1. Diana decides to purchase a US Treasury Bill for 95,000. The Treasury Bill matures in 180 days
for 100,000.
a. Calculate the quoted rate on this Treasury Bill.
Solution:

 Amount of Interest 
360
Quoted Rate = 


 Number of Days  Maturity Value 
 360   100, 000  95, 000 


  0.10000
100, 000
 180  

b. Calculate the annual effective interest rate the Diana will actually earn.
Solution:
(95, 000)(1  i)180/365  100, 000
 100, 000 
1 i  

 95, 000 
365/180
 1.109613107
i  0.10961
c. Calculate the continuously compounded interest rate the Diana will earn.
Solution:
(95, 000)e (180/365)  100, 000
 100, 000 
 100, 000 
e (180/365)  
  (180 / 365)  ln 


 95, 000 
 95, 000 
 365   100, 000 
 ln 
  0.10401
 180   95, 000 
 
2. Anis decides to purchase a Government of Canada Treasury Bill for 95,000. The Treasury Bill
matures in 180 days for 100,000.
a. Calculate the quoted rate on this Treasury Bill.
Solution:

  Amount of Interest 
365
Quoted Rate = 


 Number of Days   Purchase Price 
 365   100, 000  95, 000 


  0.10653
95, 000
 180  

b. Calculate the annual effective interest rate the Anis will actually earn.
Solution:
(95, 000)(1  i)180/365  100, 000
 100, 000 
1 i  

 95, 000 
365/180
 1.109613107
i  0.10961
c. Calculate the continuously compounded interest rate the Anis will earn.
Solution:
(95, 000)e (180/365)  100, 000
 100, 000 
 100, 000 
e (180/365)  
  (180 / 365)  ln 


 95, 000 
 95, 000 
 365 
 100, 000 
 
 ln 
  0.10401
 180   95, 000 
3. David buys a US Treasury Bill with a quoted rate of 8.25%. The Bill matures in 120 days for
10,000.
Calculate the price that David pays for the Treasury Bill.
Solution:

 Amount of Interest 
360
Quoted Rate = 


 Number of Days  Maturity Value 
 360   10, 000  Price 


  0.0825
10, 000
 120  

 120 
10, 000  Price  (0.0825) 
 (10, 000)
 360 
 120 
Price  10, 000  (0.0825) 
 (10, 000)  9725.00
 360 
4. Lily buys a Government of Canada Treasury Bill with a quoted rate of 8.25%. The Treasury Bill
matures in 45 days for 10,000.
Calculate the price that Lily pays for the Treasury Bill.
Solution:
Solution:

  Amount of Interest 
365
Quoted Rate = 


 Number of Days   Purchase Price 
 365  10, 000  Price 


  0.0825
Price
 45 

 45 
10, 000  Price  (0.0825) 
 (Price)
 365 
10, 000
 45 
Price  (0.0825) 
 9899.31
 (Price)  10, 000  Price 
 45 
 365 
1  (0.0825) 

 365 
5. Noah buys a US Treasury Bill that matures for 50,000 in 150 days. Noah will earn an annual
effective interest rate on his investment of 6.5%.
Calculate the quoted rate on this Treasury Bill.
Solution:
(Price)(1.065)150/365  50, 000
Price 
50, 000
 48, 722.60
(1.065)150/365

 Amount of Interest 
360
Quoted Rate = 


 Number of Days  Maturity Value 
 360   50, 000  48, 722.60 


  0.06132
50, 000
 150  

6. The interest rate that Anderson Bank wants to receive on a five year loan in order to deferred
consumption is an annual rate of 5% compounded continuously. For five year loans, the
percentage of loans that will default is 0.7%. For the loans where there are defaults, Anderson
Bank will be able to recover 30% of the amount owed after five years.
Calculate the credit spread calculated as an annual rate compounded continuously that
Anderson Bank needs to charge.
Solution:
Anderson wants to earn an interest rate that will compensate her for deferral of comsumption
and the cost of defaults. The annual rate for deferral of comsumption is 0.05 compounded
continuously. The annual cost of defaults is  s compounded continuously. Therefore, the
return that Anderson wants to earn if there were no defaults would be 0.05. The return necessary
given the defaults is 0.05   s .
Assuming that 1 is borrowed, the amount to be repaid prior to the cost of defaults would be
e(5)(0.05) . If you include the cost for default, then the amount to be repaid would be e5(0.05 s ) .
(1  0.007)e5(0.05 s )  (0.007)(0.30)e5(0.05 s )  e5(0.05)
(0.007)(0.30)e5(0.05 s )  e5(0.05)  0.993e5(0.05 s )
e5(0.05 s ) 
e5(0.05)
 1.290348122
(0.007)(0.30)  0.993
5(0.05   s )  ln(1.290348122)   s 
ln(1.290348122)
 0.05  0.00098
5
7. On ten year loans, Liu Loan Company wants to receive an annual rate of 4.5% compounded
continuously before taking into account defaults. This is equivalent to the rate desired in order
to defer consumption.
When taking into account expected defaults and recovery on those defaults, Liu charges an
annual rate of 4.55% compounded continuously.
For these ten year loans, Liu believes that 1.2% of the loans will default. Further, Liu expects the
loans which default will pay X% of the amount owed at the end of 10 years.
Determine X.
Solution:
Liu wants to earn an interest rate that will compensate her for deferral of comsumption
and the cost of defaults. The annual rate for deferral of comsumption is 0.045 compounded
continuously so Liu wants to earn 0.045 if there are no defaults. The return necessary
given the defaults is 0.0455.
Assuming that 1 is borrowed, the amount to be repaid prior to the cost of defaults would be
e(10)(0.045) . If you include the cost for default, then the amount to be repaid would be e(10)(0.0455) .
(1  0.012)e(10)(0.0455)  (0.012)(recovery)e(10)(0.0455)  e(10)(0.045)
(0.012)(recovery)e(10)(0.0455)  e(10)(0.045)  0.988e(10)(0.0455)
recovery 
e(10)(0.045)  0.988e(10)(0.0455)
 0.584
(0.012)e(10)(0.0455)
8. The Dakota Bank makes loans five year loans for college students. Dakota wants to receive an
annual interest rate 3.2% compounded continuously without taking into account defaults and
inflation.
Dakota knows that inflation for the next five years will be at an annual rate of 2.7% compounded
continuously.
College students have a high default rate. Dakota believes that 4.2% of the students will default
on the loan at the end of 5 years. She also believes that she will be able to recover 45% of the
amount owed on defaults.
a. Calculate the credit spread that Dakota needs to charge as an annual rate compounded
continuously.
Solution:
Dakota wants to earn an interest rate that will compensate her for deferral of comsumption,
inflation, and the cost of defaults. The annual rate for deferral of comsumption is 0.032
compounded continuously. The annual inflation is 0.027 compounded continuously. The
annual cost of defaults is  s compounded continuously. Therefore, the return that Dakota
wants to earn if there were no defaults would be 0.032  0.027  0.059. The return necessary
given the defaults is 0.032  0.027   s  0.059   s .
Assuming that 1 is borrowed, the amount to be repaid prior to the cost of defaults would be
e(5)(0.059) . If you include the cost for default, then the amount to be repaid would be e5(0.059 s ) .
(1  0.042)e(5)(0.059 s )  (0.042)(0.45)e(5)(0.059 s )  e(5)(0.059)
(0.042)(0.45)e(5)(0.059 s )  e(5)(0.059)  0.958e(5)(0.059 s )
e
(5)(0.059  s )
e(5)(0.059)

 1.374886231
(0.042)(0.45)  0.958
(5)(0.059   s )  ln(1.374886231)   s 
ln(1.374886231)
 0.059  0.004674197
5
b. Calculate the total annual interest rate compounded continuously that Dakota should
charge for the loans including the components for default and inflation.
Solution:
Dakota wants to earn an interest rate that will compensate her for deferral of consumption,
inflation, and the cost of defaults. The annual rate for deferral of comsumption is 0.032
compounded continuously. The annual inflation is 0.027 compounded continuously. The
annual cost of defaults is 0.00467 compounded continuously. Therefore, the return necessary
given the defaults is 0.032  0.027  0.00467  0.06367.
c. Sammie borrows 32,000 from Dakota to be repaid at the end of five years. Calculate the
amount that Sammie will need to repay at the end of five years.
Solution:
If we carry all the decimal places from part a, then our interest rate is 0.059  0.004674197
 0.063674197.
(32, 000)e(5)(0.063674197)  43,996.36
9. Porter makes three year loans which include inflation protection. The annual interest rate
compounded continuously that must be paid is 3.2% plus the rate of inflation.
The US government borrows 100,000 for three years from Porter. The actual annual inflation
rate during the first year was 2.4% compounded continuously. The actual annual inflation rates
for the second and third years respectively was 2.8% and 4.2% compounded continuously.
The US government is considered a risk free borrower which means there is no chance of
default.
a. Calculate the amount that the US government will owe at the end of three years.
Solution:
The first year, the government will owe 0.032+0.024 compounded continuously.
The second year, the government will owe 0.032+0.028 compounded continuously.
The third year, the government will owe 0.032+0.042 compounded continuously.
Amount Owed = 100, 000e0.0320.024 e0.032 0.028 e0.032 0.042  100, 000e0.019  120,924.96
b. Calculate the annual Real Interest Rate compounded continuously.
Solution:
The Real Interest Rate compounded continuously is the rate for deferred consumption
compounded continuously less the charge for inflation protection. This is the interest
rate compounded continuously on an inflation protected investment where there is
no default risk. Therefore, for this loan, the Real Interest Rate compounded continuously
is 0.032.
c. The annual interest rate compounded continuously that Porter wants to earn to defer
consumption is 3.5%. What is the annual cost compounded continuously of inflation
protection.
Solution:
The Real Interest Rate compounded continuously is the rate for deferred consumption
compounded continuously less the charge for inflation protection. This is the interest
rate compounded continuously on an inflation protected investment where there is
no default risk. Therefore, for this loan, the Real Interest Rate compounded continuously
is 0.032. The rate for deferral of consumption is given as 0.035. Therefore, the charge
for inflation protection must be 0.035  0.032  0.003.
10. Nick makes a seven year loan to the Government of Canada. The Government of Canada is
considered a risk free borrower with no risk of default.
The amount of the loan is 70,000. Nick wants to earn an annual rate of 3.87% compounded
continuously for deferring consumption. Nick also expects an annual inflation rate of 2.13%
compounded continuously over the next seven years.
The Government of Canada has agreed to repay 108,418.12 at the end of seven years.
a. Calculate the cost that Nick is charging for the uncertainty of the inflation rate. Express
your answer as an annual rate compounded continuously.
Solution:
The total interest rate reflects the following four components:
 Rate for deferred consumption;
 Cost of defaults;
 Rate for expected inflation; and
 Charge for inflation risk;
For this loan, the total annual interest rate compounded continuously is
 108, 418.12 
ln 
70, 000 
70,000e7 =108,418.12==>  
 0.0625
7
The rate for deferred consumption is 3.87% compounded continuously. The cost of defaults
is zero since there are no defaults. The rate for expected inflation is 2.13% compounded
continuously. Therefore, the annual charge for inflation risk compounded continuously is
0.0625  0.0387  0  0.0213  0.0025
b. Calculate the Nominal Interest Rate that Nick is earning expressed as an annual rate
compounded continuously.
Solution:
The Nominal Interest Rate is total annual interest rate compounded continuously which
is 0.0625.
Answers
1.
a. 0.10000
b. 0.10961
c. 0.10401
2.
a. 0.10673
b. 0.10961
c. 0.10401
3. 9725
4. 9899.31
5. 0.06132
6. 0.00098
7. 58.4%
8.
a. 0.00467
b. 0.06367
c. 43,996.36 Assumes that you carry all decimals for the entire calculation. If you round
you will get a slightly different answer which could differ by as much as 1.00.
9.
a. 120,924.96
b. 0.032
c. 0.003
10.
a. 0.0025
b. 0.0625