Financial Math Archives - Tricks on the HP17b

These are handy tricks which allow you to check the pricing of "non-vanilla" interest rate and currency swap features, such as delays, stubs, full coupons, etc. using a financial calculator. They are meant to help provide a "reality check" to the pricing of "non-vanilla" swaps. Topics covered include: delays, short stubs, full first coupons, upfront payments, basis swap pricing, basis point conversions, and unwind pricing.

Primer : The Present Value of a Basis Point Running

There are 2 methods of calculating the value of a basis point on your HP17b.

eg. 5 year swap, $1,000,000 notional, with an at-market rate of 6% semi-annual.

Method 1 - Time Value of Money

Select FIN, TVM (financial, time value of money on your HP17b)

Set to 2 P/YR

N = 10

PMT = 0.01% x 1,000,00 / 2 = $50

I%YR = 6.0%

FV = 0

Solve for PV = $426.51

Method 2 - Bond Method

Select FIN, BOND

TYPE = 30/360 SEMI

SETTLE = 7.182000 (July 18, 2000)

MATURITY = 7.182005 (July 18, 2005)

CPN = 6%

YLD = 6.01%

Solve for PRICE = 99.957

the difference between $100 and $99.957 is $0.04264, or $426.40 per $1,000,000

1. Upfront Payments

eg. 5 year swap, $1,000,000 notional, with an at-market rate of 6% semi-annual.

Find the rate a swap dealer would have to pay if it receives an upfront payment of $2,000.

The number of basis points running which corresponds to an upfront of $2,000 is:

$2,000 / $426.51 = 4.7 basis points

Therefore, the swap dealer would pay 6.000% + 0.047% = 6.047%

2. Delays

(a) short delays

eg. 1 month delay on a 5 year swap

1 month BA rate: 5.50%

5 year and 1 month swap rate: 6.00% semi-annual (5.93% monthly). This was found by interpolating the 5 year and 6 year semi annual swap rates.

On $1mm notional, the value of the delay can be assessed as follows:

(5.93% - 5.5%) x 1,000,000 / 12 = $358

The value of 1 basis point point running on a 5 year, $1,000,000 notional swap at 6.00% semi is $426.50.

Therefore the value of the delay is $358/$426.50 = 0.84 basis points, and the swap rate for a 5 year swap, 1 month delayed is 6.0084% semi.

(b) longer delays

eg. 5 year swap starting 5 years forward

5 year swap: 6%, 10 year swap: 6.40%

Select FIN, TVM (financial, time value of money on your HP17b)

Set to 2 P/YR

For the first 5 years:

PV = -100

N = 10

PMT = 6% x 100 / 2 = 3

I%YR = 6.4%

Solve for FV : 102.314

For the next 5 yrs

PV = -102.314

FV = 100

N = 10

I% YR = 6.4

Solve for PMT : 3.474

Therefore, the coupon on the last 5 years which gives the IRR of 6.4% semi over the full ten years is 2 x 3.474 / 100 = 6.948%

(c) delayed start basis swaps

eg. 5 year CAD-USD basis swap, 6 months forward

5 year and 6 month basis swap: BA+12

6 month basis swap BA+8

Value of a basis point on $1,000,000 notional:

6 months = approx $50

5 years, starting in 6 months = approx $420

5.5 years = approx $470

Over the entire 5.5 years, the following must hold true:

8 bps for 6 months + ? bps for 5 years = 12 bps for 5.5 years

on CAD $1mm notional

(8 x $50) + (? x $420) = (12 x $470)

? = 12.476

Therefore, the price on a 5 year basis swap 6 months forward is BA+12.476

3. Short Stubs

eg. a 5 year and 1 month swap

5 year and 1 month swap rate: 6.00% semi-annual (5.93% monthly). This was found by interpolating the 5 year and 6 year semi annual swap rates.

This 6% semi rate is applicable if the decompounded 5.93% monthly rate is used for the initial 1 monh stub. If a single "flat" rate is required on the swap, the adjustment to the overall rate can be approximated as follows:

On $1mm notional, the value of the stub can be assessed as follows:

(6% - 5.93%) x 1,000,000 / 12 = $58.33

The value of 1 basis point point running on a 5 year, 1 month, $1,000,000 notional swap at 6.00% semi is $433.

The solution is iterative. The first approximation of the value of not decompounding the stub is $58.33/$433 = 0.13 basis points running, and the swap rate for a 5 year and 1 month swap is 5.9987%.

4. Full First Coupons

eg. 5 year 1 month swap, with full first coupon on the fixed side, and 1 month short stub on the floating side.

5 year and 1 month swap rate: 6.00% semi-annual. This was found by interpolating the 5 year and 6 year semi annual swap rates.

the process for calculating the at-market rate for a full first swap is iterative. First approximation to calculate the value of 5 months of extra interest:

6% x (5/12) x 1,000,000 = 25,000

On a notional amount of $1,000,000, the value of a basis point = $432

expressed in basis points running, the value of the extra interest is $25,000/$432 = 58

2nd iteration: the current best guess is 6% - 0.58% = 5.42%

5.42% x (5/12) x 1,000,000 = $22,583

or, 22,583/432 = 52 basis points

Therefore, the at-market rate for this swap is 6% - 0.52% = 5.48%

5. Pricing Basis Swaps

Quick method for a basis swap with exchange of principle front and back (at spot fx rate)

eg. 5 year BA - LIBOR swap, Notional CAD 1,000,000, USD amount based on fx spot.

Determine what Libor+10 equates to

CAD 5yr swap = 6%, value of bp = $426.51

USD 5yr swap = 7%, value of bp = $415.83

BA-Libor Market = BA+10

basis point conversion

1 bp CAD = (426.51/415.83) x (360/365) = 1.011 bps USD

The calculation factors in (a) the ratio of the bp value, and (b) the daycount difference.

10 bps USD = 10/1.011 = 9.89 bps CAD

With BA - Libor at BA+10, the at-market CAD side of the swap would be BA+19.89

6. Unwinds

eg. swap dealer is paying 7% s.a on a swap maturing 18may, 2005

current BA setting, May 18 to Aug 18, 5.75%

Market Rates: 1 month BA rate: 5.50%

5 year and 1 month swap rate: 6.00% semi-annual (5.93% monthly). This was found by interpolating the 5 year and 6 year semi annual swap rates.

The Fixed Side is like a Bond

Select FIN, BOND

TYPE = 30/360 SEMI

SETTLE = 7.182000 (July 18, 2000) Today

MATURITY = 5.182005 (May 18, 2005)

CPN = 7%

YLD = 6%

Solve for PRICE = 104.13

Solve for ACCRU = 1.16

The NPV = Price + Accrual = 105.29

or $1,052,900 on $1,000,000 notional

or -$1,052,000 to the swap dealer

The Float side

The value is comprised of 3 components:

(a) principal of $1,000,000

(b) accrual: 5.75% x (2/12) x 1,000,000 = 9,583

(c) capital gain/loss: (5.75% - 5.5%) x (1/12) x 1,000,000 = 208

or 1,009,791 to the swap dealer

Net the swap is worth -42,209 to the swap dealer. They would have to pay this to unwind the swap.

This information is purely illustrative, using simplified examples for educational purposes only. For more information, please contact me at edmurphyconsulting@gmail.com