After completing this reading, you should be able to:
- Describe Treasury rates, LIBOR, Secured Overnight Financing Rate (SOFR) and repo rates and explain what is meant by the “risk-free” rate.
- Calculate the value of an investment using different compounding frequencies.
- Convert interest rates based on different compounding frequencies.
- Calculate the theoretical price of a bond using spot rates.
- Calculate the duration, modified duration and dollar duration of a bond.
- Evaluate the limitations of duration and explain how convexity addresses some of them.
- Calculate the change in a bond’s price given its duration, its convexity and a change in interest rates.
- Derive forward interest rates from a set of spot rates.
- Derive the value of the cash flows from a forward rate agreement (FRA).
- Calculate zero-coupon rates using the bootstrap method.
- Compare and contrast the major theories of the term structure of interest rates.
Treasury Rates, LIBOR, and Repo rates
Treasury rates are the rates earned by investors in instruments used by a government to borrow in its own currency. These include Treasury bonds and Treasury bills. Treasury rates are considered “risk-free” because they have zero risk exposure. That has much to do with the ability of the government to use a range of tools at its disposal to avoid default, including printing of cash and increased taxes. T-bill and T-bond rates are used as the benchmark for nominal risk-free rates.
LIBOR, London Interbank Offered Rate, is the rate at which the world’s leading banks lend to each other for the short-term. It’s the most widely used benchmark for short-term lending.
Repo rates are the implied rates on repurchase (repo) agreements. A repo agreement is an agreement between two parties – the seller and the buyer – where the seller agrees to sell a security to the buyer with the understanding that they (seller) will buy it back later at a higher price. The most common repo transactions are carried out overnight. The credit risk in a repo agreement depends on the term of the agreement as well as the creditworthiness of the seller.
Compounding Frequencies
Given an initial investment of \(A\) that earns an annual rate \(R\), compounded \(m\) times a year for a total of \(n\) years, then we can compute the future value, \(FV\), as follows:
$$ FV=A{ \left( 1+\frac { R }{ m } \right) }^{ m\times n } $$
In the presence of continuous compounding, then:
$$ FV=A{ e }^{ R\times n } $$
Exam tip: For any rate \(R\), the future value with continuous compounding will always be greater than the future value with discrete compounding.
Let \({ R }_{ c }\) be the continuously compounded rate that equates the future value under discrete compounding to the future value under continuous compounding:
$$ A{ \left( 1+\frac { R }{ m } \right) }^{ m\times n }=A{ e }^{ { R }_{ c }\times n } $$
$$ { R }_{ c }=m\times ln\left( 1+\frac { R }{ m } \right) $$
Alternatively, given \({ R }_{ c }\),
$$ R=\left( { e }^{ \cfrac { { R }_{ c } }{ m } }-1 \right) $$
The Theoretical Price of a Bond Using Spot Rates
The theoretical price of a bond is given by the present value of all of the bond’s cash flows. Assuming each cash flow is associated with a spot discount factor \({ z }_{ j }\), then:
$$ P=\left[ \frac { c }{ 2 } \times \sum _{ j=1 }^{ N }{ { e }^{ -\frac { { z }_{ j } }{ 2 } \times j } } \right] +FV\left( { e }^{ -\frac { { z }_{ N } }{ 2 } \times j } \right) $$
Where:
\(P\) = bond’s price
\({ z }_{ j }\)=bond equivalent spot rate correponding to \(\cfrac { j }{ 2 } \) years on a continuously compounded basis
\(FV\) = face value of the bond
\(N\) = number of semiannual payment periods
The yield of a bond is the single discount rate that equates the bond’s present value to its market price. A bond’s par yield is the discount rate that equates the bond’s price to its par value.
Deriving Forward Rates from a Set of Spot Rates
\({ y }_{ n }\) , 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 \(\left( time\quad 0 \right) \) an investment made at time \(t>0\) 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:
$$ 1+{ f }_{ t,1 }=\frac { { { \left( 1+{ y }_{ t+1 } \right) } }^{ t+1 } }{ { \left( 1+{ y }_{ t } \right) }^{ t } } $$
Example
Let’s say you have the following spot rates table:
Year |
1 |
2 |
3 |
4 |
Spot rate |
1.2% |
1.5% |
1.9% |
2.4% |
The one-year forward rate for one year can be calculated as:
$$ 1+{ f }_{ 1,1 }=\frac { { { \left( 1+{ 1.5\% } \right) } }^{2 } }{ { \left( 1+{ 1.2\% } \right) }^{ 1 } } = 1.018 $$
$$ { f }_{ 1,1 } = 1.8\% $$
Forward Rate Agreements
A forward rate agreement is an agreement between two parties to lock in an interest rate for a specified period of time starting on a future settlement date, based on a notional amount. The buyer of a forward rate agreement enters into the contract to protect himself from any future increase in interest rates. The seller, on the other hand, enters into the contract to protect himself from any future decline in interest rates. If Firm A and Firm B enter into a forward rate agreement by agreeing on an interest rate RK, the cash flows will be:
$$ Firm \quad A: L(R_{K} – R_{M} )(T_{2} -T_{1} ) $$
$$ Firm \quad B: L(R_M-R_K )(T_2-T_1 ) $$
Where:
\(L\) = principal amount
\(R_K\) = interest rate agreed to in the FRA
\(R_M\) = actual interest rate observed between T1 and T2
FRAs are cash-settled on the settlement date – the start date of the notional loan or deposit. The interest rate differential between the market rate and the FRA contract rate determines the exposure to each party. It’s important to note that as the principal is a notional amount, there are no principal cash flows.
As time passed, the agreed fixed rate \(R_K\) remains the same but the forward LIBOR rate \(R_F\) is likely to move in either direction.
Therefore, the value of the forward contract to both parties will be:
$$ Firm \quad A: V_{FRA} = L(R_{K} – R_{F} )(T_{2} -T_{1} ) e^{- R_2 T_2} $$
$$ Firm \quad B: V_{FRA} = L(R_{F} – R_{K} )(T_{2} -T_{1} ) e^{- R_2 T_2} $$
Where:
\(V_{FRA}\) = value of the forward contract
\(R_2\) = the continuously compounded risk-free rate for a maturity \(T_2\)
Example
A German bank and a French bank entered into a semiannual forward rate agreement contract where the German bank will pay a fixed rate of 4.2% and receive the floating rate on the principal of €700 million. The forward rate between 0.5 years and 1 year is 5.1%. If the risk-free rate at the 1-year mark is 6%, then what is the value of the FRA contract between the two banks?
$$ V_{FRA} = L(R_{K} – R_{F} )(T_{2} -T_{1} ) e^{- R_2 T_2} $$
$$ V_{FRA} = €700 \quad million (5.1\% – 4.2\%)(0.5 \quad years) e^{-0.06 * 1 \quad year} $$
Bond Price Derivatives
Duration
Duration, sometimes referred to as Macaulay duration, is an approximate measure of a bond’s price sensitivity to changes in interest rates. Bond prices have an inverse relationship with interest rates. When interest rates rise, bond prices fall; when interest rates fall, bond prices rise.
Duration is expressed in years. Let’s say a bond has a duration of 5 years. What does that imply? Its price will rise about 5% if its yield drops by 1% (100 basis points), and its price will fall by about 5% if its yield rises by that amount.
For a zero-coupon bond, its duration is simply its time to maturity. For a coupon bond, its duration is shorter than maturity because the cash flows have different weights. The formula for duration, given continuously compounding, is:
$$ Duration={ \Sigma }_{ i=1 }^{ n }{ t }_{ i }\left[ \frac { { c }_{ i }{ e }^{ -y{ t }_{ i } } }{ P } \right] $$
Where:
\({ t }_{ i }\)=time in years until cash flow \({ c }_{ i }\) is received
\(y\)=the continuously compounded yield (discount rate)based on a bond price \(P\)
Modified Duration
Modified duration is used in the absence of continuous compounding of the yield. If the yield, \(y\), is expressed as a rate compounded \(n\) times a year, then:
$$ modified\quad duration=\left[ \frac { Macaulay\quad duration }{ \left( 1+\frac { y }{ n } \right) } \right] $$
Note: by yield, we imply the yield to maturity.
Exam tip: Unless given, you must calculate the Macaulay duration to determine the modified duration.
Dollar duration
The dollar duration, DD, of a bond is a product of its modified duration and its market price. If we use \({ D }^{ \ast }\) to denote the modified duration, and \({ P }_{ 0 }\) to denote the bond’s market price, then:
$$ DD={ D }^{ \ast }\times { P }_{ 0 } $$
DV01 of a Bond
The dollar value of a basis point, DVBP, also represented as DV01, is the dollar exposure of a bond price for a change in yield of 0.01% (1 basis point). It is also the duration times the value of the bond and is additive across the entire portfolio.
$$ DVBP=DD\times \Delta y=\left( { D }^{ \ast }\times { P }_{ 0 } \right) \times 0.0001 $$
Effective Duration
For bonds with embedded options, neither the Macaulay duration nor the modified duration is appropriate as a measure of interest rate sensitivity. For callable and putable bonds, you ought to compute the effective duration.
$$ effective\quad duration=\frac { { BV }_{ -\Delta y }-{ BV }_{ +\Delta y } }{ 2\times { BV }_{ 0 }\times \Delta y } $$
Where:
\({ BV }_{ -\Delta y }\)=price estimate if yield decreases by a given amount,\(\Delta y\)
\({ BV }_{ +\Delta y }\)=price estimate if yield increases by a given amount,\(\Delta y\)
\({ BV }_{ 0 }\)=initially observed bond price
\(\Delta y\)=change in yield, expressed in decimal form
Convexity
Convexity is a measure of the curvature in the relationship between bond prices and bond yields. It demonstrates how the duration of a bond changes as the interest rate changes. Convexity estimates the amount of market risk affecting a bond or portfolio.
If bond \(X\) has a higher convexity than bond \(Y\), what does this imply? All else being equal, bond \(X\) will always have a higher market price than bond \(Y\) as interest rates rise and fall.
Duration assumes that interest rates and bond prices have a linear relationship. It’s, therefore, a fairly good measure of exactly how bond prices are affected by small changes in interest rate. However, the relationship between bond prices and interest rates is actually non-linear, i.e., convex. This makes convexity a better measure of risk, especially in the presence of large and frequent fluctuations in interest rates.
For the purpose of the percentage change in price triggered by convexity, i.e., the price change not explained by duration, we must calculate the convexity effect.
$$ convexity\quad effect=\frac { 1 }{ 2 } \times convexity\times \Delta { y }^{ 2 } $$
Exam tip: Convexity is always positive for regular coupon-paying bonds
Calculating the change in a bond’s price given its duration, its convexity, and a change in interest rates
Combining duration and convexity results in a far more accurate estimate of the change in the price of a bond given a change in yield.
\(Change \quad in \quad price = duration \quad effect + convexity \quad effect \)
$$ =\left[ –Duration\times price\times change\quad in\quad yield \right] +\left[ \cfrac { 1 }{ 2 } \times Convexity\times price\times { \left( change\quad in\quad yield \right) }^{ 2 } \right] $$
Questions
Question 1
A portfolio manager has a \(bond\) position worth \(CAD\quad 200\quad million\). The position has a modified duration of \(six \quad years\) and a convexity of \(120\). Assuming that the term structure is flat, by how much does the value of the position change if interest rates increase by \(50\quad basis\quad points\)?
- CAD -5,700,000
- CAD -5,000,000
- CAD -6,000,000
- CAD -54,000,000
The correct answer is A.
\(Change \quad in \quad value = duration \quad effect + convexity \quad effect \)
$$ =\left[ –Duration\times price\times change\quad in\quad yield \right] +\left[ \cfrac { 1 }{ 2 } \times Convexity\times price\times { \left( change\quad in\quad yield \right) }^{ 2 } \right] $$
$$ =-6\times 200,000,000\times 0.005+\left[ 0.5\times 120\times 200,000,000\times { 0.005 }^{ 2 } \right] $$
$$ =-5,700,000 $$
With every increase in interest rates of 50 basis point, the bond’s price will decrease by $5.7 million.
Question 2
According to a valuation model, a portfolio of \(bonds\) has a value of \($75.5\quad million\). The model also estimates that if all interest rates fell by \(50 \quad bps\), the value of the portfolio would increase to \($79.5 \quad million\). If interest rates rose by \(50 \quad bps\), the value of the portfolio would decrease to \($71.8\quad million\). The term structure is flat. Using this information, the effective duration of the bond portfolio is closest to:
- 11
- 7.8
- 5.19
- 10.20
The correct answer is D.
$$ effective\quad duration=\frac { { BV }_{ -\Delta y }-{ BV }_{ +\Delta y } }{ 2\times { BV }_{ 0 }\times \Delta y } $$
Where:
\({ BV }_{ -\Delta y }\)=price estimate if yield decreases by a given amount,\(\Delta y\)
\({ BV }_{ +\Delta y }\)=price estimate if yield increases by a given amount,\(\Delta y\)
\({ BV }_{ 0 }\)=initially observed bond price
\(\Delta y\)=change in yield,expressed in decimal form
$$ =\frac { 79.5-71.8 }{ 2\times 75.5\times 0.005 } =10.20 $$