# Second Assignment: Interrogating Bonds

Second Assignment: Interrogating Bonds
A first example exercise

Suppose that a Treasury bond with a face value of \$100 and a coupon payment of 5.2% has
one year to maturity. Thus, one year from now, the bond is scheduled to make a payment of
\$100 together with a final coupon payment of \$100 x 0.052 = \$5.2. If the bond is currently
priced at \$100.19, the yield to maturity (YTM 1 ) for the bond over the coming 1-year period is
determined from
\$100.19 =
which determines YTM 1 = = 0.050, or 5.0%. This is the discount rate, DR 1 , that the market
considers appropriate over the coming 1-year period.
Now suppose that a Treasury bond with a face value of \$100 and a coupon payment
of 7.1% has 2 years to maturity (a “2-year bond”). Thus, one year from now, the bond is
scheduled to make a payment of \$100 x 0.071 = \$7.1, and 2 years from now make a payment
of \$100 together with a final coupon payment of \$7.1. If the bond is currently priced at
\$102.99, the yield to maturity (YTM 2 ) for the 2-year bond is determined by solving:
\$102.99 =
The outcome is YTM 2 = 5.48%.

Note: YTMs must be solved using the IRR function in Excel (alternatively, you could set up
your spread-sheet for the valuation of the bond as the fundamental equation above and
gradually adjust the input YTM 2 until you get the valuation to equal \$102.99.

2
The question we now pose is: What is the discount rate, DR 2 , that the market appears
to consider appropriate for the second year? Note, that the answer is not 5.48%! – this is the
discount rate averaged over the two years (and recall that the appropriate discount rate is 5%
for the first year). In fact, to solve for the discount rate that the market is imposing in the
second year, DR 2 , we need to solve:
\$102.99 =
where YTM 1 is the yield to maturity in Year 1 (determined for one-year Treasury bonds as
above = 5.0%) and DR 2 is the appropriate discount rate for Year 2. We therefore determine
\$102.99 =
which solves to give DR 2 (in Year 2) = 0.060 (6.0%).

Approximate approach
Note, how, rather than apply the above equation in order to determine the DR 2 value in Year
2, we can approximate DR 2 by applying the simple relation:
(1+ DR 1 )(1+ DR 2 ) = (1+YTM 2 ) 2

(YTM 2 is the YTM for a 2-year bond) for which we have DR 1 = 0.05 and YTM 2 = 0.0548, so
that:
(1.05)(1+ DR 2 ) = (1.0548) 2
yielding DR 2 = (1.0548) 2 /(1.05) -1 = 0.0596 (5.96%), which compares with DR 2 = 0.060
(6.0%) above. Thus, the approximation appears sufficient.

3
Suppose now that a Treasury bond with a face value of \$100 and a coupon payment of 5.0%
has 5 years to maturity. Thus, one year from now, the bond is scheduled to make a payment
of \$100 x 0.05 = \$5.0, and thereafter, until 5 years from now make a payment of \$100
together with a final coupon payment of \$5.0. If the bond is currently priced at \$91.80, the
yield to maturity (YTM 5 ) for a 5-year bond is determined by solving:
\$91.80 =
which solves to give YTM 5 = 7.0%.
What is the discount rate, DR 3/5 that the market appears to consider appropriate for years 3 to
5?
To answer, we need to solve:
\$91.80 = where DR 1 = 5.0% and DR 2 = 6.0%, so that:
\$91.80 =
which solves to give DR 3/5 = 8.16% .

Approximate approach
Note, now, how, rather than apply the above equation in order to determine the DR 3/5 value
for years 3 – 5, we can approximate DR 3/5 by applying the relation:
(1+ DR 1 ) (1+ DR 2 ) (1+ DR 3/5 ) 3 = (1+YTM 5 ) 5
for which we have DR 1 = 0.05, DR 2 = 0.06 and YTM 5 = 0.07, so that:
(1.05) (1.06) (1+ DR 3/5 ) 3 = (1.07) 5

4
yielding DR 3/5 = [(1.07) 5 /[(1.05)(1.06)]] 1/3 – 1 = 0.080 (8.0%), which compares with DR 3/5 =
0.0816 (8.16%) above. Thus the approximation appears sufficiently accurate.

Nominal rates, inflation, and real rates
Because Treasury bonds may be regarded as effectively free from default risk, they provide a
useful benchmark for interest rates.
For example, if we predict inflation as running at, say, 6.0% per annum over 3, 4 and
5 years forward, we might deduce that bond holders require a risk-free real rate of interest of
approximately DR 3/5 as calculated above minus the inflation rate = 8.16% – 6.0% = 2.16% per
annum.
More accurately, with Eqn 3.12, we would deduce that investors anticipate a risk-free
real rate of interest per annum for 3, 4 and 5 years forward as

= = 1.02 – 1 = 0.020, or 2.0%.
Alternatively, if we considered that investors will have a required real rate of return on
Treasury bonds equal to, say, 2.5%, we would deduce the market’s prediction for inflation in
years 3 – 5 as – 1 = 5.5%.

Interrogating Bonds

5
This document is designed to guide you through the assignment, ie. what you are
expected to do for each part of the assignment. Please note that each Part is carrying an
equally weight, and a discussion on this assignment will be held in Week 5.
PART 1
Now over to you to interrogate the yields to maturity (redemption yields) (over the periods
for which you have data). You should choose the USA and Japan. You may wish to avail of
the website:
https://www.bloomberg.com/markets/rates-bonds/government-bonds/us
http://www.bloomberg.com/markets/rates-bonds/government-bonds/japan

PART 1A: If you have yields to maturity for years 1, 2, 5, 10, 20, and 30 year bonds, for
example, report the yield to maturity over these periods.
PART 1B: Now, using the APPROXIMATE method described in the tutorial above,
calculate the discount rate, DR, that the market appears to consider appropriate over (i) for 1
year, (ii) averaged over 1 – 2 years, (iii) averaged over 3-5 years, (iv) averaged over 6 – 10
years, (v) averaged 11-20 years, and (vi) averaged over 21-30 years.

PART 2
Avail of the internet (for example:
http://www.inflation.edu/, or
where /**/ above = /united-states/, etc.

6

IMF website: http://www.imf.org/external/pubs/ft/weo/2017/01/
weodata/index.aspx

to obtain the predicted rates of inflation over the periods for which you have predicted the
discount rate, DR, that the market considers appropriate (in PART 1B). Use this rate together
with the inflation rate to predict the real rates of interest over these periods. Use:

PART 3

PART 4
Comment on your results for the USA and the rates that relate to TIPs (Treasury inflation
protected bonds).

PART 5
Comment on how the yields to maturity of Treasury bonds have changed since 3 rd April 2017
as below. How does the market appear to have changed its predictions?

Year-
bonds

US US TIPS
Treasury inflation
protected securities

Australia Japan Germany UK

Cash 0.81 – 1.5% 0.10% 0% 0.25%
3-months 0.75% – – – – –
6-months 0.9% – – – – –
12-months 1.02% – – – – –

7
2-year 1.25% – 1.74% – 0.20% – 0.75% 0.11%
5-year 1.92% – 0.16% 2.23% – 0.13% – 0.39% 0.55%
10-year 2.39% 0.40% 2.69% 0.06% 0.32% 1.14%
15-year – – 3.06% – – –
20-year – 0.54% – 0.62% – –
30-year 3.01% 0.91% – 0.84% 1.1% 1.72%

PART 6
How does the “liquidity hypothesis” impact your interpretation? (namely, the idea that long
term bonds are more sensitive to changes in interest rates going forward, and hence more
risky, for which they require a “liquidity premium” – as much as an addition 1.5% on the
YTM compared with a short-term Treasury bill, all else equal).

PART 7
In the light of your findings, comment on the excerpt below from The Economist:

Buttonwood 4-10 March 2017.
If there is one aspect of the current era sure to obsess the financial historians of
tomorrow, it is the unprecedented low level of interest rates. Never before have
deposit rates or bond yields been so depressed in nominal terms, with some
governments even able to borrow at negative rates. It is taking a long time for
investors to adjust their assumptions accordingly.

8
Real interest rates (ie, allowing for inflation) are also low. As measured by inflation-
linked bonds, they are around minus 1 % in big rich economies.

PART 8
Comment on how you see the implications of your findings for the stock market.

Category:

## Description

Second Assignment: Interrogating Bonds
A first example exercise

Suppose that a Treasury bond with a face value of \$100 and a coupon payment of 5.2% has
one year to maturity. Thus, one year from now, the bond is scheduled to make a payment of
\$100 together with a final coupon payment of \$100 x 0.052 = \$5.2. If the bond is currently
priced at \$100.19, the yield to maturity (YTM 1 ) for the bond over the coming 1-year period is
determined from
\$100.19 =
which determines YTM 1 = = 0.050, or 5.0%. This is the discount rate, DR 1 , that the market
considers appropriate over the coming 1-year period. Second Assignment: Interrogating Bonds.
Now suppose that a Treasury bond with a face value of \$100 and a coupon payment
of 7.1% has 2 years to maturity (a “2-year bond”). Thus, one year from now, the bond is
scheduled to make a payment of \$100 x 0.071 = \$7.1, and 2 years from now make a payment
of \$100 together with a final coupon payment of \$7.1. If the bond is currently priced at
\$102.99, the yield to maturity (YTM 2 ) for the 2-year bond is determined by solving:
\$102.99 =
The outcome is YTM 2 = 5.48%.

Note: YTMs must be solved using the IRR function in Excel (alternatively, you could set up
your spread-sheet for the valuation of the bond as the fundamental equation above and
gradually adjust the input YTM 2 until you get the valuation to equal \$102.99. Second Assignment: Interrogating Bonds.

2
The question we now pose is: What is the discount rate, DR 2 , that the market appears
to consider appropriate for the second year? Note, that the answer is not 5.48%! – this is the
discount rate averaged over the two years (and recall that the appropriate discount rate is 5%
for the first year). In fact, to solve for the discount rate that the market is imposing in the
second year, DR 2 , we need to solve:
\$102.99 =
where YTM 1 is the yield to maturity in Year 1 (determined for one-year Treasury bonds as
above = 5.0%) and DR 2 is the appropriate discount rate for Year 2. We therefore determine
\$102.99 =
which solves to give DR 2 (in Year 2) = 0.060 (6.0%).

Approximate approach
Note, how, rather than apply the above equation in order to determine the DR 2 value in Year
2, we can approximate DR 2 by applying the simple relation:
(1+ DR 1 )(1+ DR 2 ) = (1+YTM 2 ) 2

(YTM 2 is the YTM for a 2-year bond) for which we have DR 1 = 0.05 and YTM 2 = 0.0548, so
that:
(1.05)(1+ DR 2 ) = (1.0548) 2
yielding DR 2 = (1.0548) 2 /(1.05) -1 = 0.0596 (5.96%), which compares with DR 2 = 0.060
(6.0%) above. Thus, the approximation appears sufficient. Second Assignment: Interrogating Bonds.

3
Suppose now that a Treasury bond with a face value of \$100 and a coupon payment of 5.0%
has 5 years to maturity. Thus, one year from now, the bond is scheduled to make a payment
of \$100 x 0.05 = \$5.0, and thereafter, until 5 years from now make a payment of \$100
together with a final coupon payment of \$5.0. If the bond is currently priced at \$91.80, the
yield to maturity (YTM 5 ) for a 5-year bond is determined by solving:
\$91.80 =
which solves to give YTM 5 = 7.0%.
What is the discount rate, DR 3/5 that the market appears to consider appropriate for years 3 to
5?
To answer, we need to solve:
\$91.80 = where DR 1 = 5.0% and DR 2 = 6.0%, so that:
\$91.80 =
which solves to give DR 3/5 = 8.16% .

Approximate approach
Note, now, how, rather than apply the above equation in order to determine the DR 3/5 value
for years 3 – 5, we can approximate DR 3/5 by applying the relation:
(1+ DR 1 ) (1+ DR 2 ) (1+ DR 3/5 ) 3 = (1+YTM 5 ) 5
for which we have DR 1 = 0.05, DR 2 = 0.06 and YTM 5 = 0.07, so that:
(1.05) (1.06) (1+ DR 3/5 ) 3 = (1.07) 5

4
yielding DR 3/5 = [(1.07) 5 /[(1.05)(1.06)]] 1/3 – 1 = 0.080 (8.0%), which compares with DR 3/5 =
0.0816 (8.16%) above. Thus the approximation appears sufficiently accurate. Second Assignment: Interrogating Bonds.

Nominal rates, inflation, and real rates
Because Treasury bonds may be regarded as effectively free from default risk, they provide a
useful benchmark for interest rates.
For example, if we predict inflation as running at, say, 6.0% per annum over 3, 4 and
5 years forward, we might deduce that bond holders require a risk-free real rate of interest of
approximately DR 3/5 as calculated above minus the inflation rate = 8.16% – 6.0% = 2.16% per
annum. Second Assignment: Interrogating Bonds.
More accurately, with Eqn 3.12, we would deduce that investors anticipate a risk-free
real rate of interest per annum for 3, 4 and 5 years forward as

= = 1.02 – 1 = 0.020, or 2.0%.
Alternatively, if we considered that investors will have a required real rate of return on
Treasury bonds equal to, say, 2.5%, we would deduce the market’s prediction for inflation in
years 3 – 5 as – 1 = 5.5%.

Interrogating Bonds

5
This document is designed to guide you through the assignment, ie. what you are
expected to do for each part of the assignment. Please note that each Part is carrying an
equally weight, and a discussion on this assignment will be held in Week 5. Second Assignment: Interrogating Bonds.
PART 1
Now over to you to interrogate the yields to maturity (redemption yields) (over the periods
for which you have data). You should choose the USA and Japan. You may wish to avail of
the website:
https://www.bloomberg.com/markets/rates-bonds/government-bonds/us
http://www.bloomberg.com/markets/rates-bonds/government-bonds/japan

PART 1A: If you have yields to maturity for years 1, 2, 5, 10, 20, and 30 year bonds, for
example, report the yield to maturity over these periods.
PART 1B: Now, using the APPROXIMATE method described in the tutorial above,
calculate the discount rate, DR, that the market appears to consider appropriate over (i) for 1
year, (ii) averaged over 1 – 2 years, (iii) averaged over 3-5 years, (iv) averaged over 6 – 10
years, (v) averaged 11-20 years, and (vi) averaged over 21-30 years. Second Assignment: Interrogating Bonds.

PART 2
Avail of the internet (for example:
http://www.inflation.edu/, or
where /**/ above = /united-states/, etc.

6

IMF website: http://www.imf.org/external/pubs/ft/weo/2017/01/
weodata/index.aspx

to obtain the predicted rates of inflation over the periods for which you have predicted the
discount rate, DR, that the market considers appropriate (in PART 1B). Use this rate together
with the inflation rate to predict the real rates of interest over these periods. Use:

PART 3

PART 4
Comment on your results for the USA and the rates that relate to TIPs (Treasury inflation
protected bonds).

PART 5
Comment on how the yields to maturity of Treasury bonds have changed since 3 rd April 2017
as below. How does the market appear to have changed its predictions?

Year-
bonds

US US TIPS
Treasury inflation
protected securities

Australia Japan Germany UK

Cash 0.81 – 1.5% 0.10% 0% 0.25%
3-months 0.75% – – – – –
6-months 0.9% – – – – –
12-months 1.02% – – – – –

7
2-year 1.25% – 1.74% – 0.20% – 0.75% 0.11%
5-year 1.92% – 0.16% 2.23% – 0.13% – 0.39% 0.55%
10-year 2.39% 0.40% 2.69% 0.06% 0.32% 1.14%
15-year – – 3.06% – – –
20-year – 0.54% – 0.62% – –
30-year 3.01% 0.91% – 0.84% 1.1% 1.72%

PART 6
How does the “liquidity hypothesis” impact your interpretation? (namely, the idea that long
term bonds are more sensitive to changes in interest rates going forward, and hence more
risky, for which they require a “liquidity premium” – as much as an addition 1.5% on the
YTM compared with a short-term Treasury bill, all else equal). Second Assignment: Interrogating Bonds.

PART 7
In the light of your findings, comment on the excerpt below from The Economist:

Buttonwood 4-10 March 2017.
If there is one aspect of the current era sure to obsess the financial historians of
tomorrow, it is the unprecedented low level of interest rates. Never before have
deposit rates or bond yields been so depressed in nominal terms, with some
governments even able to borrow at negative rates. It is taking a long time for
investors to adjust their assumptions accordingly. Second Assignment: Interrogating Bonds.

8
Real interest rates (ie, allowing for inflation) are also low. As measured by inflation-
linked bonds, they are around minus 1 % in big rich economies.

PART 8
Comment on how you see the implications of your findings for the stock market. Second Assignment: Interrogating Bonds.