Spot Rate vs. Forward Rates (Calculations for CFA® and FRM® Exams)

Spot Rates

A spot interest rate gives you the price of a financial contract on the spot date. The spot date is the day when the funds involved in a business transaction are transferred between the parties involved. It could be two days after a trade, or even on the same day the deal is completed. A spot rate of 5% is the agreed-upon market price of the transaction based on current buyer and seller action.

Calculating the Price of a Bond using Spot Rates

Formula

The general formula for calculating a bond’s price given a sequence of spot rates is given below:

\({ PV }_{ bond }=\frac { PMT }{ { (1+{ Z }_{ 1 }) }^{ 1 } } +\frac { PMT }{ { (1+{ Z }_{ 2 }) }^{ 2 } } +…+\frac { PMT+Principal }{ { (1+{ Z }_{ n }) }^{ n } } \)

Where:

\({ PV }_{ bond }\)  is the present value of the price of the bond;

PMT is the coupon payment per period;

\( { Z }_{ 1 },{ Z }_{ 2 }\) and \( { Z }_{ n }\) are the spot rates for periods 1,2 and n respectively; and

n is the number of evenly spaced periods to maturity.

Suppose that:

• The 1-year spot rate is 3%;
• The 2-year spot rate is 4%; and
• The 3-year spot rate is 5%.

The price of a 100-par value 3-year bond paying 6% annual coupon payment is 102.95.

$$
\begin{array}{l|cccccc}
\text{Time Period} & 1 & 2 & 3 \\
\hline
\text{Calculation} & \frac {\$6}{{\left(1+3\%\right)}^{ 1 } } & \frac { \$6 }{ { \left( 1+4\% \right) }^{ 2 } } & \frac { \$106 }{ { \left( 1+5\% \right) }^{ 3 } } \\
\hline
\text{Cash Flow} & \$5.83 & +\$5.55 & +\$91.57 & =\$102.95 \\
\end{array}
$$

Spot rates are also applied in determining the yield to maturity of a bond.

Calculating the Yield-to-maturity of a Bond using Spot Rates

Continuing on the same example, this 3-year bond is priced at a premium above par value, so its yield-to-maturity must be less than 6%. We can now use the financial calculator to find the yield-to-maturity using the following inputs:

• n = 3;
• PV = -102.95; (Since this is a cash outflow)
• PMT = 6; (Since this is a cash inflow for the investor)
• FV = 100; (Since this is a cash inflow for the investor)
• CPT => I/Y = 4.92 (Which signifies 4.92%)

The yield-to-maturity is found to be 4.92%, which we can confirm with the following calculation:

$$
\begin{array}{l|cccccc}
\text{Time Period} & 1 & 2 & 3 \\
\hline
\text{Calculation} & \frac {\$6}{{\left(1+4.92\%\right)}^{ 1 } } & \frac { \$6 }{ { \left( 1+4.92\% \right) }^{ 2 } } & \frac { \$106 }{ { \left( 1+4.92\% \right) }^{ 3 } } \\
\hline
\text{Cash Flow} & \$5.719 & +\$5.450 & +\$91.770 & =\$102.95 \\
\end{array}
$$

Forward Rates

In theory, forward rates are prices of financial transactions that are expected to take place at some future point.

A forward rate indicates the interest rate on a loan beginning at some time in the future, whereas a spot rate is the interest rate on a loan beginning immediately. Thus, the forward market rate is for future delivery after the usual settlement time in the cash market.

Deriving Forward Rates from a Set of Spot Rates

\({ Z }_{ n }\), the \(n\)-year spot rate, is a measure of the average interest rate over the period from now until \(n\) years’ time.

The forward rate, \({ f }_{ t,r }\), is a measure of the average interest rate between times \(t\) and \(t + r\). It’s the interest rate agreed today \((t=0)\) on an investment made at time \(t>0\) for a period of \(r\) years.

The one-year forward rate, \({ f }_{ t,1 }\), is therefore the rate of interest from time \(t\) to time \(t +1\). It can be expressed in terms of spot rates as follows:

$$ 1+{ f }_{ t,1 } = \frac{{{(1+{Z}_{t+1})}}^{t+1}}{{(1+{Z}_{t})}^{t}} $$

Alternatively,

Step 1: Use the formula:

$$ 1+{ f }_{ t,1 }=\frac {V_2}{V_1} $$

Where \(V_1\) is the value to which a dollar grows by time \(T_1\) and \(V_2\) is the value to which a dollar grows by \(T_1\).

Step 2: Calculate the interest rate that equates the value of one dollar at time \(T_1\) to the value of one dollar at time \(T_2\).

Example: Calculating the one-year forward rate.

You are given the following spot rates:

• 1-year spot rate: 5%;
• 2-year spot rate: 6%.

Determine the one-year forward rate one year from today, i.e., \(f_{1,1}\).

There are 2 ways to solve this question:

1. Using the Formula

$$ 1+{ f }_{ t,1 } = \frac{{{(1+{Z}_{t+1})}}^{t+1}}{{(1+{Z}_{t})}^{t}} $$

$$ (1 + \text{1-year spot}) × (1 + \text{1-year forward rate at time 1}) = (1 + \text{2-year spot})^2 $$

$$ (1.05)\ × (1\ +\ f_{1,1}) = {(1.06)}^2 $$

$$  (f_{1,1}) = 7.0095%$$

2. Using the Timeline

We use the principle of no-arbitrage. The golden rule to remember while using the timeline is that no matter which route you take to “time travel” from one period to the other- the end result (Future Value) will always be the same.

There are 2 possible routes we could take here

1. Invest for one year at the spot rate of 5% and then invest for another year at the unknown forward rate “F.”
2. Invest for two years at the spot rate of 6%.

As per the assumption of no-arbitrage, we should arrive at the same value for both.

By equating i & ii, we get F = 7%, which is the same as what we got using the formula.

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