Financial environment interest rate

Department of Finance Jagannath University 1 | P a g e

Date of Submission: 25-10-2010

Submitted To:

Md. Monzur Morshed Bhuiya Associate Professor Department of Finance Jagannath University, Dhaka.

Submitted By:

Md. Mazharul Islam. Group Representative of Finance Interface B.B.A, 3rd Batch (2nd Year, 1st Semester) Session: 2008-2009

Department of Finance Jagannath University, Dhaka.

An Assignment of Business Finance

Course Code: FIN -2101


1

Sl. No. Name Roll No.

01. Md. Mazharul Islam. (Group Representative) 091541

02. Khadizatuz Zohara. 091526


Sl. No. Contents Page No.

Problems

2-1 Yield Curves 5

2-2 Yield Curves 6

2-3 Inflation and Interest Rate 7

2-4 Rate of Interest 9

2-5 Real Risk-Free Rate, MRP and DRP 10

Exam-Type Problems

2-6 Expected Inflation Rate 12

2-7 Expected Rate of Interest 13


2-9 Interest Rate 14

2-10 Interest Rate 15


Ending Part

Formula and Necessary Illustration for Calculation 17

Summary of the Assignment 18

Table of Contents


The Financial Environment: Interest Rates

Problems 2-1:

Suppose you and most other investors expect the rate of inflation to be 7 percent next year, to fall to 5

percent during the following year, and then to remain at a rate of 3 percent thereafter. Assume that the real

risk-free rate, k*, is 2 percent and that maturity risk premium on treasury securities rise from zero on very

short-term bonds ( those that mature in few days) by 0.2 percentage points for each year to maturity, up to

a limit of 1.0 percentage point on five year or longer-term T-bonds.

a. Calculate the interest rate on one, two, three, four, five, 10 and 20 year Treasury securities, and Plot

the yield curve.

b. Now suppose IBM, a highly rated company, had bonds with the same- maturities as the Treasury

bonds. As an approximation, plot a yield curve for IBM on the same graph with the Treasury bond

yield curve, (Hint: Think about the default risk premium on IBM’s long-term versus its short-term

bonds.)

c. Now plot the approximate yield curve of Long Island Lighting Company (LILCO), a risky nuclear

utility.

Solution 2-1:

Requirement ‘a’:

Bond Type

Expected

Annual

Inflation

Rate

Real

Risk-free

Rate (k*)

Average Expected Inflation

Rate or Inflation Premium

(IP)

Average

Nominal

Interest Rate

𝑘𝑅𝐹 = k* + IP

1st year bond 7% 2% 𝐼𝑃1 = 7% 1 =7% 9%

2nd

year bond 5% 2% 𝐼𝑃2 = (7%+5%) ∕2 = 6% 8%

3rd

year bond 3% 2% 𝐼𝑃3 = (12%+3%) ∕3 = 5% 7%

4th

year bond 3% 2% 𝐼𝑃4 = (15%+3%) ∕4 =4.5% 6.5%

5th

year bond 3% 2% 𝐼𝑃5 =(18%+3%) ∕5 = 4.2% 6.2%

10th

year bond 3% 2% 𝐼𝑃10 =(21%+3%×5) ∕10=3.6% 5.6%

20th

year bond 3% 2% 𝐼𝑃20 =(36%+3%×10) ∕20=3.3% 5.3%

Bond Type Maturity Risk Premium (MRP)

1st year bond 0.2%

2nd

year bond 0.2%+0.2% =0.4%

3rd

year bond 0.4%+0.2% =0.6%

4th

year bond 0.6%+0.2% =0.8%

5th

year bond 0.8%+0.2% =1.0%

10th

year bond 1.0%

20th

year bond 1.0%


And

Bond Type 𝑘𝑅𝐹 + 𝑀𝑅𝑃 Interest Rate (k)

1st year bond 9% + 0.2% 9.2%

2nd

year bond 8% + 0.4% 8.4%

3rd

year bond 7% + 0.6% 7.6%

4th

year bond 6.5% + 0.8% 7.3%

5th

year bond 6.2% + 1.0% 7.2%

10th

year bond 5.6% + 1.0% 6.6%

20th

year bond 5.3% + 1.0% 6.3%

The yield Curve:

Requirement ‘b’:

The interest rate on the IBM bonds has the same components as the Treasury securities, except that the

IBM bonds have default risk, so a default risk premium must be included. Therefore,

𝐾𝐼𝐵𝑀 = 𝑘* + IP + MRP + DRP

For a strong company such as IBM, the default risk premium is virtually zero for short-term bonds.

However, as time to maturity increases, the probability of default, although still small, is sufficient to

warrant a default premium. Thus, the yield risk curve for the IBM bonds will rise above the yield curve for

the Treasury securities. In the graph, the default risk premium was assumed to be 1.2 percentage points on

the 20-year IBM bonds. The return should equal 6.3% + 1.2% = 7.5%.

5.0

5.5

6.0

6.5

7.0

7.5

8.0

8.5

9.0

9.5

10.0

10.5

0 2 4 6 8 10 12 14 16 18 20

Yie

ld (

%)

Yield of Maturity

Yield Curve

T - Bonds

IBM

LILCO

T - Bonds


Requirement ‘c’:

Long Island Lighting Company (LILCO) bonds would have significantly more default risk than either

Treasury securities or IBM bonds, and the risk of default would increase over time due to possible

financial deterioration. In this example, the default risk premium was assumed to be 1.0 percentage point

on the one-year LILCO bonds and 2.0 percentage points on the 20-year bonds. The 20-year return should

equal 6.3% + 2% = 8.3%.

-------------

Problem 2-2:

The following yield on U.S. Treasury securities were taken from The Wall Street Journal on January 7,

2004:

Plot a yield curve based on these data. Discuss how each term structure theory mentioned in the chapter

can explain the shape of the yield curve you plot.

Solution 2-2:

-------------

Term Rate

6 months 1.0%

1 year 1.2%

2 year 1.6%

3 year 2.5%

4 year 2.9%

5 year 3.7%

10 year 4.6%

20 year 5.1%

30 year 5.3%

4.85

4.90

4.95

5.00

5.05

5.10

5.15

5.20

5.25

5.30

5.35

0 5 10 15 20 25 30

Yie

ld (

%)

Maturity (years)

Yield Curve


Problem 2-3:

Inflation currently is about 2 percent. Last year the Fed took actions to maintain inflation at this level.

However, the economy is showing signs that it might be growing too quickly, and reports indicate that

inflation is expected to increase during the next five year. Assume that at the beginning of 2005, the rate of

inflation expected for the year is 4 percent; for 2006, it is expected to be 5 percent; for 2007, it is expected

to be 7 percent; and, for 2008 and every year thereafter, it is expected to settle at 4 percent.

a. What is the average expected inflation rate over the five year period 2005-2009?

b. What average nominal interest would, over the five-year period, be expected to produce a 2 percent

real risk-free rate of return on five-year Treasury securities?

c. Assuming a real risk-free rate of 2 percent and a maturity risk premium that starts at 0.1 percent

and increases by 0.1 percent each year, estimate the interest rate in January 2005on bond that

mature in one, two, five, 10 and 20 years and draw a yield curve based on these data.

d. Describe the general economic conditions that could be expected to produce an upward-sloping

yield curve.

e. If the consensus among investors in early 2005 is that the expected rate of inflation for every future

year is 5 percent ( 𝐼𝑡 = 5% for t = 1 to ∞), what do you think the yield curve would look like?

Consider all the factors that are likely to affect the curve. Does your answer here make you

question the yield curve you drew in part c?

Solution 2-3:

Requirement ‘a & b’:

Bond Type

Expected

Annual

Inflation

Rate

Real

Risk-free

Rate (k*)



(IP)

Average

Nominal

Interest Rate


1st year bond 4% 2% 𝐼𝑃1 = 4% 1 =4% 6%

2nd

year bond 5% 2% 𝐼𝑃2 = (4%+5%) ∕2 = 4.5% 6.5%

3rd

year bond 7% 2% 𝐼𝑃3 = (9%+7%) ∕3 = 5.33% 7.33%

4th

year bond 4% 2% 𝐼𝑃4 = (16%+4%) ∕4 =5% 7%

5th

year bond 4% 2% 𝐼𝑃5 =(20%+4%) ∕5 = 4.8% 6.8%

10th

year bond 4% 2% 𝐼𝑃10 =(24%+4%×5) ∕10=4.4% 6.4%

20th

year bond 4% 2% 𝐼𝑃20 =(44%+2%×5) ∕20=4.2% 6.2%



1st year bond 0.1%

2nd

year bond 0.1%+0.1% =0.2%

3rd

year bond 0.2%+0.1% =0.3%

4th

year bond 0.3%+0.1% =0.4%

5th

year bond 0.5%+0.1% =0.5%

10th

year bond 0.5%+(0.1%×5) =1.0%

20th

year bond 1.0%+(0.1%×10) =2.0%


0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

0 2 4 6 8 10 12 14 16 18 20

Yie

ld (

%)

Years to Maturity

Yield Curve

Pure expectations yield curve

Maturity risk

premium

Actual yield curve

Years to Maturity

Yield (%)

And

Bond Type 𝑘𝑅𝐹 + 𝑀𝑅𝑃 Estimated Interest Rate

(k)

1st year bond 6% + 0.1% 6.1%

2nd

year bond 6.5% + 0.2% 6.7%

5th

year bond 6.8% + 0.5% 7.3%

10th

year bond 6.4% + 1.0% 7.4%

20th

year bond 6.2% + 2.0% 8.2%

The Yield Curve:

Requirement ‘d’:

The ―normal‖ yield curve is upward sloping because, in ―normal‖ times, inflation is not expected to trend

either up or down, so IP is the same for debt of all maturities, but the MRP increases with years, so the

yield curve slopes up. During a recession, the yield curve typically slopes up especially steeply, because

inflation and consequently short-term interest rates are currently low, yet people expect inflation and

interest rates to rise as the economy comes out of the recession.

Requirement ‘e’:

If inflation rates are expected to be constant, then the expectations theory holds that the yield curve should

be horizontal. However, in this event it is likely that maturity risk premiums would be applied to long-term

bonds because of the greater risks of holding long-term rather than short-term bonds:


If maturity risk premiums were added to the yield curve in part e above, then the yield curve would be

more nearly normal—that is, the long-term end of the curve would be raised.

-------------

Problem 2-4:

Assume that the real risk-free rate of return, k*, is 3 percent, and it will remain at that level far into the

future. Also assume that maturity risk premiums on Treasury Bonds increase from zero for bonds that

mature in one year or less to a maximum of 2 percent, and MRP increases by 0.2 percent for each year to

maturity that is greater than one year – that is, MRP equals 0.2 percent for a two-year bond, 0.4 percent for

a three year bond, and so forth. Following are the expected inflation rates for the next five years:

Year Inflation Rate (%)

2005 3

2006 5

2007 4

2008 8

2009 3

a. What is the average expected inflation rate for a one, two, three, four and five year bond?

b. What should be the MRP for a one, two, three, four and five year bond?

c. Compute the interest rate for a one, two, three, four and five year bond?

d. If inflation is expected to equal 2 percent every year after 2009, what should be the interest rate for

a 10 and 20 year bond?

e. Plot the yield curve for the interest rates you computed in parts c and d.

Solution 2-4:


Bond Type

Expected

Annual

Inflation

Rate

Real

Risk-free

Rate (k*)



(IP)

Average

Nominal

Interest Rate


1st year bond 3% 3% 𝐼𝑃1 = 3% 1 =3% 6%

2nd

year bond 5% 3% 𝐼𝑃2 = (3%+5%) ∕2 = 4% 7%

3rd

year bond 4% 3% 𝐼𝑃3 = (8%+4%) ∕3 = 4% 7%

4th

year bond 8% 3% 𝐼𝑃4 = (12%+8%) ∕4 =5% 8%

5th

year bond 3% 3% 𝐼𝑃5 =(20%+3%) ∕5 = 4.6% 7.6%

10th

year bond 2% 3% 𝐼𝑃10 =(23%+2%×5) ∕10=3.3% 6.3%

20th

year bond 2% 3% 𝐼𝑃20 =(33%+2%×5) ∕20=2.65% 5.65%



1st year bond 0%

2nd

year bond 0%+0.2% =0.2%

3rd

year bond 0.2%+0.2% =0.4%

4th

year bond 0.4%+0.2% =0.6%

5th

year bond 0.6%+0.2% =0.8%

10th

year bond 0.8%+(0.2%×5)=1.8%

20th

year bond 2%


5.0

5.5

6.0

6.5

7.0

7.5

8.0

8.5

9.0

0 2 4 6 8 10 12 14 16 18 20

Yie

ld (

%)

Years to Maturity

Yield Curve

Requirement ‘c & d’:

Bond Type 𝑘𝑅𝐹 + 𝑀𝑅𝑃 Interest Rate (k)

1st year bond 6% + 0% 6%

2nd

year bond 7% + 0.2% 7.2%

3rd

year bond 7% + 0.4% 7.4%

4th

year bond 8% + 0.6% 8.6%

5th

year bond 7.6% + 0.8% 8.4%

10th

year bond 6.3% + 1.8% 8.1%

20th

year bond 5.65% + 2% 7.65%

Requirement ‘e’:

-------------

Problem 2-5:

Today’s edition of The Wall Street Journal reports that the yield on Treasury bills maturing in 30 days is

3.5 percent, the yield on Treasury bills maturing in 10 years is 6.5 percent, and the yield on a bond issued

by Nextel Communications that matures in six years is 7.5 percent. Also, today the Federal Reserve

announced that inflation is expected to be 2 percent during the next 12 months. There is a maturity risk

premium (MRP) associated with all bonds with maturities equal to one year or more.

a. Assume that the increase in the MRP each year is the same and the total MRP is the same for

bonds with maturities equal to 10 years and greater that is, MRP is at its maximum for bonds with

maturities equal to 10 years and greater. What is the MRP per year?

b. What is default risk premium associated with Nextel’s bond?

c. What is the real risk-free rate of return?


Solution 2-5:


Since MRP associated with all bonds with maturities equal to one year or more, so with Treasury bills

maturing in 30 days, 0% MRP is associated, then

k = k* + IP

⇒ 3.5% = k* + 2%

⇒ k* = 3.5% − 2%

∴ k* = 1.5%

At the 10 year bond:

k = k* + IP + MRP

⇒ 6.5% = 1.5% + 2% + MRP

⇒ MRP = 6.5% − 1.5% − 2%

∴ MRP = 3%

As MRP at 10 year bond is 3%. So MRP per year is (3÷10) = 0.3%.


Since 30 days T-bond and 10 years T-bond fulfills the equations:- K = k* +IP +MRP,

We have to calculate DRP from 6 years Nextel Bond:

k = k* +IP +DRP +MRP

⇒ 7.5% = 1.5% + 2% + DRP + (0.3% × 6)

⇒ 7.5% = 3.5% + DRP + 1.8%

⇒ DRP = 7.5% − 3.5% − 1.8%

∴ DRP = 2.2%


Now real risk-free rate of return

k* = 3.5% – IP

= 3.5% - 2.0%

= 1.5%

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Exam-Type Problems 2-6:

According to The Wall Street Journal, the interest rate on one-year Treasury bonds is 2.2 percent, The rate

on two-year Treasury bonds is 3.0 percent, and the rate on three-year Treasury bonds is 3.6 percent. These

bonds are considered risk free, so the rates given here are risk free rates (𝐾𝑅𝐹). The one-year bond matures

one year from today, the two-year bond matures two year from today and so forth. The real risk-free rate

(k*) for each year is 2 percent. Using the expectations theory, compute the expected inflation rate for the

next 12 months (Year 1) and (Year 2).


Solution 2-6:

Here, for one-year treasury bonds-

𝑘𝑅𝐹1= 2.2%

k* = 2%

∴ 𝐼𝑃1 = 𝑘𝑅𝐹1− k*

= 2.2% −2%

= 0.2%

For one-year Treasury bond 𝐼𝑃1 = 𝐼𝑛𝑓𝑙1 = 0.2%

As average 𝑘𝑅𝐹 for two-year bond is 3%, thus annual 𝑘𝑅𝐹 for two-year bond is ,

𝑘𝑅𝐹2=

𝑘𝑅𝐹1+ 𝑘𝑅𝐹 𝑖𝑛 𝑦𝑒𝑎𝑟 2

2

⇒ 3% =2.2% + 𝑘𝑅𝐹 𝑖𝑛 𝑦𝑒𝑎𝑟 2

2

⇒ 6% = 2.2% + 𝑘𝑅𝐹 𝑖𝑛 𝑦𝑒𝑎𝑟 2

⇒ 𝑘𝑅𝐹 𝑖𝑛 𝑦𝑒𝑎𝑟 2 = 6% − 2.2%

∴ 𝑘𝑅𝐹 𝑖𝑛 𝑦𝑒𝑎𝑟 2 = 3.8%

Since 𝑘𝑅𝐹 𝑖𝑛 𝑦𝑒𝑎𝑟 2 = 3.8% and k* = 2%.

∴ 𝐼𝑃2 = 𝑘𝑅𝐹 𝑖𝑛 𝑦𝑒𝑎𝑟 2 – k*

= 3.8% – 2%

= 1.8%

For two-year Treasury bond 𝐼𝑃2 is 1.8% ,

so 𝐼𝑛𝑓𝑙2 ={(1.8% × 2) – 0.2%)

= (3.8% – 0.2%)

= 3.4%

-------------


Suppose the annual yield on a two-year Treasury bond id 11.5 percent, while that on a one-year bond is 10

percent, k* is 3 percent, and the maturity risk premium is zero.

a. Using the expectation theory, forecast the interest rate on one-year bond during the second year.

(Hint: Under the expectation theory, the yield on a two-year bond is equal to the average yield on

one-year bonds in Year 1 and Year 2.)

b. What is the expected inflation rate in Year 1 and Year 2?


Solution 2-7:


Here,

𝑘𝑅𝐹1 = 10%

𝑘𝑅𝐹2 = 11.5%

∴ 𝑘𝑅𝐹2=


2

⇒ 11.5% = 10% + 𝑘𝑅𝐹 𝑖𝑛 𝑦𝑒𝑎𝑟 2

2

⇒ 11.5% × 2 = 10% + 𝑘𝑅𝐹 𝑖𝑛 𝑦𝑒𝑎𝑟 2

⇒ 𝑘𝑅𝐹 𝑖𝑛 𝑦𝑒𝑎𝑟 2 = 23% − 10%

∴ 𝑘𝑅𝐹 𝑖𝑛 𝑦𝑒𝑎𝑟 2 = 13%


For riskless bonds under the expectations theory, the interest rate for a bond of any maturity is 𝑘𝑅𝐹 = k* +

IP. If k* = 3%, we can solve for IP:

For year 1:

𝑘𝑅𝐹 𝑖𝑛 𝑦𝑒𝑎𝑟 1 = 𝑘* + 𝐼𝑛𝑓𝑙1

⇒ 𝑘𝑅𝐹 𝑖𝑛 𝑦𝑒𝑎𝑟 1 = 3% + 𝐼𝑛𝑓𝑙1

⇒ 10% = 3% + 𝐼𝑛𝑓𝑙1

⇒ 𝐼𝑛𝑓𝑙1 = 10% − 3%

⇒ 𝐼𝑛𝑓𝑙1 = 7%

For Year 2:

𝑘𝑅𝐹 𝑖𝑛 𝑦𝑒𝑎𝑟 2 = 𝑘* + 𝐼𝑛𝑓𝑙2

⇒ 𝑘𝑅𝐹 𝑖𝑛 𝑦𝑒𝑎𝑟 2 = 3% + 𝐼𝑛𝑓𝑙2

⇒ 13% = 3% + 𝐼𝑛𝑓𝑙2

⇒ 𝐼𝑛𝑓𝑙2 = 13% − 3%

⇒ 𝐼𝑛𝑓𝑙2 = 10%

Here, 𝐼𝑛𝑓𝑙2 is the expected inflation.

The average inflation rate is (7% + 10%)/2 = 8.5%, which, when added to k* = 3%, produces the yield on a

2-year bond, 11.5%. Therefore, all of our results are consistent.

-------------



Assume that the real risk-free rate is 4 percent and that the maturity risk premium is zero. If the nominal

rate of interest on one-year bonds is 11 percent and that on comparable-risk two-year bonds is 13 percent,

What is the one-year interest rate that is expected for year 2? What inflation rate is expected during year 2?

Comment on why the average interest rate during the two-year period differs from the one-year interest

rate expected for year 2?

Solution 2-8:

Here,

k* = 4%

𝑘𝑅𝐹1 = 10%

𝑘𝑅𝐹2 = 11.5%

MRP = 0%

∴ 𝑘𝑅𝐹2=


2

⇒ 13% = 11% + 𝑘𝑅𝐹 𝑖𝑛 𝑦𝑒𝑎𝑟 2

2

⇒ 13% × 2 = 11% + 𝑘𝑅𝐹 𝑖𝑛 𝑦𝑒𝑎𝑟 2

⇒ 𝑘𝑅𝐹 𝑖𝑛 𝑦𝑒𝑎𝑟 2 = 26% − 11%

∴ 𝑘𝑅𝐹 𝑖𝑛 𝑦𝑒𝑎𝑟 2 = 15%

∴ 𝑘𝑅𝐹 𝑖𝑛 𝑦𝑒𝑎𝑟 2 = 𝑘* + 𝐼𝑛𝑓𝑙2

⇒ 15% = 4% + 𝐼𝑛𝑓𝑙2

⇒ 𝐼𝑛𝑓𝑙2 = 15% − 4%

∴ 𝐼𝑛𝑓𝑙2 = 11%

The average interest rate during the two-year period differs from the one-year interest rate expected for Year 2

because of the inflation rate reflected in the two interest rates. The inflation rate reflected in the interest rate on any

security is the average rate of inflation expected over the security's life.

-------------


The rate of inflation for the coming year is expected to be 3 percent and the rate of inflation in year 2 and

thereafter is expected to be constant at some level above 3 percent. Assume that the real risk-free rate, k*,

is 2 percent for all maturities, and the expectations theory fully explains the yield curve, so there are no

maturity premiums. If three-year Treasury bonds yield 2 percentage points more than one-year bonds, what

rate of inflation is expected after Year 1?


Solution 2-9:

For one-year bond,

k* = 2%

IP = 3%

MRP = 0%

∴ 𝑘1 = k* + 𝐼𝑃1

= (2% +3%)

= 5%

Since for one-year bond, 𝑘1 is 5%, So for three-year bond 𝑘3 = (5% + 2%) = 7%.

For three-year bond,

𝑘3 = k* + 𝐼𝑃3

⇒ 7% = 2% + 𝐼𝑃3

⇒ 𝐼𝑃3 = 7% − 2%

∴ 𝐼𝑃3 = 5%

We know that, 𝐼𝑛𝑓𝑙1 = 𝐼𝑃1

∴ 𝐼𝑃3 = (𝐼𝑛𝑓𝑙1 + 𝐼𝑛𝑓𝑙2 + 𝐼𝑛𝑓𝑙3)/3

= ( 3% + 𝐼𝑛𝑓𝑙2 + 𝐼𝑛𝑓𝑙3)/3

Again,

𝑘3 = k* + 𝐼𝑃3

⇒ 7% = 2% + ( 3% + 𝐼𝑛𝑓𝑙2 + 𝐼𝑛𝑓𝑙3)/3

⇒ 7% − 2% = ( 3% + 𝐼𝑛𝑓𝑙2 + 𝐼𝑛𝑓𝑙3)/3

⇒ 5% = ( 3% + 𝐼𝑛𝑓𝑙2 + 𝐼𝑛𝑓𝑙3)/3

⇒ 15% = 3% + 𝐼𝑛𝑓𝑙2 + 𝐼𝑛𝑓𝑙3

⇒ 𝐼𝑛𝑓𝑙2 + 𝐼𝑛𝑓𝑙3 = 15% − 3%

∴ 𝐼𝑛𝑓𝑙2 + 𝐼𝑛𝑓𝑙3 = 12%

∴ 2 year average, 2 𝐼𝑛𝑓𝑙 = 12%

⇒ 𝐼𝑛𝑓𝑙 = 12% ÷ 2

∴ 𝐼𝑛𝑓𝑙 = 6%

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Today is January 1, 2005, and according to the result of a recent survey, investors expect the annual

interest rates for the years 2008-2010 to be:

Year One-Year Rate

2008 5

2009 4

2010 3

The rates given here include the risk-free rate, 𝑘𝑅𝐹 , and appropriate risk premiums. Today a three-year

bond – that is, a bond that matures on December 31, 2007 – has an interest rate equal to 6 percent. What is

the yield to maturity for bonds that mature at the end of 2008, 2009, 2010?


Solution 2-10:

Here,

Year One-Year Rate

2008 5

2009 4

2010 3

As, today January 1, 2005 a three-year bond that matures on December 31, 2007 has an interest rate equal

to 6%.

∴ Yield to Maturity (𝑌𝑇𝑀)2007 = 6%

∴ (𝑌𝑇𝑀)2008 = 6% × 3 + 5 /4 = 5.75%

∴ (𝑌𝑇𝑀)2009 = 6% × 3 + 5 + 4 /5 = 5.75%

∴ (𝑌𝑇𝑀)2010 = 6% × 3 + 5 + 4 + 3 /6 = 5.75%

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Suppose current interest rates on Treasury securities are as follows:

Maturity Yield

1 year 5.0

2 year 5.5

3 year 6.0

4 year 5.5

Using the expectations theory, compute the expected interest rates (yields) for each security one year from

now. What will the rates be two years from today and three years from today?

Solution 2-11:

Here,

Maturity Yield

1 year 5.0

2 year 5.5

3 year 6.0

4 year 5.5

As yield to maturity is given here, we can calculate interest rate (k) from the following table:

Year Calculations Interest rate (k)

1 Year 5% 5%

2 Year 5.5% × 2 − 5% 6%

3 Year 6% × 3 − (6% + 5%) 7%

4 Year 5.5% × 4 − (7% + 6% + 5%) 4%

In one year, the bond that matures in one year will mature (die), and the other bonds will have one less

year to maturity. Given the one-year interest rates computed above, the yields on the three remaining bonds

will be:

Original Maturity Maturity After 1 Year New Yield

1 Year Matured --------

2 Year 1 Year 6%/1 = 6%

3 Year 2 Year (7% + 6%)/2 = 6.5%

4 Year 3 Year (4% + 7% + 6%)/3 = 5.7%

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Formula:

1. 𝑘𝑅𝐹= k* + IP

2. 𝑘 = 𝑘𝑅𝐹 + 𝑀𝑅𝑃

3. DRP = k + 𝑘𝑅𝐹

Here,

k = The quoted or nominal rate of interest on a given security.

𝑘𝑅𝐹 = The quoted risk-free free rate of return.

𝑘* = The real risk-free rate of interest.

IP = Average Expected Inflation Rate or Inflation premium.

MRP= Maturity risk premium.

DRP = Default risk premium.

Illustration:

1. Nominal Risk-Free Rate: The rate of interest on a security that is free of all risk, 𝑘𝑅𝐹 is

proxied by the Treasury-bill rate or the Treasury-bond rate. 𝑘𝑅𝐹 includes an inflation

premium.

2. Real Risk-Free Rate of Interest: The rate of interest that would exist on default-free

or inflation is expected to be zero.

3. Inflation Premium: A premium for expected inflation that investors add to the real

risk-free rate of return.

4. Default Risk Premium: The difference between the quoted interest rate on a

Treasury-bond and that on a corporate bond with similar maturity, liquidity, and

other features is the default risk premium.

5. Maturity risk premium: A premium that reflects interest rate risk; bonds with

longer maturities have greater interest rate risk.

Formula and Necessary Illustration for Calculation


Summary of the Assignment

It was a very pleasant & challenging task for us to work on the topic entitled ―The Financial

Environment: Interest Rates‖ for the course entitled ―Business Finance‖ (Fin-2101).

We are very much thankful to our course instructor Md. Monzur Morshed Bhuiya for giving us the

opportunity to analyze the cost of money in different Treasury securities which is very much

helpful for us to enrich our knowledge in the field of corporate world.

In this assignment we have tried our best to deliver the most accurate & reliable information that

we have computed through our group members.

This assignment presents that how can we calculate the interest rate, MRP, DRP, IP, forecasting,

real risk-free rate of return of Treasury securities and how can we draw a yield curve and its

illustration.

It’s truly a pleasant message for us that we are now coping with the modern business calculation

such as interest rate, MRP, DRP, IP, forecasting, real risk-free rate of return of Treasury securities

through the cost of money Calculation.

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Economy & Finance

Financial environment interest rate