Abstract
Debt is one of the possible sources of capital for the company. When the debt is issued on the market, it takes the form of bonds and complements the capital obtained as debt from banks.
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Appendices
Problems
-
1.
Explain what is meant by basis risk when futures contracts are used for hedging.
-
2.
Which bond’s price is more affected by a change in interest rates, a short-term bond or a longer-term bond, with all the other features fixed? Why?
-
3.
Provide the definitions of a discount bond and a premium bond. Give examples.
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4.
All else equal, which bond’s price is more affected by a change in interest rates, a bond with a large coupon or a bond with a small coupon? Why?
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5.
What is the difference between the forward price and the value of a forward contract?
-
6.
Some have argued that airlines have no point in using oil futures given that the chance of oil price being lower than futures price in the future is the same as the chance of it being lower. Discuss this.
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7.
A futures price can be assimilated to a stock paying a dividend yield. What is the dividend yield in the futures case?
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8.
The annual effective yield on a bond is 7%. A 5-year bond pays coupons of 5% per year in semiannual payments. Calculate the duration.
-
9.
Calculate the modified duration and convexity of the bond in exercise 8.
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10.
Prove that the duration of a portfolio of many assets is the weighted average of all durations of the single assets.
-
11.
Consider the following portfolio:
Bond
Coupon
Maturity
Par value
Price value
YTM
1
7.0
5
10,000,000
9,209,000
9.0%
2
10.5
7
20,000,000
20,000,000
10.5%
3
6.0
3
30,000,000
28,050,000
8.5%
Determine the yield to maturity of the portfolio.
-
12.
Consider the two bonds in the following table:
 | Bond A | Bond B |
---|---|---|
Coupon | 8% | 9% |
Yield to maturity | 8% | 8% |
Maturity | 2Â years | 5Â years |
Par | €100.00 | €100.00 |
Price | €100.00 | €104.055 |
-
(a)
Compute the duration and modified duration for the two bonds.
-
(b)
Compute the convexity for the two bonds.
-
13.
Recall the two bonds in exercise 4.
-
(a)
The calculations of duration, modified duration, and convexity were repeated using the shortcut formula by changing the yields by 0.2%.
-
(b)
Compare the results with those found in Exercise 4 and comment.
-
(a)
-
14.
An investor holds 100,000Â units of a bond whose features are summarized in the following table. He wishes to be hedged against a rise in interest rates.
Maturity
Coupon Rate
Yield
Duration
Price
18Â years
9.5%
8%
9.50
€114.18
The characteristics of the hedging instrument, which is here a bond, are as follows:
Maturity
Coupon Rate
Yield
Duration
Price
20Â years
10%
8%
9.87
€119.79
Coupon frequency and compounding frequency are assumed to be semiannual. YTM stands for yield to maturity. The YTM curve is flat at an 8% level.
-
(a)
What is the quantity of hedging instrument that the investor has to trade? What type of position should the investor take on the hedging instrument?
-
(b)
Suppose that the YTM curve increases instantaneously by 0.1%. Calculate the corresponding new price for the two bonds.
-
(a)
-
15.
Consider the two bonds in exercise 7.
-
(a)
When the YTM curve increases instantaneously by 0.1%, what happens to the portfolio in terms of profits or losses when the portfolio is not hedged? What if it is hedged?
-
(b)
If the curves shifts by 2% instead, how does the answer to point a. change?
-
(a)
-
16.
A bank is required to pay €1,100,000 in 1 year. There are two investment options available with respect to how funds can be invested now to provide for the €1,100,000 payback. The first asset is a noninterest bearing cash fund, in which an amount x will be invested, and the second is a 2-year zero-coupon bond earning the 10% risk-free rate in the economy, in which an amount y will be invested.
-
(a)
Develop an asset portfolio that minimizes the risk that liability cash flows will exceed asset cash flows.
-
(a)
-
17.
What position is equivalent to a long forward contract to buy an asset at K on a certain date and a put option to sell it for K on that date?
-
18.
How can a forward contract on a stock with a particular delivery price and delivery date be created from options?
Appendix 5.1: Principal Component Analysis of the Term Structure
The term structure can be alternatively described by using principal component analysis (PCA). The changes in the term structure (ΔTS), by means of principal components xi, can then be defined by
Knowledge of matrix calculus is needed to apply the method. There is a unique change in the key rates for each realization of the principal components, where the latter are linear combinations of changes in the interest rates, given as
where
-
ηij are the principal component coefficients.
-
yj is the yield corresponding to maturity j.
Each component explains the maximum percentage of the total residual variance not explained by previous components. The matrix of zero-coupon rates is symmetric with m independent eigenvectors, corresponding to m nonnegative eigenvalues. Looking at eigenvalues in order of size, the highest eigenvalue corresponds to a specific eigenvector, whose elements are identified as the coefficients of the first principal component.
The second highest eigenvalue corresponds to another specific eigenvector, whose elements are identified as the coefficients of the second principal component—in addition, so on, for all eigenvalues.
Therefore, the variance of each component is given by the size of the corresponding eigenvalue, and the proportion of total variance of the interest changes explained by the i-th principal component is
From the condition of independency of eigenvectors, it follows that the matrix of coefficients ηij is orthogonal, so its inverse corresponds to the transpose. Equation (5.9) can then be inverted to obtain the interest rates as
From how the model is built, it is clear that the lowest eigenvalues play a very little role in determining the changes in interest rates. Therefore, it is possible to reduce the dimensionality of the model to the m highest eigenvalues, as given by
where
-
εi is an error term due to the approximation from reduced dimensionality.
The first k components are then able to give a sufficiently accurate approximation of the changes in interest rates. The portfolio sensitivity to these components can be used to define the IRR profile.
Difference variances for each principal component are implied by the model, with corresponding even (i.e., unitary) shift in all components, making them not equally likely.
A further step involves giving each factor a unit variance to make changes in each factor comparable. Again, from matricial calculus, the unit variance is obtained by multiplying each eigenvector by the square root of the corresponding eigenvalue so that the model obtains the form
so that in an equivalent equation, the product of the eigenvalue and eigenvector is isolated. The new factor loading in parentheses measures the impact of one standard deviation move in each principal component.
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Corelli, A. (2023). Debt Valuation. In: Analytical Corporate Finance. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-031-32319-5_5
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