# In this session, I will generally assume

Session 3 Page 1 In this session, I will generally assume annual compounding, though in one or two cases I will show how formulae simplify if one uses continuous compounding. I make the standard frictionless markets, absence of credit risk assumptions. 3.1. Forward rates Recall that the n-year spot rate, rn, is the rate of interest you get on money from now to year n with interest reinvested. So the price of a zero coupon bond with face value of 1 that matures in n years is If you short a zero coupon bond (ZCB) with maturity n and face value 1, you receive dn today and pay 1 in year n. If you reinvest the dn you receive in zero coupon bonds with maturity n+1, you will be able to buy bonds with a face value of since the cost would be The net effect of this is that you pay \$1 in year n and receive in year n+1. This is a forward deposit in effect it is like an agreement between you and the market that you will deposit \$1 for 1 year in n years time and the rate of interest is fixed now. Transaction Cash flow in year 0 +dn n n+1 Short 1 ZCB with maturity n Buy dn/dn+1 ZCBs with maturity n+1 Net cash flow -1 dn + dn/dn+1 0 -1 + dn/dn+1 The rate of interest fixed in this way is the n-year forward rate The forward rate is the answer to the question: given that I can lock in the rate rn if I invest my money for n years, and rn+1 if I invest for n+1 years, what is the marginal rate I am getting for the n+1th year? To see this algebraically, note we can rearrange the equations to give us ( ) 1 . 1 n n n d r = + d d n n+1 (d d d n n n + + 1 1 ) . \$d d n n+1 1 n 1. n d n f d + = Session 3 Page 2 Applying this repeatedly, you can see that If we had worked with continuously compounded rates, with r(t) being the continuously compounded spot rate for t years (t need not be a whole number), so the price of a ZCB which matures in t years is then the relationship between the spot rate and the instantaneous forward rate f(t) is given by This says that the spot rate to year t is simply the average of the forward rates from 0 to t. This formula holds approximately for discretely compounded rates The following example may be helpful. The forward rate start at % and increase at 1% per year till year 6 when they decline at % per year. The discount factors then decline each year at the forward rate and the spot rates are computed from the discount rates. As you can see, the 10 year spot rate of 3.49% for example is very close to the average of the forward rates for the ten years which is 3.5%. 3.2. Using the Term Structure of Forward rates The spot term structure and the forward term structure contain exactly the same information I have shown how you go from one to the other. But I want to persuade you that the forward term structure presents that information that is particularly useful for bond portfolio managers. I plot below the two curves in this example ( 1 ) 1 Value of \$1 after +1 years 1 n n n r + + Value of \$1 Return for after years final year n 1 1 ( ) 1 n n r + + ( ) 1 . n f = + = + + ( ) ( 0 1 1 ) ( ) ( ) 0 1 1 1 1 1 where is the rate from 0 to 1, or . n r n f f f n f r + = + + + ( ) , r t t e ( ) ( ) 0 . t s r t f s ds t = = 0 1 1 . n n f f f r n + + + Year (n) 0 1 2 3 4 5 6 7 8 9 10 Fwd rate (f n) 0.5% 1.5% 2.5% 3.5% 4.5% 5.5% 5.0% 4.5% 4.0% 3.5% Discount factor (d n) 1.000 0.995 0.980 0.956 0.924 0.884 0.838 0.798 0.764 0.735 0.710 Spot rate (r n) 0.50% 1.00% 1.50% 1.99% 2.49% 2.99% 3.27% 3.42% 3.49% 3.49% Term Structure Session 3 Page 3 The left hand graph shows that you can get get a higher yield from longer maturity bonds (a rising term structure) though the benefit tails as you get to 10 years. The right hand graph shows you something more interesting that you can lock in a rate today of 5.5% on six year money but only 3.5% on ten year money. As a portfolio mnager you may have your own view on what is going to happen to short term interest rates. You might think that rates will rise gradually over the next ten years from the current 0.5% to 3 or 4%. If so you might decide that 5% at 6 years looks nice, but 3.5% at 10 years does not. The forward rate curve invites you to compare what prices the market offers now with what you believe will happen to interest rates in the future. This is likely to lead you to decide which of the rates look high and which you would like to lend at, and which rates look low and you want to borrow at. I will not help you decide what you would like to do; rather I will assume you know what exposure you would like, and help you get that exposure. So suppose you decide that you would like to lock in the six year forward rate on say \$100. You already know how to do it. You go short \$100 face value of the five year ZCB, and invest the proceeds in the six year ZCB. But in practice, in many markets there are no ZCBs or they are illiquid, and you will have to use coupon bonds. You want to construct a portfolio of coupon bonds that locks in the forward rate. This is a bit more complicated than using ZCBs. You again want a portfolio that costs nothing, and that has a cash flow of -\$100 in year 5. But you have a further constraint: you want no cash flow before year 5. You have three constraints, so you will need three bonds. In the spreadsheet I assume there are 5, 6 and 7 year bonds with coupons of 3%, 2% and 5% respectively (all the numbers are arbitrary, and you can change them). I work out the 0% 1% 2% 3% 4% 5% 6% 0 2 4 6 8 10 Spot Curve 0% 1% 2% 3% 4% 5% 6% 0 2 4 6 8 10 Zero Coupon Curve Session 3 Page 4 prices of the bonds using the term structure of interest rates1. The lower panel then shows the portfolio cashflow for any arbitrary mix of the three bonds. I use solver to solve for the mix that produces what I want (zero cost, zero cashflow in years 1-4, and a cashflow of -100 in year 5. You could otherwise solve the three simultaneous equations corresponding to the three constraints, as shown using marices in the spreadsheet. The solution in this case is to go long \$100 of A and short \$71 and \$32 of B and C respectively. It is less neat that the two ZCB solution you also get a cashlow in year 7. But you do lock in a rate of 5.38%, which is an average of f6 and f7. 3.3. The Expectations Hypothesis We have seen that the forward rate fn is more than a number on a chart. It is an interest rate I can lock in on money that I promise to lend in n years. I want to look what happens when I go forward in time, so I need to extend my notation a bit. ft,n is the forward rate computed at time t for 1 year loan starting in n years time that is a loan where the money is promised at time t, lent at time t+n and repaid at time t+n+1. And the one year spot rate at time t is st. Now consider the following transactions: at time t I promise to lend 1 for one year in n years time at rate ft,n. when I get to time t+n, I borrow 1 for one year at the going rate of st+n. I have no cash flow at time t or at time t+n. At time t+n+1, I receive (1+ft,n) repayment of the money I lent, but have to repay (1+st+n) on my loan. So my net cash flow is (ft,nst+n). If the future spot rate is lower than todays forward rate, I make money. 1 In practice of course the term structure would be obtained from the bond prices, not the other way round. Bond Coupon Maturity Price 1 2 3 4 5 6 7 8 9 10 A 3% 5 102.65 3 3 3 3 103 0 0 0 0 0 B 2% 6 94.97 2 2 2 2 2 102 0 0 0 0 C 5% 7 111.71 5 5 5 5 5 5 105 0 0 0 Bond 0 1 -3.00 1.42 1.58 2 -3.00 1.42 1.58 3 -3.00 1.42 1.58 4 5 6 0.00 72.25 1.58 7 0.00 0.00 33.25 8 0.00 0.00 0.00 9 0.00 0.00 0.00 10 0.00 0.00 0.00 A -100.00 -102.65 -3.00 -103.00 B C 70.83 31.67 67.27 35.37 1.42 1.58 1.42 1.58 IRR 5.38% Total 0.00 0.00 0.00 0.00 0.00 -100.00 73.83 33.25 0.00 0.00 0.00 Cash flow in year BOND PRICE CALCULATOR PORTFOLIO CASH FLOW Cash flow in year Face Value Session 3 Page 5 Now if everyone believed that the future spot rate will be lower than todays forward rate, they would lend money forward and make money by financing the loan in the spot market. This would force down the forward rate until the deal looked insufficiently attractive to make it worth doing the trade. Conversely, if everyone believes that the future spot rate will be higher, they will borrow money on the forward market and then deposit it in the spot market. Two conclusions come from this argument: 1. forward rates reflect expectations of future spot rates; 2. if investors are not worried about risk, the expectation of rt+n at time t will be equal to ft,n. The second conclusion, that is the Expectations Hypothesis2. On theoretical grounds, there are good reasons for doubting whether the Expectations Hypothesis holds. An investor in the bond market has a choice between buying long maturity bonds and short maturity bonds. The low risk choice is to match the maturity of the bond to the horizon of the investor. If the investor is concerned by their wealth in ten years time, investing in ten year bonds is risk free. Investing in one year bonds, rolling over the investment each year, is risky because of not knowing in advance what future interest rates will do. If the investors horizon is one year on the other hand, invest in ten year bonds is risky because the price at which they can be sold in one years time depends on what interest rates are then. Much the same argument applies to issuers. If the horizons of investors and borrowers match, then it would not be unreasonable to expect the Expectations Hypothesis to hold. But there is no reason to suppose they do. Many people believe that investors have shorter horizons than borrowers. If that is so you would expect there to be a premium in long term rates to persuade investors to take the risk from going longer than they would ideally like, and to make it more attractive for issuers to issue at shorter maturities than they would like. Furthermore, it is plausible that risk premia might depend on the state of the economy. So the truth of the Expectations Hypothesis is an empirical matter. In the next two 2 Footnote to be ignored unless you are really interested. Strictly, this is only a form of the Expectations Hypothesis. Other types of transaction would give rise to slightly different versions of the Expectations Hypothesis. For example, if in the previous example instead of borrowing 1 at time t+n, you borrowed you would have zero cash flow at time t+n+1 but you would have a cash flow of at time t+n. The Expectations Hypothesis in this case would be that this profit has an expected value of zero, so With interest rates being close to zero, the two formulation are virtually identical in practice. f s t n t t n , = E [ + ], 1 1 , ( + + f s t n t n , ) ( + ) 1 ( f s s t n t n t n , + + ) ( + ) ft n , = E E t t n t n t t n [s s s + + + (1 1 1 + + )] [ ( )]. Session 3 Page 6 sections, I address two issues: does the Expectations Hypothesis hold? and why should practitioners (as opposed to academics) care about whether it holds or not? 3.4. The Empirical Evidence Tests of the Expectations Hypothesis broadly take the form of regressing st+n ft,n on factors that are known at time t. If the hypothesis is correct, difference between forward rates and future spot rates should be mean zero and unpredictable. Much of the work has been done on the US Treasury bond market because it is highly liquid and the results are not likely to be influenced significantly by transaction costs. Some of the main papers are summarised briefly in the lecture slides. There is strong evidence for the existence of a term premium that is that forward rates are generally higher than expected future spot rates, and the effect is increasing in maturity. The fact that the term structure is upward sloping at one point in time proves nothing; it could merely indicate that the market expects interest rates to rise. But the fact that it is much more frequently upward sloping than downward sloping does suggest the existence of a term premium; otherwise, if one wanted to stick with the Expectations Hypothesis, one would have to argue that the market generally (over many decades) expects future interest rates to be higher than current interest rates, which is hard to square with rational expectations. But there is also evidence of something more complicated going on. The shape of the yield curve particularly the curvature (the difference between the three year rate and the average of the one and five year rates) has power to predict the term premium (Cochrane and Piazzesi, 2005, Bond risk premia, American Economic Review 95, 138- 160). There is also evidence that macro-economic factors can predict risk premia (Duffee, 2011, Information in (and not in) the term structure, Review of Financial Studies 24(9), 2895-2934). This is a field of research that has been active for fifty years or more, and which is still developing. 3.5. Implications of the Failure of the Expectations Hypothesis The failure of the Expectations Hypothesis is not just of interest to academics; it is also important for investors and bond issuers. If the Expectations Hypothesis holds, then the implications are straightforward. Forward rates are equal to the markets expectation of future interest rates. Unless they have particular skill in forecasting future interest rates, investors should buy bonds and issuers should issue bonds that match their horizon. Deviation from their preferred horizon leads to extra risk but no extra return. In the presence of risk premia and the empirical evidence for the existence of risk premia seems strong, even if there no complete agreement on what they are the implications are more complex. As an investor, there are potential gains to be made by Session 3 Page 7 deviating from your preferred horizon, and it is worth investing effort to identify how large risk premia are at any time, and to weigh the expected returns against the extra risk you are taking on. 3.6. Conclusions The forward yield curve is a valuable way of looking at the interest rate being charged today for different periods of time in the future. It reflects market expectations of future interest rates, but there is much evidence that the forward rates also include some premium for risk, and quite strong evidence that these risk premia are not constant, but change over time. This suggests that investors should devote resources to estimating risk premia, and should be prepared to take on additional risk by deviating from a pure matching strategy in order to capture some of these premia.

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