Debt Valuation

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.

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, being all the other features fixed? Why?

• 3. Provide the definitions of a discount bond and a premium bond. Give examples.

• 4. All else equal, which bond’s price is more effected by a change in interest rates, a bond with a large coupon or a small coupon? Why?

• 5. What is the difference between the forward price and the value of a forward contract?

• 6. Someone 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.

• 7. A futures price can be assimilated to a stock paying a dividend yield. What is the dividend yield in the futures case?

• 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.

• 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	2 years	5 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. Repeat the calculations of duration, modified duration and convexity, using shortcut formula, by changing the yields by 0.2 %.

  • b. Compare the results with those found in Exercise 4 and comment.

• 14. An investor holds 100,000 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
  18 years	9.5 %	8 %	9.14	114.18 €

  Characteristics of the hedging instrument, which is here a bond are as follows:

  Maturity	Coupon rate	Yield	Duration	Price
  20 years	10 %	8 %	9.49	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.

• 15. Consider the two bonds in exercise 7.

  • a. When 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. changes?

• 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 in order to provide for the 1,100,000 € payback. First asset is a non-interest 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.

• 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: Principal Component Analysis of the Term Structure

The term structure can be alternatively described by using a Principal Component Analysis (PCA). The changes in the term structure (ΔTS), by means of principal components x _i can then be defined by

$$ \Delta TS=\left(\Delta {x}_1,\Delta {x}_2,\dots, \Delta {x}_n\right) $$

Knowledge of matrix calculus is needed in order 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

$$ \Delta {x}_i={\displaystyle \sum_{i=1}^n{\eta}_{ij}\Delta {y}_j},\ j=1,2,\dots n $$

where

• η _ij are the principal component coefficients

• y _j 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 non-negative eigenvalues. Looking at eigenvalues in order of size, the highest eigenvalue corresponds to a specific eigenvector, whose elements are identified 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. And so on, for all eigenvalues.

So 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

$$ {\sigma}_{y,i}^2=\frac{\lambda_i}{{\displaystyle \sum_{i=1}^m{\lambda}_i}} $$

(5.9)

From condition of independency of eigenvectors, it follows that the matrix of coefficients η _ij is orthogonal, so that its inverse corresponds to the transpose. Equation (5.9) can then be inverted, to get the interest rates, as

$$ \Delta {y}_j={\displaystyle \sum_{i=1}^n{\eta}_{ij}\Delta {x}_i},\ j=1,2,\dots n $$

From how the model is built, it is clear that 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

$$ \Delta {y}_j={\displaystyle \sum_{i=1}^m{\eta}_{ij}\Delta {x}_i+{\varepsilon}_i} $$

where

• ε _i 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 to each factor a unit variance, so 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 gets the form

$$ \Delta {y}_j={\displaystyle \sum_{i=1}^m\left({\eta}_{ij}\sqrt{\lambda_i}\right)\frac{\Delta {x}_i}{\sqrt{\lambda_i}}+{\varepsilon}_i} $$

so that in an equivalent equation, the product of eigenvalue and eigenvector is isolated. The new factor loading in parenthesis measures the impact of one standard deviation move in each principal component.