IB95R0 Financial Risk Management
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Paper code: IB95R0
Paper title: Financial Risk Management
Exam period: May 2023
Question 1. [120 marks in total]
A floating-rate bond is a security that pays a higher coupon as interest rates go up. In particular, a standard floating-rate bond would pay an annual coupon rate of
c(t) = t1 (t - 1, t) (1)
in year t, where t1 (t - 1, t) denotes the annually compounded one-year rate at time t - 1. In addition, the bond would also pay the face value (100) at the bond’s maturity.
(a) [15 marks] Show that (i) a floating rate bond with a remaining maturity of one year has
a price of 100; and (ii) based on this result, show that a floating rate bond with any remaining maturity T also has a price of 100, if T is a positive integer.1
An inverse floater is a security that pays a lower coupon as interest rates go up (hence the name inverse floater); in 1994, Orange County heavily invested in a (leveraged) inverse floater. For now, consider one with the following coupon rate paid at time t:
c(t) = 25% - 2 × t1 (t - 1, t), (2)
where t1 (t - 1, t) is the annually compounded rate at time t - 1. We assume this inverse floater pays an annual coupon (and, of course, the face value of 100 at maturity), and has a maturity of three years.
(b) [20 marks] Show how we can decompose the inverse floater into a portfolio of basic
securities that include (some units of) a fixed-coupon rate bond, a zero-coupon bond, and a floating-rate bond.
Using the law of one price, express the price of the inverse floater, if the prices of a fixed-rate bond, a zero-coupon bond, and a floating-rate bond are denoted by PC (0, 3), PZ (0, 3), and PFR (0, 3), respectively.
(c) [15 marks] Express the Macaulay duration of the inverse floater, if the durations of a fixed-rate bond, a zero-coupon bond, and a floating-rate bond are denoted by DC (0, 3), DZ (0, 3), and DFR (0, 3), respectively.
(d) [20 marks] Assume that today’s date is 15 February, 1994. Using your previous results and the table below that contains annually compounded interest rates for different maturities, compute the price and Macaulay duration of the inverse floater. Compare the duration to its maturity, and comment on it.
Hint: note that the duration of the floating-rate bond is simply 1, as it should be apparent from the results of part (a) .
15/02/1994 Maturity (years) R (annual) |
13/05/1994 Maturity (years) R (annual) |
1 |
5% |
0.25 |
4.2% |
2 |
5.5% |
0.5 |
4.8% |
3 |
6% |
0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 |
5% 5.5% 6% 7% 7.5% 9% 10% 10.5% 10% 9% |
(e) [15 marks] How would the value of the inverse floater change if interest rates went up by 2%? Explain, and relate the result to part (d).
(f) [15 marks] You hold one unit of the inverse floater, and are worried about interest rate
volatility. You decide to hedge it with a 3-year coupon bond paying 4% annually. How much should you go long/short on this bond in order to make your overall portfolio immune to interest rate changes? What is the value of this portfolio then?
(g) [20 marks] Assume that by 15 May, 1994, interest rates went up as suggested by the table
above. What is the value of the unhedged inverse floater now? What is the value of the portfolio that included the hedge of the inverse floater? Compare the value also to the one obtained in part (e) and comment on your findings.
Solutions.
(a) This is simply the result of (1), namely that the next coupon rate c(o), due in a year, is
set to be the current 1-year interest rate t1 (o - 1, o). With a 1-year remaining maturity,
we have
100 + 100 × c(1) 100 × (1 + t1 (0, 1))
Similarly, with a 2-year maturity, we can note that the last payment (FV and coupon) will have a value of exactly 100 1 year from now:
100 × c(1) 100 + 100 × c(2)
1 + t1 (0, 1) (1 + t2 (0, 2))2
100 × c(1) 1
1 + t1 (0, 1) 1 + t1 (0, 1)
100 × c(1) 100
1 + t1 (0, 1) 1 + t1 (0, 1)
100 + 100 × c(1) 100 × (1 + t1 (0, 1))
1 + t1 (0, 1) 1 + t1 (0, 1)
where in the 3rd row we used (1).
(b) It is straightforward that the inverse floater is a sum of: (i) a long position in one fixed
rate coupon bond with coupon rate cC (o) = 25%; (ii) a short position in two floating rate bonds with coupon rate cFR (o) = t1 (o - 1, o), and (iii) a long position in two zero coupon bonds with maturity 3; all with a face value of 100.
From here we also have PIF (0, 3) = PC (0, 3) - 2 × PFR (0, 3) + 2 × PZ (0, 3).
(c) Regarding duration, we have discussed during the course that a portfolio that invests wi , n = 1, ..., o fractions of total wealth into bonds with prices Pi and durations Di , will have an overall duration of Dp = wi Di . From here, the duration of the inverse floater becomes DIF (0, 3) = DC (0, 3) + DFR (0, 3) + PZ (0, 3) .
(d) See the Excel file, where we calculate the price and duration of the coupon bond, and we make use of the fact that PFR (0, 3) = 100, DFR = 1, and DZ (0, 3) = 3. In particular, we find that the duration of the coupon bond is 2.54, and the value weights are 1.27, -1.68, and 1.41, respectively (which add up to 1), implying that DIF = 5.77, which a lot higher than its maturity. Indeed, as interest rates go up, not only the present value of the (same) cash flows decreases, but the cash flows themselves, too.
(e) We can either calculate the exact value change using the Excel file, in which case we obtain PIF(ne)w = 102.99, or we use the duration: = - × 2% = -10.99%, which implies PIF(ne)w = 106.05. Actually when adjusting the Macaulay duration to get Modified duration, we would normally use 1/(1 + R) if the yield curve is flat, but now that it is
not, this method becomes less precise, and it is a good question how one should do the calculation based on duration.
(f) This coupon bond has a duration of 2.88 (see the Excel file). Since to hedge the interest
rate sensitivity we need to make the overall dollar duration zero, for each unit of the inverse floater we need to solve 0 = 119.15 × 5.77 + 94.72 × 2.88z, i.e., z = -2.52. From here, the overall value of the portfolio is 119.15 + 94.72z = -119.38.
(g) The value of the unhedged floater is 93.78, i.e. dropped a lot from the original price
of 119.15. This is because the asset is very sensitive to changes in rates, especially at the longer maturities. The hedged portfolio value becomes -126.31, i.e., went down a bit from -119.38, suggesting the hedge is not perfect (but still a lot better than not hedging). There are 3 components to this last observation: First, 1/4 of a year passed by, as opposed to immediate changes (and this also changed the price of the floating rate bonds). Second, the yield curve for the 3 relevant points went up on average by 2%, which is large, as opposed to a small change at which point duration hedging would be precise. Third, actually the yield curve change is not a parallel shift: long-maturity yields went up much more, which significantly impacts bond prices. For details, see the Excel file.
Question 2. [80 marks in total]
Consider a currently B-rated bond with a face value of $100 and a coupon rate of 4%. The bond has a maturity of four years. In the following table you are given the one-year transition matrix, which gives you the probabilities of migrating from one rating to another within one year. Ratings are indicated by A, B, and D (D = default). The recovery rate on this bond is 60%.
Rating at Year-End
Initial Rating A B D
A 85% 10% 5%
B 10% 85% 5%
D 0% 0% 100%
In addition, you are given the following three forward zero curves (expected interest rate curves for different maturities) for each credit rating:
Category 1 year 2 year 3 year
A 2.00% 3.00% 3.50%
B 3.00% 4.00% 4.50%
(a) [20 marks] Calculate the one-year forward distribution of the bond price and the expected
value of the bond at the end of year 1.
(b) [10 marks] Assuming that the transition probabilities are identical over time and that
consecutive transitions are independent, calculate an estimate of the probability that this B-rated bond will be upgraded by the end of the 2nd year.
(c) [10 marks] Assuming that the transition probabilities are identical over time and that consecutive transitions are independent, calculate an estimate of the probability that this B-rated bond will not be upgraded by the end of the 4th year.
(d) [20 marks] Assume that the current bond price equals $103. Calculate the 5% and 10% Credit Value at Risk (VaR) of this bond. Describe every step in your calculation. How would you calculate the Credit Expected Shortfall (ES) at the same confidence levels? Explain.
(e) [20 marks] Calculate the mean and the standard deviation of profits/losses. What would
be the 5% VaR assuming that the underlying distribution is Normal with these two moments? Explain what the difference is between this 5% VaR and the one obtained in part (d), and which one should be trusted more. (Hint: the inverse of the cdf of the standard Normal satisfies Φ _1 (0.05) = -1.64.)
Solutions.
(a-e) See xls file.
Question 3. [100 marks in total]
We will be using models for updating covariance estimates and forecasting the future level of covariances. For example, instead of the usual GARCH(1,1) volatility model given by
t(2)+1 = !v + v rt(2) + v t(2) ; (4)
we will be working with the model
covt+1 = !c + cxtyt + c covt ; (5)
where xt and yt are two daily return series with zero means and covt+1 denotes the covariance estimate for day t + 1 made at the end of day t.
(a) [15 marks] Explain the restrictions imposed on parameters !v , v , and v to have an
internally consistent and stable GARCH volatility model (4).
(b) [10 marks] Revisit the requirements listed in your answer in part (a), and explain which
one of them must be imposed on !c , c , and c to have an internally consistent and stable GARCH covariance model (5).
Suppose the covariance model parameters are given by !c = -0:0002, c = 0:02, and c = 0:95.
(c) [15 marks] Derive the long-run average covariance of the model given in (5), and calculate it in this specific numerical example.
(d) [10 marks] Explain why we need to be careful when coupling the model given in (5) with arbitrary models for variances. Hint: think about what you know of values spit out by GARCH models.
In the rest of the exercise, we will consider an EWMA (exponentially weighted moving average) model for variances and covariances jointly, described as follows:
t(2)+1,x = x t(2) + (1 - )x ; (6)
t(2)+1,y = yt(2) + (1 - )y ; and (7)
covt+1 = xtyt + (1 - )covt ; (8)
where e (0; 1).
(e) [25 marks] In an Excel file shared with you, you can find daily closing time series of two
indices, the Dow Jones (DJIA) and the FTSE 100. Assume that returns are, conditional on the volatility, normally distributed with zero mean.
Following the model described in (6)-(8), estimate a time series of the variances and the covariance of the two returns. To initialize your estimator, compute the sample average over the first 5 observations, i.e., set h6 as the simple average of the first 5 squared returns. The exponential decay is applied from that point forward. Use an exponential decay parameter of 入 = 0.96. Report the average model-implied variances and covariances of the two indices for this sample.
(f) [25 marks] Calculate the portfolio variance, and conduct a 1% VaR analysis based on the
EWMA model on a portfolio that invests half in the Dow and half in the FTSE. How often do realized portfolio returns violate the VaR calculated based on the above model? Is this a good model of volatility then? Explain.
Solutions.
(a) GARCH models are used for the conditional variance, but without further restrictions they can lead to a violation of covariance stationarity. The requirements for a GARCH(1,1) process to be covariance stationary are the following:
● Condition 1: wv > 0, av , 8v > 0, for positive variance
● Condition 2: 8v = 0 if av = 0, for identification
● Condition 3: av + 8v < 1, for covariance stationarity
(b) We would still need to have condition 2 for identification, 8c = 0 if ac = 0, and, as we will show in part (d), condition 3 for covariance stationarity, ac + 8c < 1. However, covariance does not have to be positive, and thus we can allow for wv < 0. Note that it would still make sense to keep the parts av , 8v > 0, otherwise covariance would become oscillatory.
(c) As long as ac +8c < 1, which is satisfied here, we can follow the derivation as it is normally done for a GARCH volatility model:
E[crut+1] = wc + ac E[ztyt] + 8c E[crut] = wc + ac E[Et_1 [ztyt]] + 8c E[crut] = wc + ac E[crut] + 8c E[crut] = wc + (ac + 8c )E[crut],
so if we are looking for a setting with E[crut] = E[crut+1], we obtain
wc
1 - (ac + 8c ) .
Plugging in the given numerical coefficients we obtain E[crut] = = -0.0067..
(d) One of the two main issues with variance-covariance models is that they need to satisfy positive definiteness: Left unconstrained, some multivariate volatility models do not always yield positive (semi-)definite conditional covariance matrix estimates. This is the multivariate ‘equivalent’ of a univariate volatility model predicting a negative variance. This is clearly undesirable. Imposing positive definiteness on some models leads to non-linear constraints on the parameters of the models which can be difficult to impose practically.
Translating positive definiteness to our 2-variable case, with z and y, the covariance matrix
would fail to be positive definite if and only if the correlation coefficient was outside the [-1, 1]
range, i.e. if lcru(zt , yt )l > ←7x(2),t 7y(2),t . In fact, the (5), just like the GARCH model, would have
a finite long-term mean to which it would mean revert, but it could easily diverge from it. In the particular case of setting 7x,t and 7y,t constant, we would have to put an upper threshold on the covariance to ensure positive definiteness, which cannot happen if daily returns are normally distributed and hence can take arbitrarily large values.
With multivariate models, requiring correlations between -1 and 1 would not be enough to obtain positive definiteness. In fact, one would also have to ensure that pairwise correlations are consistent with each other. E.g., having 2 variables that are perfectly negatively correlated, it cannot be the case that one of them has a perfect positive while the other has a perfect negative correlation with a 3rd variable.
(e)- (f) See the Excel file. The violation ratio is too high (>2.8%) for a 1% VaR model, but these were actually volatile times towards the end, leading up to the Global Financial Crisis.
2023-06-28