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Mini-case 1 (provide all the answers in an Excel file)
Mini-case 1 consists of a series of problems and a case study. The main purpose of this mini-case is to solidify
your understanding of interest rates.
Problem 1 (20 points) – understanding how to compute duration
Consider two bonds: bond XY and bond ZW. Bond XY has a face value of \$1,000 and 10 years to maturity
and has just been issued at par. It bears the current market interest rate of 7% (i.e. this is the yield to maturity
for this bond). Bond ZW was issued 5 years ago when interest rates were much higher. Bond ZW has face
value of \$1,000 and pays a 13% coupon rate. When issued, this bond had a 15-year, so today its remaining
maturity is 10 years. Both bonds make annual coupon payments.
a) (5 points) What is the price of Bond ZW, given that market interest rates are 7%?
b) (15 points) Compute the duration for both bonds (use Excel).
Problem 2 (35 points) – understanding the determinants of duration
In this exercise, you are going to analyze first the relationship between interest rates and bond prices, and
then the effect of time to maturity, interest rates and coupon rates on duration.
a) (5 points) First, consider a 10 year bond with a coupon rate of 7% and annual coupon payments. Draw a
graph showing the relationship between the price and the interest on this bond. The price should be on the yaxis and the interest rate on the x-axis. To compute the various prices, consider interest rates between 2% and
12% (use 0.5% increments). So your x-axis should go from 2%, then 2.5% … until 11.5% and then 12%.
Is the relationship linear (i.e. is the slope constant)? Start at 7%. If interest rates go up or down by 0.5% is
the price changing by the same amount? What type of relationship do we observe between prices and interest
rates (liner, concave, convex or something else)?
b) (5 points) Now consider the same bond with 10 year maturity, a face value or \$1,000, a coupon rate of 7%
(coupon is paid annually) and assume that the yield to maturity on the bond is 7%. Compute the duration of
this bond.
c) (5 points) Next, we are going to analyze the effect of time to maturity on the duration of the bond.
Compute the duration of a bond with a face value of \$1,000, a coupon rate of 7% (coupon is paid annually)
and a yield to maturity of 7% for maturities of 2 to 18 years in 1-year increments (so here we are going to vary
the time to maturity and see how duration changes if N=2, 3 … etc.). What happens to duration as maturity
increases?
d) (5 points) Next, we are going to analyze the effect of the yield to maturity on the duration of the bond.
Compute the duration of a bond with a face value of \$1,000, a coupon rate of 7% (coupon is paid annually)
and a maturity of 10 years as the interest rate (or yield to maturity) on the bond changes from 2% to 12%
(consider increments of 1% – so you need to compute the duration for various yields to maturity 2%, 3%, …,
12%) . What happens to duration as the interest rate increases?
e) (5 points) Finally, we are going to analyze the effect of the coupon payment on the duration of the bond.
Compute the duration of a bond with a face value of \$1,000, a maturity of 10 years and a yield to maturity of
7%. Compute the duration for coupon rates ranging from 2% to 12% (in increments of 1%). What happens
to duration as the coupon rate increases?
1
Problem 3 (5 points) – understating expected value and standard deviation
You own a \$1,000 face value, zero-coupon bond that has 5 years of remaining maturity. You plan on selling
the bond in one year and believe that the required yield next year will have the following probability
distribution:
Probability
0.1
0.1
0.6
0.1
0.1
Required Yield
5.50%
5.75%
6.00%
6.25%
6.50%
a. What is your expected price when you sell the bond?
b. What is the standard deviation?
Case study (40 points) – understanding the term structure of interest rates
Go to Harvard Business Publishing and buy the following case study:
https://hbsp.harvard.edu/import/632066
After reading the case study, answer the following questions (you also need to find Estrella’s study from the
NY Fed – a link is provided in the case study). Also, you can use other sources as long as you cite them. To
find the current yield curve (as well as historical yield curves you can go to
https://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/Historic-Yield-DataVisualization.aspx)
Please answer the following questions (your answers should be less than 1000 words or less than 2 pages):
1.
(15 points) Estrella (NY Fed) is quite certain that the yield curve is a good predictor
of
future economic activity. From the case, or the link to his FAQs, answer the following questions:
a.
How successful is the yield curve at predicting recessions?
b.
What matters most – the level of the term spread, the change in the spread, or the level of
short term interest rates?
c.
Discuss why a yield curve inversion should lead to a recession.
2.
(15 points) Dick Berner (Morgan Stanley) is a bit more skeptical about the
predictive power of the yield curve. Does he just not understand Estrella’s
overwhelming evidences, or does his skepticism rest on solid reasoning?
3.
(10 points) How is the U.S. yield curve currently sloped? What does it affect your forecast of
economic activity?
2
For the exclusive use of j. wang, 2019.
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Rev. Aug. 23, 2018
The Yield Curve and Growth Forecasts
As Arturo Rodrigo was riding the early morning Metro-North train from Manhattan to Greenwich,
Connecticut, in August 2018, his thoughts turned to the US yield curve. At this point, the US yield curve was
upward-sloping because the Federal Reserve (Fed) had kept short rates quite low, and long rates remained
historically low at around 3% (Figure 1). That said, if one took the difference between the 10-year rate and the
three-month rate as a measure of the slope of the yield curve, the current curve had been flattening over the
past few years (Figure 2).
Figure 1. Long and short rates.
10-Year Treasury Note Yield at Constant Maturity
% p.a.
Federal Funds [effective] Rate
% p.a.
20
20
16
16
12
12
8
8
4
4
10yr
Fed Funds
0
80
85
0
90
95
00
05
10
15
Source: Federal Reserve Board/Haver Analytics
Note: All figures were created by case writer. Monthly data updated through July
2018. Quarterly data updated through 2018Q2.
This case was prepared by Francis E. Warnock, the James C. Wheat Jr. Professor of Business Administration at the University of Virginia’s Darden
School of Business. Copyright  2011 by the University of Virginia Darden School Foundation, Charlottesville, VA. All rights reserved. To order copies,
send an e-mail to sales@dardenbusinesspublishing.com. No part of this publication may be reproduced, stored in a retrieval system, used in a spreadsheet, or transmitted in
any form or by any means—electronic, mechanical, photocopying, recording, or otherwise—without the permission of the Darden School Foundation. Our goal is to publish
materials of the highest quality, so please submit any errata to editorial@dardenbusinesspublishing.com.
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Figure 2. The term spread (10-year Treasury rate less three-month Treasury rate).
Slope of the US Yield Curve
10yr Treasury Rate less 3mth Treasury Rate
3.75
3.75
3.00
3.00
2.25
2.25
1.50
1.50
0.75
0.75
0.00
0.00
-0.75
-0.75
90
95
00
05
10
15
Source: Haver Analytics
Rodrigo’s task was to sift through the evidence on the predictive content of the yield curve. The US
economy was as healthy as it had been in over a decade and much of the global economy was, by all accounts,
firing on all cylinders. But did the substantial flattening of the US yield curve mean that a recession was likely?
Or did the fact that the yield curve was still positively sloped portend continued strenghths? From time to time,
the market focused on the slope of the yield curve. Rodrigo wanted to learn more about what the yield curve
could and could not tell him.1
The Yield Curve is a Good Predictor of Recessions…
Some of the strongest statements about the predictive content of the slope of the yield curve have come
from the Federal Reserve Bank of New York, primarily through the work of Arturo Estrella, an economist.2 A
summary of the information made available by the New York Fed follows:

The slope of the yield curve—the difference between long-term and short-term interest rates—predicts
subsequent US real GDP with a lead time of about four to six quarters. The relationship is positive;
when the yield curve is positively sloped (i.e., when long rates are higher than short rates), economic
growth should be strong over the next four to six quarters (Figure 3), and when the yield curve is
inverted, growth should subsequently slow. The interest rates used are those on Treasury securities, to
minimize the impact of any potential credit risk premium, and while many maturities can be used, the
10-year Treasury bond and three-month Treasury bill seem to yield the most robust results. The yield
curve has a stellar record, predicting every US recession since 1950; its only “false” signal foresaw a
recession in all but name—the credit crunch and slowdown in production of 1967.
1 For a primer on models of interest rate determination as well as discussions on yield curves and the relationship between short and long rates, see
Francis E. Warnock, “The Determinants of Interest Rates,” UVA-BP-0489 (Charlottesville, VA: Darden Business Publishing, 2006).
2
For
articles,
answers
to
FAQs,
and
a
“recession
probability”
chart
that
is
updated
monthly,
see
http://www.newyorkfed.org/research/capital_markets/ycfaq.html (accessed Aug. 16, 2018).
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Although the level of the term spread is what has predictive power, a given change (say, 50 basis points)
is more meaningful the smaller the magnitude of the spread. That is, while a change in the slope from
350 to 300 basis points might not be meaningful, the same 50-point decrease when the slope is much
flatter would greatly increase the probability of recession. Note that the slope of the yield curve is
indeed one of the Conference Board’s leading indicators of economic activity.

The source of a change in the term spread—that is, whether it was caused by movements in the short
end or the long end (or both)—does not matter. Nor does one need to wait for an inversion; it is the
level of the term spread, not whether the slope is positive or negative, that matters. Signals that last
only a day should probably be ignored, but those that last a month or more are likely important. Finally,
the yield curve’s slope does quite well on its own; including additional variables in the model does not
improve the predictive power.
Figure 3. GDP growth and the term spread.
Real Gross Domestic Product
% Change – Year to Year
Term Spread: 10yr – 3mth
3.75
6
3.00
4
2.25
2
1.50
0
0.75
-2
0.00
-4
-0.75
90
95
00
05
10
15
Source: Bureau of Economic Analysis/Haver Analytics
…Or Is It?
…the empirical linkage to weaker economic growth is tenuous, at best.
—Brian Sacki
Not everyone was convinced of the predictive power of the yield curve. In 2005, Brian Sack—at the time
an economist at Macroeconomic Advisers before moving to the New York Fed to head its Financial Markets
Group—provided evidence to the contrary. He did not deny that one could find econometric evidence
supportive of a role for the term spread. Indeed, he showed one such model in which real GDP growth over
the subsequent year is explained by real GDP growth over the previous year (to capture any inertia in economic
growth rates) and the current slope of the yield curve. He used the 10-year yield and the federal funds rate
(similar results would be found if a three-month rate was used for the short rate) and estimated the regression
using quarterly data from 1961 through 2004. Sack found that, sure enough, the slope of the yield curve and
subsequent economic growth were positively related.
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But Sack was not convinced. He noted that the standard error in such estimates was so large that the 95%
confidence interval spans seven percentage points; that is, a prediction of 3.5% growth should be read as “I am
confident that real GDP growth next year will be between 0% and 7%.” Moreover, he worried that the statistical
significance found in such regressions was not robust for two reasons. First, any statistical relationship should
be robust to the exclusion of a few observations; that is, the relationship should not be driven by a few overly
influential outliers. He noted that if the particularly sharp inversions of 1971, 1975, 1981, and 1983 were
removed, the statistical relationship disappeared. Second, any statistical relationship should be robust to being
estimated over different time periods. Sack noted that if one began the sample in 1984, the statistical relationship
disappeared.
Sack’s final concern is worthy of another paragraph or two. He noted that at time t the 10-year rate could
be written as the expected average of the next 10 one-year rates plus a term premium (tp), or (Equation 1)

𝑖
𝐸 𝑖
𝑡𝑝 ,
(1)
where the superscripts denote the maturity of the bond and E[.] denotes expected value. Subtracting the first
one-year rate from both sides of Equation 1 produces an equation (Equation 2) that shows that the slope of
the yield curve is an average of expected changes in the short rate plus the term premium:
𝑖
𝑖

𝐸 ∆𝑖
𝑡𝑝 .
(2)
Sack argued that evidence supportive of the predictive role of the slope of the yield curve focused just on
the first term of the right-hand side of Equation 2. If the slope was positive, we should expect short rates to
rise, which typically occurred when growth was strong; if the slope was negative, we should expect short rates
to fall, as often occurred during an economic slowdown.
But the second term—the term premium—is also important to consider. All else equal, if the term premium
tightened, the yield curve would flatten. Would that suggest slower growth? Sack argued the opposite—that a
compression of term premium would make a given monetary policy stance more stimulative, as long rates fell.
If changes in the slope of the yield curve were driven by changes in the term premium, yield curve flattening
would be stimulative, which is exactly the opposite of findings that a flattening precedes a growth slowdown.
So Sack came to a conclusion directly counter to the impressive evidence amassed by the New York Fed.
But he had a constructive conclusion. What really mattered, in his opinion, was the real (i.e., inflation-adjusted)
federal funds rate. If the real funds rate was high, growth should slow. When it was low (or negative), growth
should pick up. To be sure, the real funds rate and the term spread are correlated with one another—when the
real funds rate is high, the nominal funds rate is also high, and all else equal, the yield curve will be flatter—but
the relationship is not exceedingly tight (Figure 4).
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Figure 4. The term spread and the real federal funds rate.
Term Spread
10yr Treasury less 3mth Treasury
Real Federal Funds Rate
nominal fed funds less trailing one-year CPI inflation
6
6
r = -0.62
4
4
2
2
0
0
-2
-2
-4
Term Spread
Real Funds Rate
90
95
-4
00
05
10
15
Source: Haver Analytics
The Term Premium
Sack reminded us that we must consider how the term premium is evolving when assessing the predictive
ability of the yield curve. So how exactly does it evolve?
To answer that, Rodrigo first had to figure out what exactly, in a tangible sense, the term premium was.
This was a disappointing exercise. In practical, tangible terms, the term premium does not exist, but is an
unobservable construct. There were different ways to estimate a term premium, but at the end of the day, they
all produce just that—an estimate.
But even if the term premium was not observable, Rodrigo could still think about it conceptually. If the
notion that the slope of the yield curve had predictive power for future economic growth was really a statement
about the future-expected short-rates aspect of the yield curve, it would be important to be aware of changes
in the term premium component of long rates. Sack thought the term premium had compressed around 2005,
thus naturally flattening the yield curve a bit. Richard Berner, chief US economist at Morgan Stanley, agreed.ii
In 2005, Berner noted that institutions’ newfound demand for long-duration bonds squeezed the term
premium. In his own work, he focused on demand by pension funds for long-dated assets; a surge in such
demand would compress the term premium. He also noted work that originated at the Federal Reserve Board
(FRB) on foreign governments’ increased demand for Treasury bonds.3 The increased demand for long-dated
assets would, as both Berner and the FRB research noted, tend to flatten the yield curve. All else equal, rather
than signaling poor economic growth prospects, this flattening would be stimulative because it would put
downward pressure on borrowing costs.
3 The FRB work was first released as Francis E. Warnock and Veronica Cacdac Warnock, “International Capital Flows and U.S. Interest Rates,”
International Finance Discussion Papers 840 (September 2005), available at http://www.federalreserve.gov/Pubs/Ifdp/2005/840/default.htm (accessed Feb.
7, 2018), later published in Journal of International Money and Finance 28 (2009): 903–19.
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In 2005, Berner put forward another factor behind a compression in the term premium: the Fed’s conduct
and policy. The Great Moderation,4 with its historically low inflation and its stability, meant that investors did
not need to be compensated as much for holding long-dated fixed-rate assets. If volatility was low, you would
not require as much protection against price and reinvestment risk. Moreover, the Fed’s move toward greater
transparency also eliminated some uncertainty. Put it all together, and Berner argued that the term premium
had been permanently compressed.
Subsequent estimates of the term premium from researchers at the Federal Reserve Bank of New York
suggested that the term premium had not been permanently compressed, but was somewhat volatile (Figure 5).
Readily apparent in the ACM term premium was the compression during the conundrum years of 2004 to 2006
as well as during the Fed’s quantitative-easing policies. Indeed, other than during the one-year period following
the May 2013 “taper tantrum,” the ACM term premium had been near zero since year-end 2011.
Figure 5. The ACM term premium.5
ACM Term Premium: 10 Year
EOP, %
6
6
4
4
2
2
0
0
-2
-2
80
85
90
95
00
05
10
15
Source: Federal Reserve Bank of New York/Haver Analytics
The Yield Curve in the Darke …
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