1.
a.
You need to borrow $250,000 to buy your first home. Your mortgage banker will lend you the money at
an APR of 4 percent with the expectation that you make fixed monthly payments for the next 30 years.
Draw the cash flows for this loan and explain how you would calculate your payments. Your answer
should use future value and present value principles to explain your answer. Specify all of the arguments
that you would substitute into the appropriate Excel functions.
Excel uses the following relationship to calculate the payment. The loan amount equals the present value
of the 360 future payments. There is no future value lump sum because the loan is fully amortized. We
find the size of the payment so that the following is true:
Amount of Mortgage = PresentValue of Payments
360
PMT
i 1
1  APR 12 

360
PMT
i 1
1  4% 12 
$250, 000  
b.
i
i
The calculations in part (a) give a payment of $1,193.54 a month. Explain and calculate the first line in
an amortization table for this loan.
The beginning balance of the loan is the amount of the mortgage or $250,000. At the monthly
interest rate of 0.33%, then the interest that must be paid is $250,000 multiplied by 0.33%. We then
subtract the interest from the payment to determine how much principal has been paid off. Finally,
we subtract the principle from the beginning balance to get the ending balance.
NPER
RATE
PV
PMT
FV
360
0.333%
$250,000
($1,193.54)
$0
Payment Beginning Balance
1
$250,000.00
2
$249,639.80
3
$249,278.39
4
$248,915.78
5
$248,551.96
c.
Payment Interest
$1,193.54
$1,193.54
$1,193.54
$1,193.54
$1,193.54
Principal
$833.33
$832.13
$830.93
$829.72
$828.51
$360.20
$361.41
$362.61
$363.82
$365.03
Ending Balance
$249,639.80
$249,278.39
$248,915.78
$248,551.96
$248,186.93
Calculate and explain the EAR of this loan.
The APR must be converted into an EAR by taking into consideration the compounding
frequency. This is done using the following formula that is implemented in the EFFECT function:
12
APR 

EAR   1 
 1
12 

12
 4.00 
 1 
  1  4.07%
12 

d.
If your mortgage banker allows you to amortize your loan over 30 years but also requires a balloon
payment at the end of 5 years, diagram the cash flows and explain how you would calculate the size
of the balloon. Your answer should use future value and present value principles to explain your
answer. Specify all of the arguments that you would substitute into the appropriate Excel functions.
The solution using the FV function gives a balloon payment of $226,118.78. This amount is calculated
solving for the balloon using the following relationship:
Amount of Mortgage = PresentValue of Payments  PresentValue of Balloon
60
PMT
i 1
1  APR 12 

60
PMT
i 1
1  4% 12 
$250, 000  
e.
i
i


Balloon
1  APR 12 
60
Balloon
1  4% 12 
60
If your mortgage broker quotes you an interest rate of 3.5% with 3 points for a 30-year loan, explain
how you would calculate the APR and EAR for this loan. Your answer should use future value and
present value principles to explain your answer. Specify all of the arguments that you would
substitute into the appropriate Excel functions.
First we need to calculate the payment using the quoted rate or 3.5% annually or 3.5% / 12.
Solving for the PMT is almost identical to the part (a) except we have a different quoted interest rate
and solve for the payment in the following equation:
Amount of Mortgage = PresentValue of Payments
360
PMT
i 1
1  APR 12 

i
360
PMT
i 1
1  3.5% 12 
$250, 000  
i
Note: This assumes that the loan amount is for $250,000 and then the lender only gives the borrower
a discounted amount. An alternative would be to have the borrow receive $250,000 and then the
mortgage liability is increased to take into consideration the number of points. This calculation would
be:
$250, 000
1  0.03
 $257, 731.96
Loan Amount 
As Excel solves for the rate, it is determining the interest rate that makes the following expression
true:
Amount Received = PresentValue of Payments
360
PMT
i 1
1  APR 12 

360
$250, 000  
i 1
i
$1,122.61
1  APR 12 
i
The APR must be converted into an EAR by taking into consideration the compounding using the
following formula that is implemented in the EFFECT function:
12
APR 

EAR   1 
 1
12 

12
 3.75 
 1 
  1  3.815%
12 

2.
The 2/15/2046 Treasury bond has a coupon rate of 2.5%, an asked price of 95.2969, and an asked yieldto-maturity of 2.731%.
a.
Draw a diagram that represents the cash flows if you are the purchaser of this bond.
b.
Explain how the yield to maturity is related to the price of this bond. Use formulas and present and
future value concepts to carefully explain your answer.
The yield to maturity is that discount rate that causes the present value of the future cash flows to
equal the price of the bond. Alternative, the price of the bond is the present value of the future cash
flows discounted at the yield to maturity. The equation for the relationship is
n
Price  
Coupon

FaceValue
n
ytm 

1 

2 

60
100
$95.2969  

i
60
2.731%   2.731% 
i 1 
1 
 1 

2  
2 

i 1
c.
ytm 

1 

2 

1.25
i
Explain why you would only be willing to pay $95.2969 for this bond which has a face value of $100.
The market interest rate is given by the yield to maturity which is 2.731%. This is less than the coupon
rate on this bond which is 2.5%. The coupon payments are small relative to other bonds. Therefore,
investors won’t want to buy this bond at par, so its price will decrease enough until the YTM on this
bond exactly matches the market rate of 2.731%. When the price is $95.2969, then the bond offers
the same interest rate as all other bonds of this maturity.
3.
You graduate from the MPA program on your 25th birthday today and want to start saving for your
anticipated retirement at age 65. You think that in addition to social security, you will be able to visit your
posterity, go on missions, and travel if you receive an annuity payment of $40,000 a year. You invest your
money in the local credit union, which offers 6 percent interest per year. You plan to make equal annual
payments on each birthday into the account established at your credit union for your retirement fund.
You start making these deposits on your 26th birthday and continue to make deposits until you are 65
(the last deposit will be on your 65th birthday)
a.
Draw a cash flow diagram for the retirement phase of your life.
We weren’t given the number of years that you expected to live. Let’s assume that you expect to live
until you are 90 years old. Therefore, you will have 25 years of retirement.
b.
Draw a cash flow diagram for the savings phase of your life.
c.
Provide step by step details about how you would complete all of the calculations necessary to
determine how much you need to save each year in order to be able to make your anticipated
$40,000 withdrawals during retirement. Explain how you would use Excel functions to complete all of
the calculations. Your explanation should explicitly specify the values for the Excel financial functions.
Your answer should also include how you are using present and future value concepts to move cash
flows forward or backward along the time line.
The first step is to determine the amount of money that you will need to have in the credit union
when you retire so that we will be able to make withdrawals of $40,000 a year. We have an annuity
so we want to calculate the present value of these payments. We would use the PV function in Excel.
The equation for these calculations is:
25
$40,000
i 1
1  6% 
Bank Deposit  
i
The second step is to figure out how much to save each year so that we have the required
amount in the bank when retirement arrives at age 65. This is the future value of the annuity
and would be calculated using the PMT function in Excel. This function is performing the
following calculation:
40
FutureValue  Amount in Account   1  6%   Annual Deposit
i
i 1
We know the amount that must be in the account from the previous problem. Therefore, we must
solve the above equation for the amount of the annual deposit