Step 1

Specify values for the settlement date, maturity dates, coupon rates, and market prices for 10 U.S. Treasury Bonds. This data allows you to price a five-year swap with net cash flow payments exchanged every six months. For simplicity, accept default values for the end-of-month payment rule (rule in effect) and day-count basis (actual/actual). To avoid issues of accrued interest, assume that all Treasury Bonds pay semiannual coupons and that settlement occurs on a coupon payment date.

Settle   = datenum('15-Jan-1999');

BondData = {'15-Jul-1999'  0.06000   99.93
            '15-Jan-2000'  0.06125   99.72
            '15-Jul-2000'  0.06375   99.70
            '15-Jan-2001'  0.06500   99.40
            '15-Jul-2001'  0.06875   99.73
            '15-Jan-2002'  0.07000   99.42
            '15-Jul-2002'  0.07250   99.32
            '15-Jan-2003'  0.07375   98.45
            '15-Jul-2003'  0.07500   97.71
            '15-Jan-2004'  0.08000   98.15};

BondData is an instance of a MATLAB^® cell array, indicated by the curly braces ({}).

Next assign the date stored in the cell array to Maturity, CouponRate, and Prices vectors for further processing.

Maturity   = datenum(char(BondData{:,1}));
CouponRate = [BondData{:,2}]';
Prices     = [BondData{:,3}]';
Period     = 2; % semiannual coupons

Step 2

Now that the data has been specified, use the term structure function zbtprice to bootstrap the zero curve implied from the prices of the coupon-bearing bonds. This implied zero curve represents the series of zero-coupon Treasury rates consistent with the prices of the coupon-bearing bonds such that arbitrage opportunities will not exist.

ZeroRates = zbtprice([Maturity CouponRate], Prices, Settle)

ZeroRates =

    0.0614
    0.0642
    0.0660
    0.0684
    0.0702
    0.0726
    0.0754
    0.0795
    0.0827
    0.0868

The zero curve, stored in ZeroRates, is quoted on a semiannual bond basis (the periodic, six-month, interest rate is doubled to annualize). The first element of ZeroRates is the annualized rate over the next six months, the second element is the annualized rate over the next 12 months, and so on.

Step 3

From the implied zero curve, find the corresponding series of implied forward rates using the term structure function zero2fwd.

ForwardRates = zero2fwd(ZeroRates, Maturity, Settle)

ForwardRates =

    0.0614
    0.0670
    0.0695
    0.0758
    0.0774
    0.0846
    0.0925
    0.1077
    0.1089
    0.1239

The forward curve, stored in ForwardRates, is also quoted on a semiannual bond basis. The first element of ForwardRates is the annualized rate applied to the interval between settlement and six months after settlement, the second element is the annualized rate applied to the interval from six months to 12 months after settlement, and so on. This implied forward curve is also consistent with the observed market prices such that arbitrage activities will be unprofitable. Since the first forward rate is also a zero rate, the first element of ZeroRates and ForwardRates are the same.

Step 4

Now that you have derived the zero curve, convert it to a sequence of discount factors with the term structure function zero2disc.

DiscountFactors = zero2disc(ZeroRates, Maturity, Settle)

DiscountFactors =

    0.9704
    0.9387
    0.9073
    0.8739
    0.8416
    0.8072
    0.7718
    0.7320
    0.6945
    0.6537

Step 5

From the discount factors, compute the present value of the variable cash flows derived from the implied forward rates. For plain interest rate swaps, the notional principal remains constant for each payment date and cancels out of each side of the present value equation. The next line assumes unit notional principal.

PresentValue = sum((ForwardRates/Period) .* DiscountFactors)

PresentValue =

    0.3460

Step 6

Compute the swap's price (the fixed rate) by equating the present value of the fixed cash flows with the present value of the cash flows derived from the implied forward rates. Again, since the notional principal cancels out of each side of the equation, it is assumed to be 1.

SwapFixedRate = Period * PresentValue / sum(DiscountFactors)

SwapFixedRate =

    0.0845

The output for these computations is:

  Zero Rates  Forward Rates
    0.0614        0.0614
    0.0642        0.0670
    0.0660        0.0695
    0.0684        0.0758
    0.0702        0.0774
    0.0726        0.0846
    0.0754        0.0925
    0.0795        0.1077
    0.0827        0.1089
    0.0868        0.1239

  Swap Price (Fixed Rate) = 0.0845

All rates are in decimal format. The swap price, 8.45%, would likely be the mid-point between a market-maker's bid/ask quotes.