4.1.3 Interest Rates_1.5*0.980488+101.5*d1=98.5

3. Interest Rates

3.1 Fundamentals

3.1.1 Compounding Frequency

Future value with different compounding frequencies. Initial investment $ $100$ at an $8\%$ rate grows.

Present value with different compounding frequencies $ $100$ received in five years when the interest rate is $8\%$.

3.1.2 Continuous Frequency

In the limit of the compounding frequency, we obtain continuous compounding.

Future value: If an interest rate $R$ is continuously compounded, it can be shown that an amount $A$ grows to $A{e}^{RT}$ by time $T$.

Present value: The present value of an amount $A$ received at time $T$ is $A{e}^{-RT}$.

3.1.3 Equivalent Rate

Convert $R_1$ (compounded $m_1$ times per annum) to the equivalent rate $R_2$(compounded $m_2$ times per annum):
$$(1+\frac{R_1}{m_1}{)}^{m_1}=(1+\frac{R_2}{m_2}{)}^{m_2}$$

Convert $R_m$ (compounded $m$ times per annum) to the equivalent rate $R_c$ (continuously compounded rate).

$$e^{R_c}=(1+\frac{R_m}{m}{)}^m$$

In the United States and some other countries, bonds which last longer than one year from date of issue normally pay interest every six months. The yield provided by a bond is therefore normally expressed with semi-annual compounding.

3.1.4 Discount Factor

Discount Factor is the present value of one unit of currency to be received at the end of that term. The discount factor is a declining function of maturity due to the time-value of money phenomenon)

$d(1.5)=0.9825$, means that the present value of $1 to be received in 18 months is $0.9825 today.

Discount factor from T-bills(零息债券)

A Treasury bill maturing in 6 months is worth USD $995,912.50$ now, and its par value is USD 1 million.What is the six-month discount factor?

$d(0.5)=\frac{995,912.50}{1,000,000}=0.9959125$

Discount factor from coupon-bearing bonds(付息债券)

A six-month bond that pays coupons at the rate of $5\%$ per year is currently worth $100.5$. What is the six-month discount factor?

$d(0.5)=\frac{100.5}{100+2.5}=0.980488$

In addition to the bond in the example above, a one-year bond that pays coupons every six months at the rate of $3\%$ per year is currently worth $98.5$. What is the one-year discount factor?

$1.5\ast d(0.5)+101.5\ast d(1)=98.5\to d(1)=0.955953$

3.1.5 Annuity

Annuity is a finite set of level sequential cash flows.

An investor deposits $ $2000$ at the beginning of the following four years into an account that pays nominal annual interest of $6\%$ (compounding monthly). The value of the account at the end of four years is closest to:

The equivalent annual rate = $(1+6\%/12{)}^{12}-1=6.1678\%$

Calculator: [2nd][PMT], [2nd][ENTER] $\to$ BGN mode

$N=4$, $1/Y=6.1678$, $PV=0$,$PMT=2000$ $\to$ $CPTFV=9312$

Annuity Factor: The sum of the discount factors, assume semi-annual basis
$$AT=\sum _{t=1}^{2n}d(\frac{t}{2})$$

Perpetuity is a security that pays coupons forever. The price of a perpetuity is simply the coupon divided by the yield.

$$\text{Perpetuity}=\frac{C}{y}$$

3.2. Spot Rates

The spot rate is the interest rate earned when cash is received at just one future time. It is also referred to as the zero-coupon interest rate, or just the “zero”.

3.2.1 Determine Spot Rates

Direct way: derive spot rates from money market instruments lasting less than one year.

Bootstrap: instruments lasting longer than one year usually make regular payments prior to maturity.

One way of calculating the zero-coupon rates implied by these instruments is by working forward and fitting the zero-coupon rates to progressively longer maturity instruments.

Calculate the zero rates for 6 months, one year, 18 months and two years using bootstrapping.

$\frac{100}{1+z_{0.5}/2}=98.5\to z_{0.5}=3.05\%$

$\frac{100}{(1+z_1/2{)}^2}=95.9\to z_1=4.23\%$

$\frac{2}{1+z_{0.5}/2}+\frac{2}{(1+z_1/2{)}^2}+\frac{102}{(1+z_{1.5}/2{)}^3}=98\to z_{1.5}=5.44\%$

$\frac{3}{1+z_{0.5}/2}+\frac{3}{(1+z_1/2{)}^2}+\frac{3}{(1+z_{1.5}/2{)}^3}+\frac{103}{(1+z_{1.5}/2{)}^3}=100.6\to z_2=5.73\%$

3.2.2 Spot Rates v.s. Discount Factors

Spot rate give the same information as discount factors.

Suppose the discount factor for $t$ years is $d(t)$ and that the $t$-year spot rate is $z(t)$ with semi-annual compound
$$d(t)=\frac{1}{(1+\frac{z(t)}{2}{)}^{2t}}\to z(t)=2[(\frac{1}{d(t)}{)}^{\frac{1}{2t}}-1]$$

Computing spot rates from STRIPS prices or discount factors for semi-annual compounding.

$d(0.5)=\frac{1}{(1+\frac{Z_{0.5}}{2})}=0.992063\to Z_{0.5}=1.6\%$

$d(1)=\frac{1}{(1+\frac{Z_1}{2}{)}^2}=0.979326\to Z_1=2.1\%$

$d(1.5)=\frac{1}{(1+\frac{Z_{1.5}}{2}{)}^3}=0.964132\to Z_1=2.45\%$

3.2.3 Bond Valuation Based on Spot Rates

Law of one price assets that produce identical future cash flows regardless of future events should have the same price.

The valuation of bond involves identifying its cash flows and discounting them at the interest rates corresponding to their maturities.

Use a sequence of spot rates that correspond to the cash flow dates to calculate the bond price.
$$P=\sum _{t=1}^n\frac{C}{(1+Z_t{)}^t}+\frac{FV}{(1+Z_n{)}^n}$$

• $Z_n$: spot rate for period $n$.

Assume a 1.5-year bond with a face value of $ $100$ pays a $3.5\%$ coupon on a semiannual basis. What is the price of the bond according to the following spot rates?

$$\text{Price}=\frac{1.75}{(1+2.2\%/2)}+\frac{1.75}{(1+2.25\%/2{)}^2}+\frac{101.75}{(1+2.3\%/2{)}^3}=101.7611$$

3.3 Forward and Par Rates

3.3.1 Forward Rates

Forward rates are interest rates corresponding to a future period implied by the spot curve.

All forward rates are computed using spot rates.

Annual compounding $z_1$ is 3%, $z_2$ is 4%. The implied forward rate from the end of year one to the end of year two is:

$(1+3\%)(1+F)=(1+4\%{)}^2\to F=5.01\%$

Semi-annual compounding $z_{0.5}$ is 1.6%, $z_1$ is 2.1%. The implied forward rate from the end of half-year to the end of year one is:

$(1+\frac{1.6\%}{2})(1+\frac{F}{2})=(1+\frac{2.1\%}{2}{)}^2\to F=2.6\%$

Continuous compounding $z_3$ is 4.5%, $z_4$ is 5%. The implied forward rate from the end of year three to year four is:

$e^{0.045\times 3}\times e^{F\times 1}=e^{0.05\times 4}\to F=6.5\%$

When rates are expressed with ordinary compounding, the forward rate for the period between time $T_1$ and $T_2$ is

$$(1+R_1{)}^{T_1}(1+F_{1,2}{)}^{T_2-T_1}=(1+R_2{)}^{T_2}$$

When rates are expressed with continuous compounding, the forward rate for the period between time $T_1$ and $T_2$ is
$$F=\frac{R_2{T}_2-R_1{T}_1}{T_2-T_1}$$

3.3.2 Trading Strategies

An investor can borrow funds for time $T_1$ and invest for $T_2$. If the rates for the period between $T_1$ and $T_2$ are less than the forward rate, the investor’s total financing cost will be less than the investor’s return, producing a profit.

Suppose the two-year rate is $4\%$ and the three-year rate is $5\%$, with both rates continuously compounded. So the forward rate for the third year is $7\%$.

• If an investor feels confident that the rate for the third year will be less than $7\%$, the investor can borrow for two years at $4\%$ and invest for three years at $5\%$。
• If the realized rate is expected to be greater than the forward rate, the investor should borrow for three years and invest for two years.

3.3.3 Bond Valuation Based on Forward Rates

Forward rates can be used to value a bond in the same manner as spot rates because they are interconnected.

Discount bond cash flows one period by one period with forward rates.

Example: suppose an analyst needs to value a 1.5 year, 3% semi-annual coupon payment bond.What is the price of the bond according to the following forward rates?

$$\frac{1.5}{1+\frac{2.5\%}{2}}+\frac{1.5}{(1+\frac{2.5\%}{2})(1+\frac{3.5\%}{2})}+\frac{101.5}{(1+\frac{2.5\%}{2})(1+\frac{3.5\%}{2})(1+\frac{3.78\%}{2})}=99.6327$$

3.3.4 Par Rates

Par rate is the coupon rate which bond is priced at par value.

For an asset with a par amount of one unit that makes semiannual payments and matures in $T$ years

$$\frac{p}{2}\sum _{t=1}^{2t}d(\frac{t}{2})+d(T)=1\to A(T)=\sum _{t=1}^{2t}d(\frac{t}{2})$$

• $p$: the par rate
• $A(T)$: the annual factor

Assume a 1.5-year bond pays semiannual coupons and has a par value of $ $100$, please compute the 1.5-year par rate.

$$\frac{p\times 100}{2}[d(0.5)+d(1)+d(1.5)]+d(1.5)\times 100=100\to p=2.44\%$$

3.3.5 Relationship among Spot, Forward and Par Rates

Based on the assumption of semi-annual compounding, calculate the spot rate in 1.5 years

From discount factor

$$d(1.5)=\frac{1}{(1+\frac{Z_{1.5}}{2}{)}^3}\to Z_{1.5}=2.45\%$$

From forward rate
$$(1+\frac{Z_{1.5}}{2}{)}^3=(1+\frac{f_{0-0.5}}{2})(1+\frac{f_{0.5-1}}{2})(1+\frac{f_{1-1.5}}{2})$$

From par rate
$$100=\frac{2.44/2}{(1+\frac{1.6\%}{2})}+\frac{2.44/2}{(1+\frac{2.1\%}{2}{)}^2}+\frac{100+2.44/2}{(1+\frac{Z_{1.5}\%}{2}{)}^3}$$

Term structure
If spot curve is upward-slopping

If spot curve is downward-slopping

If the term structure is flat(with all the spot rates the same), all par rates and all forward rates equal to the spot rate.

3.4. Other Rates

3.4.1 Government Borrowing Rates

Government borrowing rates is interest rate paid by a government on its borrowings in its own currency. In the U.S., this is referred to as the Treasury Rate.

Government debt from developed countries is considered to be risk-free and the interest rates on these borrowings are generally below those on other borrowings in the same currency.

3.4.2 Repo Rate

In a repo agreement, securities are sold by Party A and Party B for a certain price with the intention of being repurchased at a later time at higher price.

If Party A fails to repurchase the securities as agreed, Party B can simply keep the securities. This means that Party B takes very little risk, provided that:

• The value of the securities equals (or is very close to) the initial price for the security.
• This value is fairly stable.

3.4.3 Libor

Libor (London Interbank Offered Rate): an unsecured borrowing rate between banks. Libor rates are quoted for several different currencies and for borrowing periods ranging one day to one year.

• AA-rated global banks estimate the rates (forward-looking estimates for rates at which banks think they could borrow money) .
• the highest 25% and lowest 25% of the estimates are
  discarded.

Libor is a less than ideal benchmark because it is based on estimates that can be manipulated.

Two replacement benchmark rate:
The repo overnight rate: The United States has proposed the use of the repo-based Secured Overnight Financing Rate (SOFR).

Overnight interbank borrowing rate: arises from unsecured borrowing and lending between banks at the end of each day to keep cash in reserve with central bank.

• Effective federal funds rate: In the United States, the weighted average of the rates in brokered transactions
• Sterling overnight index average (SONIA): In the U.K.
• Euro overnight index average (EONIA): In the Eurozone

3.4.4 Swap Rates

Swap rate is the fixed portion of a swap as determined by its particular market and the parties involved.

Floating rates are based on some short-term reference interest rate, such as three-month or six-month dollar Libor(London Interbank Offered Rate).

The swap market therefore defines par rates. For example, the five-year swap rate defines a five-year bond selling for par. The par rates can be used to determine discount factors and therefore spot rates.

• Frequency: swap - every three months; par bonds - usually considered in Treasury markets pay coupons every six months.
• Swap rates also include a degree of counter-party default risk, whereas Treasury rates are usually free of default risk.

3.4.5 Overnight Indexed Swap

Overnight index swap(OIS): the geometric average of overnight rates is exchanged for a fixed rate every three months for five years.

OIS rates are the fixed rates in overnight indexed swaps.

The floating rate per three-months is
$$(1+r_1{d}_1)(1+r_2{d}_2)\dots (1+r_n{d}_n)-1$$

• $r_i$: is the overnight rate per day on day $i$.
• $d_i$: is the number of calendar days where rate r_i applies
• $n$: is the number of trading days in the three-month period.

The risk-free rates used to value derivatives are determined from overnight interbank rates using overnight indexed swaps. Treasury rates are not used because they are considered to be artificially low.

• Banks are not required to keep capital to support an
  investment in Treasury instruments, but they are required
  to keep capital for other very low risk instruments.
• In some countries, the income from Treasury instruments is given favorable tax treatment.

3.5 Term Structure

3.5.1 Term Structure Shift

Parallel shift: The yields on all maturities increase or decrease by the same number of basis points maintaining its prior slope and shape.

Non-Parallel Shift: The yields on all maturities increase or decrease by different number of basis points and non-parallel shift changes the slope of the yield curve

A flattening term structure occurs

• long- and short-maturity rates both move down, but long-maturity rates move down by more (bull flattener)
• long- and short-maturity rates both move up, but short-maturity rates move up by more (bear flattener)
• short short-maturity, long long-maturity.
  

A steepening term structure occurs

• long- and short-maturity rates both move down, but short-maturity rates move down by more (bull steepening).
• long- and short-maturity rates both move up, but short-maturity rates move up by less(bear steepening).
• long short-maturity, short long-maturity.
  

3.5.2 Theories of the Term Structure

The market segmentation theory: argues that short-, medium-, and long-maturity instruments attract different types of traders.

• Short-maturity rates would only attract traders interested in short-maturity investments.
• Medium- and long-maturity instruments would only attract traders interested in those investments.
• Unrealistic: Market participants do not focus on just one segment of the interest rate term structure.

The expectations theory: argues that the interest rate term structure reflects where the market is expecting interest rates to be in the future.

• If the market expects interest rates to rise, then the term structure of interest rates will be upward-sloping.
• If the market expects interest rates to decline, then the term structure will be downward-sloping.
• The expectations theory argues that forward rates should be equal to expected future spot rates.

The liquidity preference theory: if the interest rate term structure reflects what the market expects interest rates to be in the future, most investors will choose a short-term investment over a long-term investment. This is because of liquidity consideration. Liquidity considerations therefore lead to lenders wanting to lend for short periods of time and borrowers wanting to borrow for long periods of time.

In order to match borrowers and lenders, financial intermediaries must increase long-term rates relative to the market’s expectations about future short-term rates.

• Long-term borrowing becomes less attractive, because short-term rates are more attractive to borrowers
• Short-term lending becomes less attractive because long-term rates are more attractive to lenders