp. offers a lifetime annuity to retiring professors. For a payment of $80,000 at age 65, the firm will pay the retiring professor $600 a month until death. a. If the professor’s remaining life expectancy is 20 years, what is the monthly interest rate on this annuity? b. What is the effective annual interest rate? c. If the monthly interest rate is .5%, what monthly annuity payment can the firm offer to the retiring professor?

Solution:

We will solve each part step by step.

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Part (a): Finding the Monthly Interest Rate

We are given:

• Present Value (PV) = $80,000 (initial payment)
• Monthly Payment (C) = $600
• Number of Months (t) = $20 \times 12 = 240$ months
• Interest Rate per Month (r) = ??? (to be found)

The annuity formula for present value is:

$$PV = C \times \left(1 - \frac{1}{(1 + r)^t} \right) \div r$$
$$80,000 = 600 \times \left(1 - \frac{1}{(1 + r)^{240}} \right) \div r$$

This equation needs to be solved for $r$. We will solve it numerically.

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Part (b): Effective Annual Interest Rate (EAR)

Once we have the monthly interest rate $r$, the EAR is given by:

$$EAR = (1 + r)^{12} - 1$$

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Part (c): Monthly Payment with $r = 0.5\%$

If the monthly interest rate is given as $0.5\% = 0.005$, we use the annuity formula again:

$$C = PV \times \frac{r}{1 - \frac{1}{(1 + r)^t}}$$ $$C = 80,000 \times \frac{0.005}{1 - \frac{1}{(1.005)^{240}}}$$

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Solving Numerically

Now, let’s compute these values.

Results

(a) The monthly interest rate on the annuity is 0.5479% (or 0.00548 in decimal form).
(b) The effective annual interest rate (EAR) is 6.78%.
(c) If the monthly interest rate is 0.5%, the monthly annuity payment would be $573.14 instead of $600.

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Analyzing A Bank's "Perpetual Annuity" Offer: A 6% Interest Rate Calculation

A local bank advertises the following deal: “Pay us $100 a year for 10 years and then we will pay you (or your beneficiaries) $100 a year forever.” Is this a good deal if the interest rate is 6%?

Solution:

To determine if this is a good deal, we need to compare the present value (PV) of what you pay and the present value of what you receive using a discount rate of 6%.

Step 1: Calculate the PV of Payments (Outflows)

You will pay $100 per year for 10 years. The present value of this annuity (PV of payments) is given by the annuity formula:

$$PV_{\text{payments}} = C \times \left( 1 - \frac{1}{(1 + r)^t} \right) \div r$$

where:

• $C = 100$ (annual payment),
• $r = 0.06$ (6% interest rate),
• $t = 10$ (number of years of payment).

$$PV_{\text{payments}} = 100 \times \left( 1 - \frac{1}{(1.06)^{10}} \right) \div 0.06$$
$$PV_{\text{payments}} = 100 \times \left( 1 - \frac{1}{1.7908} \right) \div 0.06$$
$$PV_{\text{payments}} = 100 \times \left( 1 - 0.5584 \right) \div 0.06$$
$$PV_{\text{payments}} = 100 \times 0.4416 \div 0.06 = 100 \times 7.36 = 736$$

So the present value of your payments is $736.

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Step 2: Calculate the PV of Receipts (Inflows)

After 10 years, the bank pays $100 per year forever. The present value of a perpetuity starting in year 11 is:

$$PV_{\text{perpetuity at year 10}} = \frac{C}{r} = \frac{100}{0.06} = 1666.67$$

Since this amount is received starting in year 11, we discount it back 10 years to today:

$$PV_{\text{perpetuity today}} = \frac{1666.67}{(1.06)^{10}}$$ $$PV_{\text{perpetuity today}} = \frac{1666.67}{1.7908} \approx 931$$

So the present value of what you receive today is $931.

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Step 3: Compare Inflows and Outflows

• PV of Payments (Outflows): $736
• PV of Receipts (Inflows): $931
• Net Present Value (NPV):

$$NPV = PV_{\text{receipts}} - PV_{\text{payments}}$$ $$NPV = 931 - 736 = 195$$

Since the NPV is positive ($195$), this is a good deal!

Final Answer:

Yes, this is a good deal because the present value of what you receive ($931) is greater than what you pay ($736), giving a positive net benefit of $195.

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Evaluating Good Investments

A factory costs $400,000. You forecast that it will produce cash inflows of $120,000 in year 1, $180,000 in year 2, and $300,000 in year 3. The discount rate is 12%. 

• a. What is the value of the factory? 
• b. Is the factory a good investment?

Solution:

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How To Evaluate Investments In Properties

You can buy property today for $3 million and sell it in five years for $4 million. 

(You earn no rental income on the property.) 

• a. If the interest rate is 8%, what is the present value of the sales price? 
• b. Is the property investment attractive to you? 
• c. Would your answer to part (b) change if you also could earn $200,000 per-year rent on the property? The rent is paid at the end of each year.

Solution:

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Finding the Value of an Investment Made in 1880

In 1880 five aboriginal trackers were each promised the equivalent of 100 Australian dollars for helping to capture the notorious outlaw Ned Kelley. In 1993 the granddaughters of two of the trackers claimed that this reward had not been paid. The Victorian prime minister stated that if this was true, the government would be happy to pay the $100. However, the granddaughters also claimed that they were entitled to compound interest. 

• a. How much was each granddaughter entitled to if the interest rate was 4%? 
• b. How much was each entitled to if the interest rate was 8%?

Solution:

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Calculating The Value of Investment Made for a Child

Your wealthy uncle established a $1,000 bank account for you when you were born. For the first eight years of your life, the interest rate earned on the account was 6%. Since then, rates have been only 4%. Now you are 21 years old and ready to cash in. How much is in your account?

Solution:

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How To Calculate Interest On Savings

You deposit $1,000 in your bank account. 

• a. If the bank pays 4% simple interest, how much will you accumulate in your account after 10 years?
• b. How much will you accumulate if the bank pays compound interest?

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How To Make Stock Return Analysis

A stock is selling today for $40 per share. At the end of the year, it pays a dividend of $2 per share and sells for $44. 

1. What is the total rate of return on the stock? 
2. What are the dividend yield and percentage capital gain?
3. Now suppose the year-end stock price after the dividend is paid is $36. What are the dividend yield and percentage capital gain in this case?

Solution Below:

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Investment Growth Analysis

Suppose that the value of an investment in the stock market has increased at an average compound rate of about 5% since 1900. It is now 2020. a. If your great-grandfather invested $1,000 in 1900, how much would that investment be worth today? b. If an investment in 1900 has grown to $1 million, how much was invested in 1900?

Solution:

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