Fixed income securities valuation Spot rates and forward rates and bond equivalent yields 1

Learning outcomes At the end of this lesson, the student should be able to: Calculate spot rates Calculate forward rates Understand how to link spot and forward rates Calculate spot rates and forward rates and use them to price fixed income securities 2

Spot rates Yield to maturity is the single discount rate that makes the present value of a bond’s promised cash flows equal to its market price. Actually, the appropriate discount rates for cash flows that come at different points in time are typically not all the same (unless the yield curve is flat, cash flows are identical and there is no liquidity premium) 3

Conceptually, spot rates are the discount rates for (yields on) zero-coupon bonds, securities that have only a single cash flow at a future date. Annual spot yield curves are often published by the financial press or by central banks The spot yield curve can be used to estimate the price or value of a bond. 4

In general, any bond can be viewed as a package of zero-coupon instruments. That is, each zero-coupon instrument in the package has a maturity equal to its coupon payment date or, in the case of the principal, the maturity date. Obviously, in the case of each coupon bond the value or price of the bond is equal to the total value of its component zero-coupon instruments. 5

If the value of the bond does not equal the value of all the component zero-coupon instruments, it would be possible for a market participant to generate riskless profits by stripping off the coupon payments and creating stripped securities. Continued arbitrage of this kind will restore prices of the bond and bond strips to equilibrium 6

Example Consider an annual-pay bond with a 10% coupon rate and three years to maturity. This bond will make three payments. For a sh1,000 bond these payments will be sh100 in one year, sh100 at the end of two years, and sh1,100 three years from now. Suppose we are given the following spot rates: 1 year = 8% 2 year = 9% 3 year = 10% 7

Discounting each promised payment by its corresponding spot rate, we can value the bond as: 100/ / / =

In the same question, calculate the forward rates for year 2 and 3, and use these to discount the cash flows and obtain an identical answer 9

The corresponding 1 year forward rates are 8%, 10.01% and 12.03% for year 1, 2 and 3 respectively. The first period spot rate and first period forward rate are identical 10

Example 2 Calculate the 6 month, 1 yr and 18 month annualized spot rates 11

We are dealing with semi-annual periods here so the first spot rate is 2.5% or 5% BEY The second spot rate is found by equating the market value to the present value of the coupons as 100= 3/ /(1+S 2 /2) 2 S 2 /2=3.008% S 2 = 6.015% on a BEY basis 12

The third spot rate is found as 100= 3.5/ / /(1+S 3 /2) 3 S 3 /2= S 3 =7.048% on a BEY basis 13

Example 3 Given the following spot rates (in Bond Equivalent Yield form): 0.5 years = 4% 1.0 years = 5% 1.5 years = 6% Calculate the value of a 1.5 year, 8% Treasury bond. 14

Answer The value of the bond is then found as follows: Taking a par value of sh100 and semi-annual pay, V= 4/ / / = sh

Forward rates We can say that the spot rate is the yield to maturity of a zero-coupon bond that has a stated maturity, where zero-coupon bonds are sold at a discount from their par value and pay no coupons. We could also say that the forward rate is the yield to maturity of a zero-coupon bond that an investor agrees to purchase at some future specified date. 16

A concept related to the forward rate is the forward contract, which is an agreement between a buyer and a seller to trade something in the future at a price negotiated today A forward contract is obligatory to both the buyer and the seller. The forward contract for interest rates is known as a forward rate agreement, FRA 17

An FRA is an OTC derivative contract based on a notional principal representing a future borrowing amount that the borrower would wish to lock in now. If we look at future spot rates as a product of expected forward rates then we can determine which rate of interest can be locked in now for a predetermined future period This rate is what we call the forward rate 18

For example, a forward contract to buy $1 million of par value of Treasury bills at a 6% discount rate (which determines the bond’s price) in six months obliges the buyer to purchase the T-bills at a 6% discount rate; it also obliges the seller to sell the T-bills at the same price. If T-bills are selling for a 7% discount rate in six months when the forward contract matures and the buyer delivers the bills, the buyer of the contract will make a loss as the T-bills have a lower market price. 19

A loss arises because the buyer is obliged to purchase the T-bills at 6% despite the fact that a 7% rate is available in the market. At a 7% rate, the buyer could purchase the T-bills at the lower market price, but the buyer must buy them at the 6% rate (a higher price) to comply with the forward contract. A range of actively traded forward contracts is available in interest rates, currencies and energy products 20

The forward interest rates can be derived from the spot rates of bonds with various maturities. To understand this, consider the following example: 21

Suppose we wish to invest for two years. Consider the following investment strategies: Strategy 1: invest in a two-year zero-coupon bond and earn 5.8% (the two-year spot rate). Strategy 2: invest in a one-year zero-coupon bond and earn 5%. Also enter into a one-year forward rate agreement (FRA) to invest in one year. What interest rate on the FRA will make strategies 1 and 2 equivalent? 22

It will be the forward rate that results in an overall annual rate of return of 5.8% for two years. This is the same as the compounded annual yield of the two year zero-coupon bond in strategy 1 To understand this, consider investing $1 in each bond, and let R i denote the spot rate and f i denote the forward rate for each year, i = 1, 2. 23

Strategy 1: $1(1 + R 2 ) 2 = $1( ) 2 = $ Strategy 2: $1(1 + R 1 ) = $1( ) = $1.05. Then invest $1.05 in the FRA. The forward rate that makes strategies 1 and 2 equivalent is $1.05(1 + f 2 ) = $ , or f 2 = , or 6.606%. 24

Thus, in equilibrium we have (1 + R 2 ) 2 = (1 + R 1 )(1 + f 2 ) Note that if f 2 is higher than 6.606%, then all investors will be better off not buying the two-year bond. Its price will fall, and R 2 will go up until the equation (1 + R 2 ) 2 = (1 + R 1 )(1 + f 2 ) holds. The opposite is true if f 2 is smaller than 6.606%. 25

Similarly, for a three-year period there are three alternative strategies: Strategy 1: invest in a zero-coupon bond with three years to maturity and earn 6.3% (the three-year spot rate). Strategy 2: invest in a one-year zero-coupon bond, enter into a one-year FRA to invest in one year, and enter again into a one-year FRA to invest in two years. Strategy 3: invest in a two-year zero-coupon bond and enter into a one-year FRA to invest in two years. 26

Following the same analysis as before, the return on all of these strategies must be the same. Hence, we arrive at the following equilibrium: (1 + R 3 ) 3 = (1 + R 1 )(1 + f 2 )(1 + f 3 ) 27

However, because in equilibrium, as we have seen before, (1 + R 2 ) 2 = (1 + R 1 )(1 + f 2 ), this can be rewritten as: (1 + R 3 ) 3 = (1 + R 2 ) 2 (1 + f 3 ) and for the given spot rates R 2 and R 3, f 3 can be determined. 28

This type of analysis could be conducted for n periods in order to arrive at the following general expression of equilibrium: (1 + R n ) n = (1 + R 1 )(1 + f 2 )(1 + f 3 )... (1 + f n ) Or (1 + R n ) n = (1 + R n−1 ) n−1 (1 + f n ) 29

Spot interest rates can be thought of as a portfolio of agreements for forward contracts. If the yield curve is upward-sloping, then the implied forward rates are higher than the short-term spot rate. 30

Concept check 1 If the current 1-year rate is 2%, the 1-year forward rate ( 1 f 1 ) is 3% and the 2-year forward rate ( 1 f 2 ) is 4%, what is the 3-year spot rate? 31

By current 1 year rate we mean the interest rate you pay if you borrow now for a one year period By 1 year forward we mean the rate you pay if you borrow in 1 years time for a one year period By 2 year forward, we mean the rate you pay if you borrow in 2 years time for a one year period S 3 = [(1.02)(1.03)(1.04)] 1/3 – 1 = 2.997% 32

This can be interpreted to mean that a shilling compounded at 2.997% for three years has the same ending value as a shilling that earns compound interest of 2% the first year, 3% the next year, and 4% for the third year. 33

Concept check 2 The 2-period spot rate, S 2, is 8% and the current 1- period (spot) rate is 4% (this is both S 1 and 1 f 0 ). Calculate the forward rate for one period, one period from now, 1 f 1. 34

(1 + S 2 ) 2 = (1 + 1 f 0 )(1 + 1 f 1 ) (1.08) 2 = (1.04)(1 + 1 f 1 ) 1 f 1 = =12.154% 35

In other words, investors are willing to accept 4.0% on the 1-year bond today (when they could get 8.0% on the 2-year bond today) only because they can get % on a 1-year bond one year from today. This future rate that can be locked in today is a forward rate. 36

Concept check 3 Let’s extend the previous example to three periods. The current 1-year spot rate is 4.0%, the current 2- year spot rate is 8.0%, and the current 3-year spot rate is 12.0%. Calculate the 1-year forward rates one and two years from now. 37

(1 + S 2 ) 2 = (1 + S 1 )(1 + 1 f 1 ) (1.08) 2 = (1.04)(1 + 1 f 1 ) 1 f 1 = % (1 + S 3 ) 3 = (1 + S 1 )(1 + 1 f 1 ) (1 + 1 f 2 ) Is the same as (1 + S 3 ) 3 = (1 + S 2 ) 2 (1 + 1 f 2 ) (1.12) 3 = (1.08) 2 × (1 + 1 f 2 ) 1 f 2 =20.45% 38

Concept check 4 We can also calculate implied forward rates for loans for more than one period. E.g. Given spot rates of: 1-year = 5%, 2-year = 6%, 3-year = 7%, 4-year = 8% we can calculate 2 f 2. 39

The implied forward rate on a 2-year loan two years from now is: ( / ) = =10.038% 40

Concept check 5 Assume the following spot rates (as BEYs). i. What is the 6-month forward rate one year from now? ii. What is the 1-year forward rate one year from now? iii. What is the value of a 2-year, 4.5% coupon Treasury note? Assume a par value of $100 41

i. =6.205% in BEY ii. =6.405% in BEY iii. $

End of section exercises 1-3 (Basic) 1. Based on semiannual compounding, what would the YTM be on a 15-year, zero-coupon, sh1,000 par value bond that’s currently trading at sh331.40? 2. An analyst observes a bond with an annual coupon that’s being priced to yield 6.350%. What is this issue’s bond equivalent yield? 3. An analyst determines that the cash flow yield of a fund is 0.382% per month. What is the bond equivalent yield? 43

End of section exercises 4-8 (Intermediate) Calculate the five year spot and forward rates assuming annual compounding 44 Years to maturity Par coupon YTM Calculated spot rates Calculated forward rates

5 From the following treasury yields as at January , calculate the implied forward one-year rate of interest on January Term to maturityYield to maturity 1 year3.50% 2 years4.50% 3 years5.00% 4 years5.50% 5 years6.00% 10 years6.60%

6 Compute the two-year implied forward rate three years from now 46 Term to maturityCurrent coupon YTMSpot rate of Interest 1-year treasury5.25% 2-year treasury year treasury year treasury year treasury year treasury

7 T he following is a list of prices for zero-coupon bonds of various maturities. Calculate the yields to maturity of each bond and the implied sequence of forward rates. 47

8 Consider the following $1,000 par value zero-coupon bonds: According to the expectations hypothesis, what is the expected 1-year interest rate 3 years from now? 48

References Investments by Bodie, Kane and Marcus 10ed Investment Analysis and Portfolio Management 10ed by Reilly and Brown CFA Curriculum Level 1 Fixed Income