if observed yield is plotted as a function of time to maturity for a variety of bonds within a fixed risk class, the result is a scatter of points that can be approximated by a curve—the yield curve This curve typically rises gradually with increasing maturity, reflecting the fact that long maturity bonds typically offer higher yields than short maturity bonds. The shape of the yield curve varies continually, and occasionally it may take on an inverted shaped, where yields decrease as the time to maturity increases,
Fixed-income securities are best understood through the concept of the term structure of interest rates. In this structure there is, at any time, a specified interest rate for every maturity date. This is the rate, expressed on an annual basis, that would apply to a zero-coupon bond of the specified maturity These underlying interest rates are termed spot rates, and if they are plotted as a function of time to maturity, they determine a spot rate curve, similar in character to the yield curve, However, spot rates are fundamental to the whole interest rate market—unlike yields, which depend on the payout pattern of the particular bonds used to calculate them Once spot rates are determined, it is straightforward to define discount factors for every time, and the present value of a future cash flow is found by discounting that cash flow by the appropriate discount factor. Likewise, the present value of a cash flow stream is found by summing the present values of the individual flow elements.
A series of forward rates can be inferred from a spot rate curve. The forward rate between future times /s and h is the interest rate that would be charged for borrowing money at time and repaying it at time r2> but at terms arranged today These forward rates are important components of term structure theory
There are three main explanations of the characteristic upward sloping spot rate curve. The first is expectations theory. It asserts that the current implied forward rates for 1 year ahead—that is, the forward rates from year 1 to future dates—are good estimates of next year's spot rates. If these estimates are higher than today's values, the current spot rate curve must slope upward The second explanation is liquidity preference theory. It asserts that people prefer short-term maturities to long-term maturities because the interest rate risk is lower with short-term maturities This preference drives up the prices of short-term maturities The third explanation is the market segmentation theory According to this theory, there are separate supply and demand forces in every range of maturities, and prices are determined in each range by these forces Hence the interest rate within any maturity range is more or less independent of that in other ranges Overall it is believed that the factors in all three of these explanations play a role in the determination of the observed spot rate curve
Expectations theory forms the basis of the concept of expectations dynamics, which is a particular model of how spot rates might change with time According to expectations dynamics, next year's spot rates will be equal to the current implied forward rates for i year ahead—the rates between year 1 and future years In other words, the forward rates for 1 year ahead actually will be realized in I year. "This prediction can be repeated for the next year, and so on This means that all future spot rates are determined by the set of current forward rates Expectations dynamics is only a model, and future rates will most likely deviate from the values it delivers; but it provides a logical simple prediction of future rates As a special case, if the current spot rate curve is flat—say, at 12%—then according to expectations dynamics, the spot rate curve next year will also be flat at 12%. The in variance theorem states that if spot rates evolve according to expectations dynamics, the interest earned on funds committed to the interest rate market for several years is independent of how those funds are invested
Present value can be calculated by the running method, which starts from the final cash flow and works backward toward the first cash flow At any stage k of the process, the present value is calculated by discounting the next period's present value using the short rate at time k that is implied by the term structure This backward moving method of evaluation is fundamental to advanced methods of calculation in various areas of investment science
Duration can be extended to the term structure framework The key idea is to consider parallel shifts of the spot rate curve, shifts defined by adding a constant A to every spot rate. Duration is then defined as (—1/P) dP/dA evaluated at A — 0 Fisher-Weil duration is based on continuous-time compounding, which leads to a simple formula In discrete time, the appropriate, somewhat complicated formula is termed quasi-modified duration.
Once duration is defined, it is possible to extend the process of immunization to the term structure framework. A portfolio of assets designed to fund a stream of obligations can be immunized against a parallel shift in the spot rate curve by matching both the present values and the durations of the assets and the obligations.
L (One forward rate) If the spot rates for 1 and 2 years are s\ = 6 3% and .v2 — 6 9%, what is the forward rate f\ 2'?
2. (Spot update) Given the (yearly) spot rate curve s = (5 0, 5 3, 5 6, 5 8, 6 0, 6 I), find the spot rate curve for next year
3. (Construction of u zero) Consider two 5-year bonds: one has a 9% coupon and sells for 101.00; the other has a 7% coupon and sells ior 93 20 Find the price of a 5-year zero-coupon bond
4. (Spot rate projects) It is November 5 in the year 2011 The bond quotations of Table 4 6 are available Assume that all bonds make semiannual coupon payments on the 15th of the month T he fractional part of a bond's price is quoted in I/32nd's Estimate the (continuous-time) term structure in the form of a 4th-order polynomial, i (/) = r/(> + a\t + d2t2 + «:)/'' +
Coupon |
Maturity |
Ask price |
«i |
Feb-2012 |
100:0 |
H |
Feb-2012 |
100:22 |
■n 7 ' 8 |
Aug-2012 |
100:24 |
H |
Aug-2012 |
101:1 |
»J |
Feb-2013 |
101:7 |
81 |
Feb-2013 |
101:12 |
8 |
Aug-2013 |
100:26 |
8J |
Aug-2013 |
102:1 |
«Î |
Feb-2014 |
98:5 |
»1 |
Feb-2014 |
102:9 |
«ä |
Aug-2014 |
97:13 |
M |
Aug-2014 |
101:23 |
Feb-2015 |
99:5 | |
»i |
Feb-2015 |
109:4 |
si |
Aug-2015 |
101:13 |
loi |
Aug-2015 |
107:27 |
n |
Feb-2016 |
99:13 |
H |
Feb-2016 |
103:0 |
where t is time in units of years from today. The discount rate for cash Hows at time t is accordingly d{t) = e~rU)1 Recall that accrued interest must be added to the price quoted to get the total price Estimate the coefficients of the polynomial by minimizing the sum of squared errors between the total price and the price predicted by the estimated term structure curve Plot the curve and give the five polynomial coefficients
5,. (Instantaneous rates o) Let $(/), 0 < / < oo, denote a spot rate curve; that is, the present value of a dollar to be received at time I is e~x{nt For /] < t2, let f {t\, h) be the forward rate between /j and 12 implied by the given spot rate curve
(b) Let /■(/) = lim,,-, f(t,h) We can call / (/) the instantaneous interest rate at time t Show that r(/) = $(r) + *'(/)/.
(c) Suppose an amount .vo is invested in a bank account at i = 0 which pays the instantaneous rate of interest r(t) at all / (compounded) Then the bank balance ,v(/) will satisfy dv(/)/d/ = /(/).v(/). Find an expression for .v{/) [Hint. Recall in general that vdz + -dv ~ d(y^).
6. (Discount conversion) At time zero the one-period discount rales da.\,d\ 2,d2i, , i/5.6 are known to be 0 950, 0.940, 0 932, 0 925, 0.919, 0 913 Find the time zero discount factors d{) 2, , i/a<,
7., (Bond taxes) An investor is considering the purchase of 10-year U S Treasury bonds and plans to hold them to maturity Federal taxes on coupons must be paid during the year they are received, and tax must also be paid on the capital gain realized at maturity (defined as the difference between face value and original price) Federal bonds are exempt from state taxes This investor's federal tax bracket rate is / = 30%, as it is for most individuals There are two bonds that meet the investor's requirements Bond I is a 10-year, 10% bond with a price (in decimal form) of P\ = 92 21 Bond 2 is a 10-year, 7% bond with a price of P2 = 75 84 Based on the price information contained in those two bonds, the investor would like to compute the theoretical price of a hypothetical 10-year zero-coupon bond thai had no coupon payments and required tax payment only at maturity equal in amount to 30% of the realized capital gain (the face value minus the original price) This theoretical price should be such that the price of this bond and those of bonds I and 2 are mutually consistent on an after-tax basis Find this theoretical price, and show that it does not depend on the tax rate t. (Assume ail cash flows occur at the end of each year)
8. (Real zeros) Actual zero-coupon bonds are taxed as if implied coupon payments were made each year (or realty every 6 months), so tax payments are made each year, even though no coupon payments are received The implied coupon rate for a bond with n years to maturity is (100— fit)/«, where P(\ is the purchase price If the bond is held to maturity, there is no realized capital gain, since ail gains are accounted for in the implied coupon payments Compute the theoretical price of a real 10-year zero-coupon bond. This price is to be consistent on an after-tax basis with the prices of bonds 1 and 2 of Exercise 7
9. (Flat forwards) Show explicitly that if the spot rate curve is flat [with .v(A-) = / for all k], then all forward rates also equal /
10. (Orange County blues) Orange County managed an investment pool into which several municipalities made short-term investments A total of $7.5 billion was invested in this pool, and this money was used to purchase securities Using these securities as collateral, the pool borrowed $12 5 billion from Wall Street brokerages, and these funds were used to purchase additional securities The $20 billion total was invested primarily in long-term fixed-income securities to obtain a higher yield than the short-term alternatives. Furthermore, as interest rates slowly declined, as they did in 1992-1994, an even greater return was obtained Things fell apart in 1994, when interest rates rose sharply
Hypothetrcally, assume that initially the duration of the invested portfolio was 10 years, the short-term rate was 6%, the average coupon interest on the portfolio was 8.5% of face value, the cost of Wall Street money was 7%, and short-term interest rates were falling at per year
(a) What was the rate of return thai poo! investors obtained during lliis early period? Does it compare favorably with the 6% that these investors would have obtained by investing normally in shori-term securities?
(b) When interest rates had fallen two percentage points and began increasing at 2% per year, what rate of return was obtained by the poo!?
11. (Running PV example) A (yearly) cash How stream is x = (-40, 10, 10, 10, 10, 10, 10). The spot rates are those of Exercise 2.
(i/) Find the current discount factors di)k and use them to determine the (net) present value of the stream
(h) Find the series of expectations dynamics short-rate discount factors and use the running present value method to evaluate the stream.
12. (Pure duration o) It is sometimes useful to introduce variations of the spot rates that are different from an additive variation Let s° ~ (y1/, s", v", ., v") be an initial spot rate sequence (based on m periods per year). Let s(i) = (vi, v2, . , v„) be spot rates parameterized by À, where
for A- = 1,2, , ».. Suppose a bond price P(X), is determined by these spot rates Show that
is a pure duration; that is, find D and describe it in words
13. (Stream immunization A company faces a stream of obligations over the next 8 years as shown: where the numbers denote thousands of dollars The spot rate curve is that of
Year |
1 2 |
3 |
4 |
5 |
6 |
7 8 |
500 900 |
600 |
500 |
100 |
100 |
100 50 |
Example 4 8 Find a portfolio, consisting of the two bonds described in that example, that has the same present value as the obligation stream and is immunized against an additive shift in the spot rate curve.
14. (Mortgage division) Often a mortgage payment stream is divided into a principal payment stream and an interest payment stream, and the two streams sue sold separately We shall examine the component values Consider a standard mortgage of initial value M = M(0) with equal periodic payments of amount B If the interest rale used is r per period, then the mortgage principal after the kth payment satisfies
M(k) = (I +r) M(k - 1) -for k — 0,1, This equation has the solution
Let us suppose that the mortgage has n periods and B is chosen so that M(n) = 0; namely,
The Arth payment has an interest component of f(k) ~ t M{k - 1)
and a principal component of
(a) Find the present value V (at rate r) of the principal payment stream in terms of B,t,n,M-
(c) What is the present value IV oi the interest payment stream? {d) What is the value of V as n -»■ oo?
(e) Which stream do you think has the larger duration—principal or interest?
15. {Short rate sensitivity) Gavin Jones sometimes has flashes of brilliance. He asked his instructor if duration would measure the sensitivity of price to a parallel shift in the short rate curve (That is, rk -*■ >k +A. ) His instructor smiied and told him to work it out He was unsuccessful at first because his formulas became very complicated Finally he discovered a simple solution based on the running present value method Specifically, letting Pk be the present value as seen at time k and $k — dA-/dA[}.-<>, the St's can be found recursively by an equation of the form — ~ak Pk 4- bk Sk. while the Pk's are found by the running method Find ak and bk
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