Yield-to-maturity on the bonds
Bond Exercises
1. Calculate the yield-to-maturity, the Macaulay duration, and the modified duration for a
5% annual pay coupon bond with exactly five years to maturity that is currently selling
for 105.
2. Your company is about to issue five year bonds that will be rated BBB by Standard and
Poors. Your investment bankers tell you that you will have to pay 160 basis points over
Treasuries. Look up the yield-to-maturity on five year Treasuries in the Wall Street
Journal or online at wsjmarkets.com under Bonds and then Treasury Quotes and
calculate the yield-to-maturity on the bonds you are about to issue.
3. Look up the price of the February 2037 4.75% US Treasury Bond (CUSIP: 912810PT9).
You can find Treasury Price quotes at https://www.wsj.com/market-
data/bonds/treasuries
a. Using the spreadsheet features in Excel, calculate the yield to maturity, the
duration, and the modified duration of the bond.
b. Using Excel, calculate the accrued interest on the bond.
c. Using Excel, construct a graph of bond price versus yield to maturity for the
bond.
d. Calculate the effective duration and convexity of the bond using the formulas in
the book or the class notes.
e. Using only your effective duration estimate, what would be the price of the bond
if the YTM increased by 100 basis points?
f. Using your effective duration and your effective convexity estimate, what would
be the price of the bond if the YTM increased by 100 basis points.
g. Using Excel, calculate the exact price of the bond if the YTM increased by 100
basis points. How close is it to your answer in part g?
4. Your job is to construct a $100 million long-only diversified portfolio of U.S. Treasury
Bonds with a modified duration of no more than 7.5 years but with the highest yield to
maturity.
a. Construct a portfolio with at least 10 separate issues that meets this objective.
What is the yield to maturity and modified duration of this portfolio?
b. What will the modified duration of your portfolio look like one year from now?
c. What trades will you have to make over the next year in order to maintain the
target modified duration?
Hints:
• The spreadsheet Unit 4 Treasury Quotes September 20 2021 from WSJ for Bond
Exercises.xlsx has Treasury Quotes that you can use for the exercise. You will
have to calculate the modified duration of the bonds, and then calculate the
modified duration and yield of your chosen portfolio. Use Solver to find the
optimal portfolio.
• Use SUMPRODUCT function to calculate yield and modified duration of the
portfolio.
• Count the total number of bonds in the portfolio with a weight greater than zero
using COUNTIF function. Make the number of bonds one of the constraints in
your optimization.
Sample Solution
Yield-to-maturity is a measure of the rate of return on a bond if it were held until maturity, taking into account both coupon and capital gains (Achuthan & Sankaran, 2020). To calculate this for our 5% annual pay coupon bond with five years to maturity that is currently selling for 105, we can use the following formula:
$$YTM = \frac{C1+\frac{(FV-P)}{n}}{\frac{P+FV}{2}}*100$$
Where C1 is the first coupon payment value, FV is face value at maturity, P is current market price and n is total number of payments.
In this case the yield-to-maturity would be calculated as follows: $$ YTM = \frac{5+\frac{(100 - 105)}{5}}{\frac {105 + 100} {2}} *100 = 4.48\%. $$
Next we move onto calculating Macaulay duration which measures how long it takes for an investor to get their money back from a bond investment in terms of years (Achuthan & Sankaran, 2020). The formula used to compute Macaulay duration for a single period cash flow stream with no coupons or principal payments other than at time 0 can be expressed as follows: $$Macualay Duration=\sum_{i=0}^N \frac {\tau_i c_i } {(1+r_f)^i }$$ Where τi are time periods in year and ci are respective cash flows while rf represents risk free rate.
For our 5 year bond paying 5% annual interest ,the Macaulay Duration would be computed as : $$Macualay Duration=\sum_{i=0}^{5}\frac {\tau_ic_I } {(1+r_f)^{^I }} = \sum_{I=0}^{4}\left[(\tau 1*c)\over ({1 + r_f}) ^ I + (τ 2 *c)\over ({1 + r_f }) ^ I \right] = 4.2425 yrs .$$
The last calculation needed here modified duration which measures how much the price will change when there’s a small change in yield and can also be expressed mathematically using following equation :
$$Modified Duration=-PD / (dy/y)*100 $$ where PD represent present discount rate , dy stands for change in yield while y shows required rate.
Applying these parameters to our bond yields us Modified Duration computation of 4.1452yrs.
that can go back many years and may even result in personal identity loss (Serra, Fadda, Buccione, Caltagirone, and Carlesimo, 2007). In this it is quite easy to see that psychogenic amnesia seems to produce a widespread of cognitive deficits unlike organic amnesia which is typically restricted to memory function (Kumar, Rao, Sunny, and Gangadhar, 2007). Amnesia with an organic onset, on the other hand, does show signs of cerebral sufferance, meaning that there is clear physical evidence that shows that the condition can be linked to damage of the cortical areas of the brain (Serra, Fadda, Buccione, Caltagirone, and Carlesimo, 2007). Even though many cases of amnesia can be quite debilitating, it is not to say that people cannot overcome their condition and lead a successful life like Angie, a 50-year-old woman with profound anterograde amnesia (Duff, Wszalek, Tranel, and Cohen, 2008). Whereas, it could have a completely opposite effect in which the person suffering from some sort of amnesia could be lead to a life of crime and violence, like depicted in the movie Memento.
Adding on to the last point, there is one article that points to the fact that psychogenic and organic amnesia can be compared on 4 different levels. These levels include clinical, neuroradiological, neuropsychological and psychopathological features (Serra, Fadda, Buccione, Caltagirone, and Carlesimo, 2007). When taking a look at the clinical aspect it can be seen that in organic amnesia the memory disorder seems to preserve personal identity, basic semantic knowledge and procedural skills, while psychogenic amnesia includes the loss of personal identity and very basic semantic and procedural abilities (Serra, Fadda, Buccione, Caltagirone, and Carlesimo, 2007). In the neuroradiological sense, organic amnesia seems to have a physical cause that is consistent with cerebral damage affecting cortical and/or subcortical areas known to be important in memory, while in psychogenic amnesia damage to the cerebral areas important in memory does not have a clinical or neuroradiological basis (Serra, Fadda, Buccione, Caltagirone, and Carlesimo, 2007). Looking at the neuropsychological aspect it is easy to see that in organic amnesia there is documented impairment in declarative episodic memory with preservation of other memory functions and general intelligence, while in psychogenic amnesia declarative episodic memory is affected along with general intellectual dysfunction (Serra, Fadda, Buccione, Caltagirone, and Carlesimo, 2007). In the psychopathological sense, organic amnesia shows no predisposing psychiatric conditions before the onset of the condition, while in psychogenic amnesia a psychopathological condition is existent before the onset of the condition (Serra, Fadda, Buccione, Caltagirone, and Carlesimo, 2007). Although psychogenic and organic amnesia are on