403-FIBasicIR.Rmd

# Simple Interest Rate Derivatives

```{r, IRD,echo=FALSE,results='hide'}
  library(timeDate)
  library(empfin)
  library(tufte)
```

`r newthought("In")` this chapter we cover the pricing of simple interest rate
derivatives:

-   Forward rate agreements (FRA)

-   Euro-Currency Futures

-   Short-term (T-Bill) Futures

## Spot and Forward Interest Rates

Definition:

Spot rate

:   The interest rate for an investment starting today, paid back at a
    future time $T$.

Forward rate

:   The interest rate that applies to an investment to be made in the
    future.

Forward rates can be infered from spot rates. Let

$B(t,T)$

:   Discount factor from $T$ to $t$

$r_{t,T}$

:   Continuously compounded rate from $t$ to $T$.

We have:

$$B(t,T) = e^{-r_{t,T}(T-t)}$$

A simplified calculation of forward rates from zero-coupon rates can be
performed using the equations:

$$\begin{aligned}
  B(t_1, t_2) &=& \frac{B(0, t_2)}{B(0, t_1)} \\
  r_{t_1, t_2} &=& -\frac{\ln(B(t_1, t_2)}{t_2-t_1}\end{aligned}$$

The calculation is illustrated by the following example.
Table \[tab:zc-1\] provides a sample term-structure of zero-coupon
rates.

   Year   Spot Rate
  ------ -----------
    1       10.0
    2       10.5
    3       10.8
    4       11.0
    5       11.1

Using these spot rates (continuously compounded), we can compute the
1-year forward rates with the following script:

```{r, label=ZC-1, echo=TRUE}
  sr <- c(10.0, 10.5, 10.8, 11.0, 11.1)/100
  t <- seq(1,5)
  B <- exp(-sr*t)
  fr <- c(sr[1], -log(B[2:5]/B[1:4]))
```

The resulting forward rate curve is shown in figure \@ref(fig:zc-2).

```{r, label=ZC-2, echo=F, results='hide'}
  plot(t, fr, type='l', xlab='Time (year)', ylab='Rate', lwd=2, col='blue')
  lines(t, sr, type='l', lwd=2, col='green')
  legend('bottomright', c("Forward", "Spot"),lty=c(1,1), col=c('blue', 'green'))
```

```{r, label=fig-zc-2,echo=FALSE, fig.width=6, fig.height=6}
  <<ZC-2>>
```

The basic interest rate derivatives provide a vehicule for locking a
short term interest rate a some point in the future. There are many such
vehicules:

1.  FRA: Forward rate agreement (OTC)

2.  T-Bill Futures: Futures on a US 90-day Treasury Bill

3.  Euro-currency Futures: Futures on an 3-month interest rate (LIFFE,
    EUREX)

In addition, Government Bond Futures, to be considered in the next
chapter, provide a vehicule for locking a long term future yield.

![image](img/IR-Futures.pdf)

\@ref(fig:zc-2\]

## Forward Rate Agreement (FRA)

Define $r_{t_0, t_1, t_2}$ the interest rate between $t_1$ and $t_2$,
observed at $t_0$.

An OTC contract where at time $t_0$:

1.  Buyer to pay fixed rate $r_{t_0, t_1, t_2} = r_f$

2.  Seller to pay variable rate $r_{t_1, t_1, t_2} = r_l $


Cash settlement (buyer’s perspective) at $t_1$ is present value of
difference between fixed rate and rate observed at $t=t_1$:

$$100 \left[e^{r_l(t_2-t_1)} - e^{r_f(t_2-t_1)} \right] e^{-r_{t_1, t_1, t_2}(t_2-t_1)}$$



Value at time $t_0$ (from buyer’s perspective):

$$100 \left[e^{r_l(t_2-t_1)} - e^{r_f(t_2-t_1)} \right] e^{-r_{t_0, t_0, t_2}(t_2-t_0)}$$

Value is zero if

$$r_f = \frac{r_{t_0, t_2} t_2 - r_{t_0, t_1} t_1}{t_2 - t_1}$$

Recall that the forward rate $r_{t, t_1, t_2}$ is defined by

$$r_{t, t_1, t_2} (t_2-t_1) + r_{t, t_1}(t_1-t) = r_{t, t_2}(t_2-t)$$

Thus, the present value of the FRA can be computed by assuming that the
current forward rate will be realized.


As an example, assume today is 1Apr2010, and a firm needs to borrow:

1.  100 M €

2.  from 1dec2010 to 1dec2011

Hedge:

1.  Buy a “9x12” FRA.

2.  $r_f = 3\%$ for 12 months, start in 9 months.

Table \[tab:FRA\] summarizes the outcomes under two rate scenarios at
horizon. In all cases, the effective borrowing rate is $3\%$.

                                     Case 1                          Case 2
  ---------------------- ------------------------------- -------------------------------
          $r_f$                        .02                             .04
        Settlement        $100 \frac{2\%- 3\%}{1.02} =$   $100 \frac{4\%- 3\%}{1.04} =$
      PV of Interest                 - 1.96                           -3.84
      FRA Settlement                  -.98                             .96
   PV of Effective Rate               -2.94                           -2.88
      Effective Rate        $\frac{2.94}{1.02} = 3.0$      $\frac{-2.88}{1.04} = 3.0$


## T-Bill Futures


A futures contract on a short rate instrument.

1.  Underlying asset: 90-days TB

2.  Notional amount: USD 1 million

Price of underlying T-Bill:

$$V_t = 100 e^{-r_{t_0, t_2} (t_2 - t)}$$

Futures price:

$$F_t = V_t e^{r_{t_0, t_1}(t_1-t_0)}$$

or,

$$F_t = 100 e^{-r_{t_0, t_1, t_2} (t_2-t_1)}$$

### Arbitrage

Simplified example, all rates continuously compounded.

1.  45-day T-Bill rate is 10%

2.  135-day T-Bill rate is 10.5%

3.  implied rate from T-Bill Futures: 10.6%

Arbitrage:

1.  Sell Futures contract

2.  Borrow fund for 45 days at 10%

3.  Invest for 135 days at 10.5%

### Quotes


Quotation in yield:

$$F_t = 100 - \mbox{TB yield}$$

Cash price calculation: $$100 - \frac{90}{360} \mbox{TB Yield}$$

Price of Futures converges towards price of underlying T-Bill.


Buy a TB futures quoted 96.83. (Buy forward the underlying T-Bill at a
yield of

$$y_t = (100 - 96.83) = 3.17\%$$

Delivery price:

$$1,000,000 (100 - 3.17 \frac{90}{360}) = 992,075$$


P&L at maturity: difference between delivery price and current T-Bill
price. if yield is now 4%:

T-Bill price:

$$1,000,000 (100 - 4.00 \frac{90}{360}) = 990,000$$

P&L = -2075.

## Euro-Currency Futures

Euro-currency Futures are Futures contracts on rates, not on price as
the T-Bill contract. The price converges to underlying interest rate.

Three-month Euribor contracts are quoted on EUREX.

1.  1 million € notional

2.  Cash settlement, one day after Final Settlement Day

3.  Price $F_t$ = 100 - 3-month Euribor rate

The EuroDollar Futures contract quoted on the CME has similar features.

Invoice amount (Euribor)

$$10,000 (100-\frac{1}{4}(100- F_t))$$


Eurodollar Rate Versus Forward Rate
-----------------------------------

We next provide a simple method for calculating the spread between the
Eurodollar rate and the forward rate. This spread is a direct
consequence of the impact of discounting mentioned earlier.

The PV of a forward swap that receives fixed is:

$$PV = (r_f-r_l) \frac{n}{360} B(0, T)$$

With:

$r_f$

:   Fixed rate

$r_l$

:   Floating rate

$n$

:   tenor in days

$B(0,T)$

:   discount factor from the swap payment date to today.

The change in PV resulting from a change $\Delta r_l$ in the floating
rate and $\Delta B(0,T)$ in the discount factor is

$$\Delta PV = - \Delta r_l \frac{n}{360} (B(0, T) + \Delta B(0, T))$$

The corresponding change in value of a Eurodollar contract is

$$\frac{90}{360} \Delta r_l$$

The quantity of Eurodollar futures needed to hedge against changes in
$r_l$ is $$H = - \frac{n}{90} B(0, T)$$

To eliminate a riskless profit, the swap position hedged with eurodollar
contracts must have a zero expected profit:

$$E \left[\Delta r_l (B(0,T) + \Delta B(0,T)) \right] = e \left[ B(0,T) (\Delta r_l + s) \right]$$

Solve for the spread $s$:

$$E\left[s\right] = E\left[\Delta r_l \frac{\Delta B(0, T)}{B(0,T)} \right]$$

Finally,

$$E\left[s\right] = \sigma(r_l) \sigma(\frac{\Delta B(0, T)}{B(0,T)}) <r_l, \frac{\Delta B(0, T)}{B(0,T)}>$$

This adjustment applies to each quarter. The cumulative adjustment can
be significant for long-dated forward rate.

## Convexity Adjustment

Calculation of expected forward rate. In a risk-neutral world, the
expected price of a bond is its forward price. But the expected rate is
not the forward rate, because of the non-linear relationship between
rate and prices.

Define:

$y_t$

:   Forward bond yield observed at time $t$ for forward contract with
    maturity $T$

$B_T(y_T)$

:   Bond price at time $T$, function of its yield

$\sigma_T$

:   Volatility of forward bond yield

A Taylor expansion of $B_T(y_T)$ around $y_0$ gives:

$$B_T(y_T) = B_T(y_0) + (y_T-y_0)G'(y_0) + \frac{1}{2} (y_T-y_0)^2 G''(y_0)$$

Let $E_T()$ be the forward risk-neutral expectation. Applied to both
sides, we get:

$$E[B_T(y_T)] = B_T(y_0) + E(y_T-y_0) G'(y_0) + \frac{1}{2} E[(y_T-y_0)^2] G''(y_0)$$

By definition of the expectation,

$$E[B_T(y_T)] = B_T(y_0)$$

Thus:

$$E(y_T-y_0) G'(y_0) + \frac{1}{2} E[(y_T-y_0)^2] G''(y_0) = 0$$

We use the approximation

$$E[(y_T-y_0)^2] = \sigma_T^2 y_0^2 T$$

and obtain after after some algebra:

$$E(y_T) = y_0 - \frac{1}{2}  \sigma_T^2 y_0^2 T \frac{G''(y_0)}{G'(y_0)}$$

### Special Case of Natural Time Lag

Consider an interest rate derivative where the payoff depends on a
$\tau$-period rate, and where the same duration $\tau$ occurs between
the observation of the rate and the occurence of the payoff. In such
case, $\tau$ is named a “natural time lag”. This is the case for many
vanilla interest derivatives such as LIBOR swaps. In this case, the need
for a convexity adjustment vanishes, as illustrated by the following
example:

Let $R_T$ be the interest rate, maturity $T+\tau$ observed at time $T$.
It determines a cash flow $R\tau$ also to be paid at time $T+\tau$. The
present value of this cash flow at time $T$ is:

$$\frac{T\tau}{1+R\tau} = 1-\frac{1}{1+R\tau}$$

Let $F$ be the forward rate between $T$ and $T+\tau$ and $B_T$ the
corresponding forward bond price. By definition of the forward rate:

$$B_T = \frac{1}{1+F\tau}$$

By definition of the risk neutral forward risk measure:

$$B_T = E[\frac{1}{1+R\tau}]$$

Thus:

$$E[\frac{1}{1+R\tau}] = \frac{1}{1+F\tau}$$

Finally,

$$\begin{aligned}
E[\frac{R\tau}{1+R\tau}] &=& 1-\frac{1}{1+F\tau} \\
&=& \frac{F\tau}{1+F\tau}\end{aligned}$$

Thus, instruments that feature a “natural time lag”, such as LIBOR swaps
and FRA can be priced assuming that the expected futures rate is the
forward rate.

### Convexity Adjustment Examples

To Do...