Spot And Forward Rates, Part 1

Fixed Income 6 min readModule 1 of 6Save

What you'll learn

LOS 23.a
Describe relationships among spot rates, forward rates, yield to maturity, expected and realized returns on bonds, and the shape of the yield curve.
LOS 23.b
Describe how zero coupon rates (spot rates) may be obtained from the par curve by bootstrapping.

Spot Rates

The price today of $1 par, zero coupon bond is known as the discount factor, which we will call $P_{T}$ . Because it is a zero coupon bond, the spot interest rate is the yield to maturity of this payment, which we represent as $S_{T}$ . The relationship between the discount factor $P_{T}$ and the spot rate $S_{T}$ for maturity T can be expressed as:

P_{T} = \frac{1}{( 1 + S _{T} ) ^{T}}

The term structure of spot rates, the graph of the spot rate $S_{T}$ versus the maturity T, is known as the spot yield curve or spot curve. The shape and level of the spot curve changes continuously with the market prices of bonds.

Forward Rates

The annualized interest rate on a loan to be initiated at a future period is called the forward rate for that period. The term structure of forward rates is called the forward curve. (Note that forward curves and spot curves are mathematically related, we can derive one from the other.)

We will use the following notation:

Yield to Maturity

As we’ve discussed, the yield to maturity (YTM) or yield of a zero coupon bond with maturity T is the spot interest rate for a maturity of T. However, for a coupon bond, if the spot rate curve is not flat, the YTM will not be the same as the spot rate.

Worked example: Spot rates and yield for a coupon bond

Compute the price and yield to maturity of a three year, 4% annual pay, $1,000 face value bond given the following spot rate curve: $S_{1}$ = 5%, $S_{2}$ = 6%, and $S_{3}$ = 7%.

Solution

Calculate the price of the bond using the spot rate curve:

Price = \frac{40}{( 1.05 )} + \frac{40}{( 1.06 ) ^{2}} + \frac{1040}{( 1.07 ) ^{3}} = $922.64

Calculate the yield to maturity ( $y_{3}$ ):

N = 3; PV = −922.64; PMT = 40; FV = 1,000; CPT I/Y → 6.94 $y_{3}$ = 6.94%

Note that the yield on a three year bond is a weighted average of three spot rates, so in this case we would expect $S_{1}$ < $y_{3}$ < $S_{3}$ . The yield to maturity $y_{3}$ is closest to $S_{3}$ because the par value dominates the value of the bond and therefore $S_{3}$ has the highest weight.

Expected and Realized Returns on Bonds

Expected return is the ex ante holding period return that a bond investor expects to earn.

The expected return will be equal to the bond’s yield only when all three of the following are true:

The bond is held to maturity. All payments (coupon and principal) are made on time and in full.

All coupons are reinvested at the original YTM.

The second requirement implies that the bond is option free and there is no default risk.

The last requirement, reinvesting coupons at the YTM, is the least realistic assumption. If the yield curve is not flat, the coupon payments will not be reinvested at the YTM and the expected return will differ from the yield.

Realized return on a bond refers to the actual return that the investor experiences over the investment’s holding period. Realized return is based on actual reinvestment rates.

The Forward Pricing Model

The forward pricing model values forward contracts based on arbitrage free pricing.

Consider two investors.

Investor A purchases a $1 face value, zero coupon bond maturing in j+k years at a price of $P_{(j + k)}$ .

Investor B enters into a j year forward contract to purchase a $1 face value, zero coupon bond maturing in k years at a price of $F_{(j, k)}$ . Investor B’s cost today is the present value of the cost: PV[ $F_{(j, k)}$ ] or $P_{j}$ $F_{(j, k)}$ .

Because the $1 cash flows at j+k are the same, these two investments should have the same price, which leads to the forward pricing model:

P_{(j + k)} = P_{j} F_{(j, k)}

Therefore:

F_{(j, k)} = \frac{P _{(j + k)}}{P _{j}}

Worked example: Forward pricing

Calculate the forward price two years from now for a $1 par, zero coupon, three year bond given the following spot rates.

The two year spot rate, $S_{2}$ = 4%.

The five year spot rate, $S_{5}$ = 6%.

Solution

Calculate discount factors $P_{j}$ and $P_{(j + k)}$ .

P_{j} = P_{2} P_{(j + k)} = P_{5} = 1/ (1 + 0.04)^{2} = 0.9246 = 1/ (1 + 0.06)^{5} = 0.7473

The forward price of a three year bond in two years is represented as $F_{(2, 3)}$

F_{(j, k)} F_{(2, 3)} = P_{(j + k)} / P_{j} = 0.7473/0.9246 = 0.8082

In other words, $0.8082 is the price agreed to today, to pay in two years, for a three year bond that will pay $1 at maturity.

The Forward Rate Model

The forward rate model relates forward and spot rates as follows:

[1 + S_{(j + k)}]^{(j + k)} = (1 + S_{j})^{j} [1 + f (j, k)]^{k}

[1 + f (j, k)]^{k} = [1 + S_{(j + k)}]^{(j + k)} / (1 + S_{j})^{j}

This model is useful because it illustrates how forward rates and spot rates are interrelated.

This equation suggests that the forward rate f(2,3) should make investors indifferent between buying a five year zero coupon bond versus buying a two year zero coupon bond and at maturity reinvesting the principal for three additional years.

Worked example: Forward rates

Suppose that the two year and five year spot rates are $S_{2}$ = 4% and $S_{5}$ = 6%.

Calculate the implied three year forward rate for a loan starting two years from now [i.e., f(2,3)].

Solution

\begin{aligned}[1+f(j,k)]^k&=[1+S_{(j+k)}]^{(j+k)}/(1+S_j)^j\\[1+f(2,3)]^3&=[1+0.06]^5/[1+0.04]^2\\f(2,3)&=7.35\%\end{aligned}

Note that the forward rate f(2,3) > $S_{5}$ because the yield curve is upward sloping.

If the yield curve is upward sloping, [i.e., $S_{(j + k)}$ > $S_{j}$ ], then the forward rate corresponding to the period from j to k [i.e., f(j,k)] will be greater than the spot rate for maturity j+k [i.e., $S_{(j + k)}$ ]. The opposite is true if the curve is downward sloping.

A par rate is the yield to maturity of a bond trading at par. Par rates for bonds with different maturities make up the par rate curve or simply the par curve. By definition, the par rate will be equal to the coupon rate on the bond.

Generally, par curve refers to the par rates for government or benchmark bonds.

By using a process called bootstrapping, spot rates or zero coupon rates can be derived from the par curve. Bootstrapping involves using the output of one step as an input to the next step. We first recognize that (for annual pay bonds) the one year spot rate ( $S_{1}$ ) is the same as the one year par rate.

We can then compute $S_{2}$ using $S_{1}$ as one of the inputs. Continuing the process, we can compute the three year spot rate $S_{3}$ using $S_{1}$ and $S_{2}$ computed earlier. Let’s clarify this with an example.

Worked example: Bootstrapping spot rates

Given the following (annual pay) par curve, compute the corresponding spot rate curve:

Maturity	Par Rate
1	1.00%
2	1.25%
3	1.50%

Solution

$S_{1}$ = 1.00% (given directly).

If we discount each cash flow of the bond using its yield, we get the market price of the bond. Here, the market price is the par value. Consider the 2 year bond.

100 = \frac{1.25}{( 1.0125 )} + \frac{101.25}{( 1.0125 ) ^{2}}

Alternatively, we can also value the 2 year bond using spot rates:

100100 98.7624 = \frac{1.25}{( 1 + S _{1} )} + \frac{101.25}{( 1 + S _{2} ) ^{2}} = \frac{1.25}{( 1.01 )} + \frac{101.25}{( 1 + S _{2} ) ^{2}} = 1.2376 + \frac{101.25}{( 1 + S _{2} ) ^{2}} = \frac{101.25}{( 1 + S _{2} ) ^{2}}

Multiplying both sides by $[(1 + S_{2})^{2} /98.7624]$ , we get $(1 + S_{2})^{2} = 1.0252$ .

Taking square roots, we get (1 + $S_{2}$ ) = 1.01252. $S_{2}$ = 0.01252 or 1.252%

Similarly,

100 = \frac{1.50}{( 1 + S _{1} )} + \frac{1.50}{( 1 + S _{2} ) ^{2}} + \frac{101.50}{( 1 + S _{3} ) ^{3}}

Using the values of $S_{1}$ and $S_{2}$ computed earlier,

100100 97.0517 (1 + S_{3})^{3} (1 + S_{3}) = \frac{1.50}{( 1.01 )} + \frac{1.50}{( 1.01252 ) ^{2}} + \frac{101.50}{( 1 + S _{3} ) ^{3}} = 2.9483 + \frac{101.50}{( 1 + S _{3} ) ^{3}} = \frac{101.50}{( 1 + S _{3} ) ^{3}} = 1.0458 = 1.0151 and hence S_{3} = 1.51%

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