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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.

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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.

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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%.

We now ask:

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

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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%.

YOUR ASSIGNMENT

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

http://www.tradingeconomics/united-states/inflation-cpi

https://www.tradingeconomics.com/japan/inflation-cpi

where /**/ above = /united-states/, etc.

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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

Compare your answers for the USA with the country of your choice. Second Assignment: Interrogating Bonds.

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% – – – – –

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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.

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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.