A.6.4 Spot and Forward Rates - Appendix: Mathematics of the Mean

Appendix: Mathematics of the Mean–Variance Model

I. A.6.4 Spot and Forward Rates

Most consumers are familiar with financial transactions involving market spot rates. When financing a car or a home, loan payments are generally determined using the prevailing interest rates. But there are situations when a consumer is interested in what will happen to interest rates in the future. Suppose you purchase a house and have two financing options: a 30-year fixed rate at 6.4% or a 7-year adjustable-rate mortgage at 5.5%. If you select the 7-year adjustable rate mortgage, you are concerned with what happens to interest rates in the future as the interest rate applied to the outstanding principal changes in line with variations in a benchmark interest rate such as LIBOR.

Similarly, in many commercial transactions borrowers and lenders agree now to make a loan in the future. A forward rate contract on interest rates is known as a forward rate agreement and is discussed in Section I.A.7.3.

Consider again the four zero-coupon investment strategies introduced in Section I.A.6.2.

Suppose you knew with 100% certainty that if you were to buy a 2-year bond one year from now it would cost you $780, while a year bond would cost you $740. Also, if you were to buy a 1-year bond two 1-years from now it would cost you $840. Within minutes you have all your calculations done. You can see how you got the yields shown in Table I.A.6.2. For example, the yield of a 1-year bond bought one year from now is given by:

1,2

1,1

1 1000 1 35.1%

740 B

Table I.A.6.2: Bond yields versus maturity Matures at t =

Bought at t = 1 2 3

0 11.1% 11.8% 12.6%

1 35.1% 13.2%

2 19.0%

So, if you were to invest $1 today, and possibly reinvest your payoff, depending on your choice, at the end of 3 years your total payoff would be any of the four shown in Figure I.A.6.6.

Figure I.A.6.6: Returns for four alternative scenarios of investing in bonds for 3 years

t = 0 t= 1 t= 2 t= 3 Scenario

$1.428 (1)

$1.000

$1.250 $1.487 (2)

$1.111 $1.424 (3)

$1.111 $1.501 $1.786 (4)

12.6% p.a. for 3 yrs

19.0% p.a. for 1 yr 35.1% p.a. for 1 yr

11.8% p.a. for 2 yrs 19.0% p.a. for 1 yr

11.1% p.a. for 1 yr

13.2% p.a. for 2 yrs

In a world with fixed or certain future interest rates, your investment choice would be easy:

scenario (4) in Figure I.A.6.6 would give you the highest payoff for your investment of $1.786.

This equates to an average annual yield of ( 1.786 1) 21.3%³ versus 12.6%, 14.1% and 12.5%

for scenarios (1)–(3), respectively.

There are two types of interest rates illustrated in the example above.

The spot rate is the prevailing interest rate on a zero-coupon bond. As in Example I.A.6.3, r0,tdenotes a spot rate that is observed today and continues until time t. In Figure I.A.6.6, the spot rate for a 1-year bond at time 0 is 11.1% and the spot rate for a 3-year bond at time 0 is 12.6%. When valuing bonds, the spot rate is commonly used to calculate the present value of the cash flows because there is no concern about the reinvestment rate for the coupons received over time. We also use the notation rx,t for x > 0, to denote the spot rate that is observed at time x > 0 and prevails until time t.

A short rate is a one-period interest rate that prevails at some current or future time t. The current one-period spot interest rate is r0,1 and, more generally, the short rate at some future time t – 1 that prevails during the tth year is rt –1,t. Future short rates are unknown, but in a world with no uncertainty a future short rate rt –1,t that is ‘solved’ or ‘implied’ by the current spot rates is also called the forward rate, ft –1,t.³ Note that forward rates are marginal rates, whereas a yield is an average rate.

3 Section I.A.6.5 will examine possible relationships between forward rates and future short rates when future short rates are uncertain.

Table I.A.6.3: Forward rates versus maturity Forward

rates (%)

Maturity (years)

2 1

4 2

5 3

Let us explore the relationship between the spot rate and short rates, assuming that future short rates are known. Suppose that in the interest-rate market today, the forward rates in Table I.A.6.3 are available. The current spot rates are implied by the known forward rates. One dollar invested at the 2-year spot rate will be worth $1 at the end of the 2-year term.

Similarly, if you invest $1 at the 1-year spot rate, r

(1r0,2)

0,1, and roll over the proceeds at the prevailing forward rate, r1,2, at the end of the 2-year period, your investment is worth $1 . These two investment alternatives need to produce the same payoff, under the assumption of known forward rates, otherwise a risk-free profit can be made. Hence

0,1 1,2

(1r )(1r )

$1(1r_0,2)² $1(1r_0,1)(1 f_1,2)

r or

2 .

$1(1r0,2) $1(1 0.02)(1 0.04)

Solving the above equation gives the current spot rate a value of 2.995% derived from the known forward rates.

Example I.A.6.4

Find the 3-year spot rate given the forward rates in Table I.A.6.3.

Investing $1 at the prevailing spot rate for a 3-year period has a value of $1(1 + r0,3)³. Similarly, investing the $1 for 1-year periods with the proceeds rolled over at the prevailing short rate will be worth $ at the end of the 3-year period. Both investments must produce the same payoff, therefore

0,1 1,2 2,3

1(1r )(1r )(1r )

$1(1r_0,3)³ $1(1r_0,1)(1r_1,2)(1 _2,3) or

$1(1r_0,3)³ $1(1 0.02)(1 0.04)(1 0.05) .

Solving this gives a spot rate of 3.659%.

There are two important observations from the above examples. First, the spot rate is a geometric average of the short rates. Second, the yield curve is a graph of the spot rates for bonds with identical risk and different maturities.

Table I.A.6.4: Spot rates versus maturity Spot rates

(%)

Maturity (years)

2 1

4 2

6 3

Now consider turning the situation around – we know the spot rates and want to find the forward rates. Assume the spot rates in Table I.A.6.4. Investing $1 for a 2-year term at 4% must be equivalent to investing for a year at the 1-year short rate, 2%, and reinvesting the proceeds for a second year at the short rate, r1,2.

0,2 0,1 1,2

$1(1r ) $1(1r )(1 r )

r or

$1(1 0.04) $1(1 0.02)(1 r1,2).

Solving for r1,2 gives a 1-year short rate starting in year 2 of 6.04%.

Example I.A.6.5

Find the 1-year short rate starting in year 2 using the data in Table I.A.6.4.

Investing $1 for 3 years at the prevailing spot rate must result in the same payoff as investing at the certain short rates for 1-year terms:

0,3 0,1 1,2 2,3

$1(1r ) $1(1r )(1r )(1 ) or

$1(1 0.04) $1(1 0.02)(1 0.0604)(1 r2,3). Solving for r2,3 shows the short rate at the end of year 2 is approximately 4%.

We need to clarify that the forward rate for year t is a rate implied from a spot yield curve. While it is not always an observable rate of interest, people use the forward rate to denote what the

‘prevailing’ interest rate for a certain year ‘will be’. Suppose the short rates for years 1, 2 and 3 are known and denoted as r0,1, r1,2 and r2,3, respectively; then the following relationship will hold:

If we do not know a certain short rate, we can ‘discover’ it by solving the above equations. For example, if we knew the yields for a 1-year bond bought at time t = 0 and a 2-year bond bought at time t = 0, we can use:

to get the short rate at the end of year 1 that will prevail during year 2:

In general, the short rate for year t is given by the recursive formula:

In the real world, which is dynamic, the yields of bonds of any maturity change over time and as such the forward rate also changes. Suppose you wanted today to calculate the forward rate that will prevail during the third year down the road. You find that the yield of a bond bought today and maturing in 3 years is 12.3%, while the yield of a bond bought today and maturing in 2 years is 11.6%. Your calculation gives you:

Three months later, the prices of these same bonds have changed and their yields are now found to be 12.7% and 11.2%. Repeating your calculation gives you:

So a change in the bond yields of the order of only 3-4% results in more than a 15% increase in the short rate. You may have a hint now that our implied values for future short rates are valid only at the moment we make the calculation. For this reason, in Section I.A.6.5 we will model future short rates as random variables.

Example I.A.6.6: Estimating spot rates from coupon paying US Treasury bonds

Suppose that you see in the Wall Street Journal a list of yields to maturity for US Treasury bonds as in Table I.A.6.5.

Table I.A.6.5: US Treasuries yields to maturity

Maturity (years) Price Yield to maturity

1 100 5.00%

2 100 5.35%

3 100 5.75%

The spot rates can be determined from the yield-to-maturity data of coupon-paying Treasury bonds by using an iterative process commonly referred to as bootstrapping. The price of a bond is simply the present value of the cash flows. The 2-year spot rate r0,2can be found by solving the following equation:

1 2

2 2 2

0,1 0,2 0,2

coupon face coupon 5.35 100 5.35

(1 ) (1 ) 1 0.05 (1 )

P r r

 r

Solving for r0,2 gives:

0.5

0,2

105.35

1 0.053592 100 5.35/1.05

r §¨© ·¸¹ |

or approximately 5.3592%, which is 0.0092% higher than the yield. Similarly, the 3-year spot rate can be found by solving the following equation for r0,3:

2 3

0,3

5.75 5.75 105.75

100 1 0.05(1 0.053592) (1 r )

Solving the equation for r0,3 gives approximately 5.7804%, which is 0.0304% higher than the yield.

Note that all spot rates are higher than the yield to maturity and that the difference between the yield to maturity and the spot rate increases as the maturity of the bond increases.

In document PRM_Handbook (Page 189-195)