You are planning to retire in 14 years and would like to invest all your current savings in bonds issued by your state. The bonds have the face value of $5,000, annual coupon rate of 3.8 percent, and maturity of 26 years. The bonds are currently priced to yield 5.3 percent.

You are planning to retire in 14 years and would like to invest all your current savings in bonds issued by your state. The bonds have the face value of $5,000, annual coupon rate of 3.8 percent, and maturity of 26 years. The bonds are currently priced to yield 5.3 percent.
You expect to be able to reinvest the semiannual coupon payments you will receive over your investment horizon at the annual rate of 4.4 percent. You also expect that the annual yield for comparable bonds at the time you plan to sell your bonds in 14 years (end of your investment horizon) will be 4.9 percent.
 

What is the expected annual realized compound yield of these bonds? (Do not round intermediate calculations and round your final answers to 3 decimal places. (e.g., 32.167))

Total return

%

Suppose the real rate is 4.0 percent and the inflation rate is 5.6 percent.

  

What rate would you expect to see on a Treasury bill? (Round your answer to 2 decimal places. (e.g., 32.16))

Treasury bill rate

%

Explanation:

The Fisher equation, which shows the exact relationship between nominal interest rates, real interest rates, and inflation is:

(1 + R) = (1 + r)(1 + h)

R = (1 + 0.040)(1 + 0.056) – 1 = 0.0982, or 9.82%

You’ve just joined the investment banking firm of Dewey, Cheatum, and Howe. They’ve offered you

You’ve just joined the investment banking firm of Dewey, Cheatum, and Howe. They’ve offered you two different salary arrangements. You can have $83,000 per year for the next two years, or you can have $72,000 per year for the next two years, along with a $28,000 signing bonus today. The bonus is paid immediately, and the salary is paid in equal amounts at the end of each month.

  

If the interest rate is 8 percent compounded monthly, what is the PV for both the options? (Do not round intermediate calculations and round your final answers to 2 decimal places. (e.g., 32.16))

   

	PV
Option 1	$
Option 2	$

rev: 08_21_2012


Explanation:

Here we need to compare two cash flows, so we will find the value today of both sets of cash flows. We need to make sure to use the monthly cash flows since the salary is paid monthly. Doing so, we find:

PVA1 = $83,000/12({1 – 1/[1 + (0.08/12)]24} / (0.08/12)) = $152,931.26

PVA2 = $28,000 + $72,000/12({1 – 1/[1 + (0.08/12)]24} / (0.08/12)) = $160,663.26

  

Calculator Solution:

Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.

 

Enter

2 × 12

8% / 12

$83,000 / 12

N

I/Y

PV

PMT

FV

Solve for

$152,931.26

   

Enter

2 × 12

8% / 12

$72,000 / 12

N

I/Y

PV

PMT

FV

Solve for

$132,663.26

$132,663.26 + 28,000 = $160,663.26

You have 34 years left until retirement and want to retire with $3.6 million. Your salary is paid

You have 34 years left until retirement and want to retire with $3.6 million. Your salary is paid annually, and you will receive $52,000 at the end of the current year. Your salary will increase at 2.2 percent per year, and you can earn a 15.2 percent return on the money you invest. If you save a constant percentage of your salary, what percentage of your salary must you save each year? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))

 

Percent of salary to save

%

Explanation:

We need to find the lump sum payment into the retirement account. The present value of the desired amount at retirement is:

   

PV = FV/(1 + r)t

PV = $3,600,000/(1 + 0.152)34

PV = $29,303.41

    

This is the value today. Since the savings are in the form of a growing annuity, we can use the growing annuity equation and solve for the payment. Doing so, we get:

   

PV = C {[1 – ((1 + g)/(1 + r))t ] / (r – g)}

$29,303.41 = C{[1 – ((1 + 0.022)/(1 + 0.152))34 ] / (0.152 – 0.022)}

C = $3,875.56

  

This is the amount you need to save next year. So, the percentage of your salary is:

  

Percentage of salary = $3,875.56/$52,000

Percentage of salary = 0.0745, or 7.45%

 

Note that this is the percentage of your salary you must save each year. Since your salary is increasing at 2.2 percent, and the savings are increasing at 2.2 percent, the percentage of salary will remain constant.

You have just arranged for a $1,620,000 mortgage to finance the purchase of a large tract of land. The

You have just arranged for a $1,620,000 mortgage to finance the purchase of a large tract of land. The mortgage has an APR of 6.2 percent, and it calls for monthly payments over the next 22 years. However, the loan has an eight-year balloon payment, meaning that the loan must be paid off then.

 

How big will the balloon payment be? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))

 

Balloon payment

$

Explanation:

The monthly payments with a balloon payment loan are calculated assuming a longer amortization schedule, in this case, 22 years. The payments based on a 22-year repayment schedule would be:

  

PVA = $1,620,000 = C({1 – [1 / (1 + 0.062/12)264]} / (0.062/12))

C = $11,258.10

  

Now, at Time = 8, we need to find the PV of the payments which have not been made. The balloon payment will be:

  

PVA = $11,258.10({1 – [1 / (1 + 0.062/12)12(14)]} / (0.062/12))

PVA = $1,262,223.38

   

Calculator Solution:

Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.

Enter

22 × 12

6.2% / 12

$1,620,000

N

I/Y

PV

PMT

FV

Solve for

$11,258.10

   

Enter

14 × 12

6.2% / 12

$11,258.10

N

I/Y

PV

PMT

FV

Solve for

$1,262,223.38

An insurance company is offering a new policy to its customers. Typically, the policy is bought by a

An insurance company is offering a new policy to its customers. Typically, the policy is bought by a parent or grandparent for a child at the child’s birth. The details of the policy are as follows: The purchaser (say, the parent) makes the following six payments to the insurance company:

 

First birthday:	$	790
Second birthday:	$	790
Third birthday:	$	890
Fourth birthday:	$	850
Fifth birthday:	$	990
Sixth birthday:	$	950

 

After the child’s sixth birthday, no more payments are made. When the child reaches age 65, he or she receives $290,000. The relevant interest rate is 10 percent for the first six years and 7 percent for all subsequent years.

Find the future value of the payment at the child's 65th birthday. (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))

 

Future value

$

Explanation:

We need to find the FV of the premiums to compare with the cash payment promised at age 65. We have to find the value of the premiums at year 6 first since the interest rate changes at that time. So:

  

FV1 = $790(1.10)5 = $1,272.30

  

FV2 = $790(1.10)4 = $1,156.64

  

FV3 = $890(1.10)3 = $1,184.59

  

FV4 = $850(1.10)2 = $1,028.50

  

FV5 = $990(1.10)1 = $1,089.00

 

Value at Year 6 = $1,272.30 + 1,156.64 + 1,184.59 + 1,028.50 + 1,089.00 + 950

Value at Year 6 = $6,681.03

  

Finding the FV of this lump sum at the child’s 65th birthday:

FV = $6,681.03(1.07)59 = $361,814.88

The policy is not worth buying; the future value of the deposits is $361,814.88, but the policy contract will pay off $290,000. The premiums are worth $71,814.88 more than the policy payoff.

   

Note, we could also compare the PV of the two cash flows. The PV of the premiums is:

PV = $790/1.10 + $790/1.102 + $890/1.103 + $850/1.104 + $990/1.105 + $950/1.106

PV = $3,771.27

  

And the value today of the $290,000 at age 65 is:

PV = $290,000/1.0759 = $5,354.95

PV = $5,354.95/1.106 = $3,022.73

   

The premiums still have the higher cash flow. At time zero, the difference is $748.54. Whenever you are comparing two or more cash flow streams, the cash flow with the highest value at one time will have the highest value at any other time.

    

Here is a question for you: Suppose you invest $748.54, the difference in the cash flows at time zero, for six years at a 10 percent interest rate, and then for 59 years at a 7 percent interest rate. How much will it be worth? Without doing calculations, you know it will be worth $71,814.88, the difference in the cash flows at time 65!

   

Calculator Solution:

Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.

Value at Year 6:

Enter

5

10%

$790

N

I/Y

PV

PMT

FV

Solve for

$1,272.30

   

Enter

4

10%

$790

N

I/Y

PV

PMT

FV

Solve for

$1,156.64

   

Enter

3

10%

$890

N

I/Y

PV

PMT

FV

Solve for

$1,184.59

   

Enter

2

10%

$850

N

I/Y

PV

PMT

FV

Solve for

$1,028.50

  

Enter

1

10%

$990

N

I/Y

PV

PMT

FV

Solve for

$1,089.00

   

So, at Year 5, the value is: $1,272.30 + 1,156.64 + 1,184.59 + 1,028.50 + 1,089.00 +  950 = $6,681.03

At Year 65, the value is:

 

Enter

59

7%

$6,681.03

N

I/Y

PV

PMT

FV

Solve for

$361,814.88

  

The policy is not worth buying; the future value of the deposits is $361,814.88 but the policy contract will pay off $290,000.00.

Prepare an amortization schedule for a five-year loan of $60,000. The interest rate is 9 percent per

Prepare an amortization schedule for a five-year loan of $60,000. The interest rate is 9 percent per year, and the loan calls for equal annual payments. (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))

 

Year	Beginning Balance	Total Payment	Interest Payment	Principal Payment	Ending Balance
1	$	$	$	$	$
2
3
4
5

 

How much interest is paid in the third year? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))

 

Interest paid

$

The interest rate is 9 percent per year, and the loan calls for equal annual payments. How much total interest is paid over the life of the loan? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))

 

Total interest paid

$

Explanation:

The payment for a loan repaid with equal payments is the annuity payment with the loan value as the PV of the annuity. So, the loan payment will be:

   

PVA = $60,000 = C {[1 – 1 / (1 + 0.09)5] / 0.09}

C = $15,425.55

  

The interest payment is the beginning balance times the interest rate for the period, and the principal payment is the total payment minus the interest payment. The ending balance is the beginning balance minus the principal payment. The ending balance for a period is the beginning balance for the next period.

 

In the third year, $3,514.19 of interest is paid.

Total interest over life of the loan = $5,400 + 4,497.70 + 3,514.19 + 2,442.17 + 1,273.67

Total interest over life of the loan = $17,127.74

A 15-year annuity pays $1,900 per month, and payments are made at the end of each month. If the

A 15-year annuity pays $1,900 per month, and payments are made at the end of each month. If the interest rate is 10 percent compounded monthly for the first seven years, and 6 percent compounded monthly thereafter, what is the present value of the annuity? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))

 

Present value

$

Explanation:

This question is asking for the present value of an annuity, but the interest rate changes during the life of the annuity. We need to find the present value of the cash flows for the last eight years first. The PV of these cash flows is:

 

PVA2 = $1,900[{1 – 1 / [1 + (0.06/12)]96} / (0.06/12)] = $144,580.91

Note that this is the PV of this annuity exactly seven years from today. Now we can discount this lump sum to today. The value of this cash flow today is:

PV = $144,580.91 / [1 + (0.10/12)]84 = $72,005.31

   

Now we need to find the PV of the annuity for the first seven years. The value of these cash flows today is:

PVA1 = $1,900 [{1 – 1 / [1 + (0.10/12)]84} / (0.10/12)] = $114,449.67

The value of the cash flows today is the sum of these two cash flows, so:

PV = $72,005.31 + 114,449.67 = $186,454.98

   

Calculator Solution:

Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.

   

Enter

84

10% / 12

$1,900

N

I/Y

PV

PMT

FV

Solve for

$114,449.67

  

Enter

96

6% / 12

$1,900

N

I/Y

PV

PMT

FV

Solve for

$144,580.91

   

Enter

84

10% / 12

$144,580.91

N

I/Y

PV

PMT

FV

Solve for

$72,005.31

  

$114,449.67 + $72,005.31 = $186,454.98

What is the value today of $5,000 per year, at a discount rate of 9 percent, if the first payment is

What is the value today of $5,000 per year, at a discount rate of 9 percent, if the first payment is received 5 years from today and the last payment is received 15 years from today? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))

 

Value today

$

Explanation:

We want to find the value of the cash flows today, so we will find the PV of the annuity, and then bring the lump sum PV back to today. The annuity has 11 payments, so the PV of the annuity is:

  

PVA = $5,000{[1 – (1/1.0911)] / 0.09} = $34,025.95

  

Since this is an ordinary annuity equation, this is the PV one period before the first payment, so this is the PV at t = 4. To find the value today, we find the PV of this lump sum. The value today is:

  

PV = $34,025.95 / 1.094 = $24,104.84

   

Calculator Solution:

Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.

   

Enter

11

9%

$5,000

N

I/Y

PV

PMT

FV

Solve for

$34,025.95

    

Enter

4

9%

$34,025.95

N

I/Y

PV

PMT

FV

Solve for

$24,104.84

 

The present value of the following cash flow stream is $8,550 when discounted at 9.3 percent

The present value of the following cash flow stream is $8,550 when discounted at 9.3 percent annually. What is the value of the missing cash flow? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))

  

Year				Cash Flow
	1			$	2,150
	2
	3				2,750
	4				3,350

Explanation:

We are given the total PV of all four cash flows. If we find the PV of the three cash flows we know, and subtract them from the total PV, the amount left over must be the PV of the missing cash flow. So, the PV of the cash flows we know are:

 

PV of Year 1 CF: $2,150 / 1.093  = $1,967.06

PV of Year 3 CF: $2,750 / 1.0933 = $2,106.07

PV of Year 4 CF: $3,350 / 1.0934 = $2,347.28

 

So, the PV of the missing CF is:

$8,550 – 1,967.06 – 2,106.07 – 2,347.28 = $2,129.59

  

The question asks for the value of the cash flow in Year 2, so we must find the future value of this amount. The value of the missing CF is:

$2,129.59(1.093)2 = $2,544.12

  

Calculator Solution:

Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.

  

	CF0	$0
	C01	$2,150
	F01	1
	C02	$0
	F02	1
	C03	$2,750
	F03	1
	C03	$3,350
	F04	1
	I = 9.3%
	NPV CPT
	$6,420.41

PV of missing CF = $8,550 – 6,420.41 = $2,129.59

Value of missing CF:

 

Enter

2

9.3%

$2,129.59

N

I/Y

PV

PMT

FV

Solve for

$2,544.12