After completing this reading, you should be able to:

- Calculate and interpret the impact of different compounding frequencies on a bond’s value.
- Calculate discount factors given interest rate swap rates.
- Compute spot rates given discount factors.
- Interpret the forward rate, and compute forward rates given spot rates.
- Define par rate and describe the equation for the par rate of a bond.
- Interpret the relationship between spot, forward, and par rates.
- Assess the impact of maturity on the price of a bond and the returns generated by bonds.
- Define the “flattening” and “steepening” of rate curves and describe a trade to reflect expectations that a curve will flatten or steepen.

## Different Compounding Frequencies and Their Effect on Bond Value

Besides annual interest payments, most securities on today’s market have much shorter accrual periods. For example, interest may be payable monthly, quarterly (every 3 months), or semiannually.

This, in turn, leads to different present values or future values of an investment depending on the frequency of compounding employed. Here’s how to calculate the PV and the FV of an investment with multiple compounding periods per year.

The most important thing is to ensure that the interest rate used corresponds to the number of compounding periods present per year.

### Future value

$$ FV=PV{ \left\{ \left( 1+\frac { { r }_{ q } }{ m } \right) \right\} }^{ m\times n } $$

Where:

\({ r }_{ q }\) is the quoted annual rate,

\(m\) represents the number of compounding periods (per year)

Lastly, \(n\) is the number of years

### Present value

Suppose you make \(PV\) the subject of the above formula. You should find that

$$ PV=FV{ \left\{ \left( 1+\frac { { r }_{ q } }{ m } \right) \right\} }^{ -m\ast n } $$

**Example**

Suppose you wish to have $20,000 in your savings account at the end of the next 4 years. Assume that the account offers a return of 10 percent per year, compounded monthly. How much would you need to invest now so as to have the specified amount after the three years?

Solution

First, we write down the formula to use,

$$ PV=FV{ \left\{ \left( 1+\frac { { r }_{ q } }{ m } \right) \right\} }^{ -m\ast n } $$

Second, we establish the components that we already have:

\({ r }_{ q }\)=0.10

\(m\)=12

\(n\)=4 years

Then we factor everything into the equation to find our \(PV\)

$$ PV=20000{ \left\{ \left( 1+\frac { 0.1 }{ 12 } \right) \right\} }^{ -12\times 4 }=$13,429 $$

Therefore, you will need to invest at least $13,429 in your account to ensure that you have $20,000 after three years.

Note that this it could be done with the financial calculator with the following inputs:

N = 12*4 = 48; I/Y = 10/12 = 0.833; PMT = 0; FV = 20,000;

CPT => PV = -13,429

Here, we can see that the PV on the financial calculator is a negative value since it’s a cash outflow. The FV has a positive sign since it’s a cash inflow.

We can also rearrange the future value formula to obtain the holding period return (HPR) as follows:

$$ { r }_{ q }=m\left[ { \left( \frac { FV }{ PV } \right) }^{ \cfrac { 1 }{ mn } }-1 \right] $$

## Calculating Discount Factors Given Interest Rate Swap Rates

Given a series of interest rate swap rates, it is possible to derive discount factors. The size of both fixed and floating leg payments is determined by the notional amount which is technically never exchanged between counterparties.

If the notional amount was actually exchanged, the fixed leg of the swap would resemble a fixed coupon-paying bond, with fixed leg payments acting like semiannual, fixed coupons, and the notional amount acting like the principal payment. Floating rate payments would act like coupon payments of floating rate bond.

The discount factor for \(t\) years is denoted as \(d\left( t \right) \)

The methodology used to come up with discount factors when dealing with interest rate swaps is similar to that used to find discount factors when dealing with bonds.

**Example of calculating discount factors**

Compute the discount factors for maturities ranging from six months to two years, given a notional swap amount of $100 and the following swap rates:

$$

\begin{array}{|l|l|}

\hline

Maturity \quad (years) & Swap \quad Rates \\ \hline

0.5 & 0.75\% \\ \hline

1.0 & 0.85\% \\ \hline

1.5 & 0.98\% \\ \hline

2.0 & 1.20\% \\ \hline

\end{array}

$$

The four discount factors \(d\left( 0.5 \right) \), \(d\left( 1.0 \right) \), \(d\left( 1.5 \right) \), and \(d\left( 2.0 \right) \) can be calculated as follows:

$$ \left( 100+\frac { 0.75 }{ 2 } \right) d\left( 0.5 \right)=100 $$

$$ d\left( 0.5 \right) =\frac { 100 }{ 100.375 } =0.9963 $$

$$ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots $$

$$ \frac { 0.85 }{ 2 } d\left( 0.5 \right) +\left( 100+\frac { 0.85 }{ 2 } \right) d\left( 1.0 \right) =100 $$

$$ 0.425\times 0.9963+100.425d\left( 1.0 \right)=100 $$

$$ d\left( 1.0 \right) =\frac { 100-0.4234 }{ 100.425 } =0.9916 $$

$$ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots $$

$$ \frac { 0.98 }{ 2 } d\left( 0.5 \right) +\frac { 0.98 }{ 2 } d\left( 1.0 \right) +\left( 100+\frac { 0.98 }{ 2 } \right) d\left( 1.5 \right) =100 $$

$$ 0.49\times 0.9963+0.49\times 0.9916+100.49d\left( 1.5 \right)=100 $$

$$ d\left( 1.5 \right) =\frac { 100-0.4882-0.4859 }{ 100.49 } =0.9854 $$

$$ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots $$

$$ \frac { 1.20 }{ 2 } d\left( 0.5 \right) +\frac { 1.20 }{ 2 } d\left( 1.0 \right) +\frac { 1.20 }{ 2 } d\left( 1.5 \right) \left( 100+\frac { 1.20 }{ 2 } \right) d\left( 2.0 \right) =100 $$

$$ 0.6\times 0.9963+0.6\times 0.9916+0.6\times 0.9854+100.60d\left( 2.0 \right)=100 $$

$$ d\left( 2.0 \right) =\frac { 100-0.5978-0.5950-0.5912 }{ 100.6 } =0.9763 $$

$$ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots $$

$$

\begin{array}{|l|l|}

\hline

Maturity \quad (years) & Discount \quad Factor \\ \hline

0.5 & 0.9963 \\ \hline

1.0 & 0.9916 \\ \hline

1.5 & 0.9854 \\ \hline

2.0 & 0.9763 \\ \hline

\end{array}

$$

## Computing Spot Rates Given Discount Factors

A \(t\)-period spot rate is the yield to maturity on a zero-coupon bond that matures in \(t\) years, assuming semiannual compounding. The \(t\)-periodic spot rate is denoted as \(z\left( t \right)\).

Spot rates and discount factors are related as shown in the following formula, assuming semiannual coupons:

$$ z\left( t \right) =2\left[ { \left( \frac { 1 }{ d\left( t \right) } \right) }^{ \frac { 1 }{ 2t } }-1 \right] $$

## Spot Rate Vs Forward 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 financial transaction are transferred between the parties involved. It could be two days after a trade, or even on the same day, a trade is completed. A spot rate of 5% is the agreed-upon market price of the transaction based on current buyer and seller action.

Forward rates are theorized prices of financial transactions that might take place at some point in the future. The spot rate tells you “how much it would cost to execute a financial transaction today”. The forward rate, on the other hand, tells you “how much would it cost to execute a financial transaction at a future date X”.

The point to note here is that spot and forward rates are agreed to in the present. The only difference comes in the timing of execution.

**Example of converting spot rates into forward rates**

Compute the six-month forward rate in six months, given the following spot rates:

\( z\left( 0.5 \right) = 1.6\%\)

\( z\left( 1.0 \right) = 2.2\%\)

Solution:

The six-month forward rate, \( f\left( 1.0 \right)\), on an investment that matures in one year, must solve the following equation:

$$ { \left( 1+\frac { 0.022 }{ 2 } \right) }^{ 2 }={ \left( 1+\frac { 0.016 }{ 2 } \right) }^{ 1 }\times { \left( 1+\frac { f\left( 1.0 \right) }{ 2 } \right) }^{ 1 } $$

$$ 1.0221=1.008\times { \left( 1+\frac { f\left( 1.0 \right) }{ 2 } \right) }^{ 1 } $$

$$ 1.01399-1=\frac { f\left( 1.0 \right) }{ 2 } $$

$$ f\left( 1.0 \right) =0.02797=2.8\% $$

## Par Rate

The par rate is the rate at which the present value of a bond equals its par value. It’s the rate you’d use to discount of all a bond’s cash flows so that the price of the bond is 100 (par). For a 100 par value, two-year bond that pays semiannual coupons, the 2-year par rate can easily be calculated, provided we have the discount factor for each period.

$$ \frac { par\quad rate }{ 2 } \left[ d\left( 0.5 \right) +d\left( 1.0 \right) +d\left( 1.5 \right) +d\left( 2.0 \right) \right] +100d\left( 2.0 \right) =100 $$

In general, for any maturity \({ C }_{ T }\), and assuming a par value of $1,

$$ \frac { { C }_{ T } }{ 2 } \times \sum _{ t=1 }^{ 2T }{ d\left( \frac { t }{ 2 } \right) } +d\left( T \right) =1 $$

The sum of the discount factors is called the annuity factor, \({ A }_{ 1 }\), and is given by:

$$ \sum _{ t=1 }^{ 2T }{ d\left( \frac { t }{ 2 } \right) } $$

Therefore,

$$ \frac { { C }_{ T } }{ 2 } \times { A }_{ T }+d\left( T \right) = $$

**Impact of Maturity on the Price of a Bond and the Returns Generated by Bonds**

In general terms, bond prices will tend to increase with maturity whenever the coupon rate is above the forward rate over the period of maturity extension. The opposite holds: bond prices will tend to decrease with maturity whenever the coupon rate is below the forward rate over the period of maturity extension.

To help you understand just how this happens, assume we have two investors with the opportunity to invest in either STRIPS or a five-year bond. The investor who opts for the 5-year bond (which utilize forward rates) will have a simple task: they will invest at the onset and then wait to receive regular scheduled coupon payments, plus the principal amount at maturity. The investor who opts for STRIPS (which utilize spot rates) will roll them over as they mature throughout the 5-year period (Rolling over implies that when one STRIP expires, the investor will use the proceeds to invest in the next six-month contract, and so on for 5 years.

In market conditions where short-term rates are above the forward rates utilized by bond prices, the investor who rolls over the STRIPS will tend to outperform the investor in the 5-year bond. The opposite is true: In market conditions where short-term rates are below the forward rates utilized by bond prices, the investors who rolls over the STRIPS will tend to underperform the investor in the 5-year bond.

**Flattening and Steepening of Rate Curves**

A yield curve represents the yield of each bond along a maturity spectrum that’s plotted on a graph. The most and widely accepted yield curves pit the three-month versus two-year T-bonds, or the five-year versus ten-year T-bonds. On occasion, the Federal Funds Rate versus the 10-year Treasury note may be used.

The yield curve typically **slopes upwards** indicating that the rate of interest on long-term bonds is higher than the rate on short-term bonds. This reflects the investors’ demands to be compensated for taking on more risk by investing long-term. Such a curve is said to be normal.

Other than a normal yield curve, we could have a flattening yield curve or a steepening yield curve.

### Flattening yield curve

A flat yield curve indicates that little if any, difference exists between short-term and long-term rates for similarly rated bonds. It may manifest as a result of long-term interest rates falling more than short-term interest rates or short-term rates increasing more than long-term rates.

A flattening curve reflects expectations of investors about the macroeconomic outlook. It may be the result of the following events:

- A decrease in expected inflation, causing the premium loaded on long-term rates to decrease; or
- An impending increase in the Federal Funds Rate. (An increase in the FFR could cause an increase in short term rates while long-term rates remain relatively stable, causing the curve to flatten.)

A trader who anticipates a flattening of the yield curve can buy a long-term rate and sell short a short term rate because they expect bond prices to rise in the long-term.

### Steepening yield curve

A steepening yield curve indicates a widening gap between the yields on short-term bonds and long-term bonds. A steepening curve could occur when long-term rates rise faster than short-term rates. Sometimes, short-term rates can also show some defiance by decreasing even as long-term rates rise.

For example, assume that a two-year note was at 2.3% on July 15, and the 10-year was at 3.3%. By 30^{th} August, the two-year note could have risen to 2.38% and the 10-year to 3.5%. The difference would effectively go from 1 percentage point to 1.12 percentage points, resulting in a steeper yield curve.

A steepening curve reflects expectations of investors about the macroeconomic outlook. It may be the result of the following events:

- An increase in expected inflation, causing the premium loaded on long-term rates by investors to increase; or
- An impending decrease in the Federal Funds Rate.

A trader who anticipates a steepening of the yield curve can sell short a long-term rate and buy a short-term rate because they expect bond prices to fall in the long-term (bond prices fall as rates increase).

## Questions

### Question 1

Given the following spot rates, compute the 6-month forward rate in 1 year.

\( z\left( 1.0 \right) = 3.25\%\)

\( z\left( 1.5 \right) = 3.60\%\)

- 4.2%
- 1.5%
- 4.3%
- 5.1%

The correct answer is **C**.

Solution:

The 6-month forward rate on an investment that matures in 1.5 years must solve the following equation:

$$ { \left( 1+\frac { 0.036 }{ 2 } \right) }^{ 3 }={ \left( 1+\frac { 0.0325 }{ 2 } \right) }^{ 2 }\times { \left( 1+\frac { f\left( 1.5 \right) }{ 2 } \right) }^{ 1 } $$

$$ 1.05498=1.03276\times { \left( 1+\frac { f\left( 1.5 \right) }{ 2 } \right) }^{ 1 } $$

$$ 1.02152-1=\frac { f\left( 1.5 \right) }{ 2 } $$

$$ f\left( 1.5 \right) =0.04303=4.3\% $$

### Question 2

Consider a bond with par value of USD 1,000 and maturity in four years. The bond pays a coupon of 4% annually. The spot rate curve is as follows:

$$

\begin{array}{|l|l|}

\hline

n-year & Spot \quad rate \\ \hline

1-year & 5\% \\ \hline

2-year & 6\% \\ \hline

3-year & 7\% \\ \hline

4-year & 8\% \\ \hline

\end{array}

$$

The value of the bond is closest to:

- $1,160
- $500
- $870.78
- $850

The correct answer is **C**.

The value of the bond is the present value of all future cash flows (coupons plus principal), discounted at the various spot rates.

$$ Each \quad coupon=4\% \times 1,000=$40 $$

$$ PV=\frac { 40 }{ { \left( 1+0.05 \right) }^{ 1 } } +\frac { 40 }{ { \left( 1+0.06 \right) }^{ 2 } } +\frac { 40 }{ { \left( 1+0.07 \right) }^{ 3 } } +\frac { 40+1000 }{ { \left( 1+0.08 \right) }^{ 4 } } $$

$$ =38.10+35.60+32.65+764.43=$870.78 $$