Bond Pricing: The Yield Curve

This is a series of short posts examining the bond markets and some of the key pricing, risk and quoting concepts (with EXAMPLES!) –
        The Yield Curve
        Duration/Convexity and DV01
        Spread Quoting
        Z-Spread
        Roll PnL

As discussed in a previous post, bonds give investors a way to lock up their capital in return for a stream of coupon payments, with the full initial payment returned at maturity. Mathematically, the undiscounted payoff is very straightforward:

    \[\text{Payoff} = \sum_{i=1}^N CF(t_i)\]

where the cashflows CF(t_i) are typically annual or bi-annual, fixed percentages of the notional – and include the return of the notional at expiry.

After issuance, bond markets set the prices of bonds at each maturity. Typically government bonds – ‘govvies’ – in each currency set the overall ‘Yield Curve’ for the currency, which is the relationship between the rate of return against the length of investment (govvies are usually thought to be ‘risk free’ if denominated in the local currency as the government can print money to make repayments). Corporate bonds in each currency will trade at higher prices than govvies, with the ‘spread’ between a bond and a govvie determined by the market’s view of its credit-riskyness.

Bonds with higher coupons will typically be worth more to investors as they result in larger payments. However, if markets are efficient we expect that regardless of the coupons, $100 invested in two bonds from the same issuer that mature at the same time should yield the same return (otherwise, investors would pile out of one and into the other). The ‘Yield-to-Maturity’ (YtM) is a concept that captures this – it is the annually compounded rate of return expected from buying a given bond:

    \[C(t) = \sum_{i=1}^N CF(t_i) \cdot \frac{1}{(1 + r_y)^{t_i}}\]

Where C(t) is the market price of the bond, and r_y is the YtM of the bond, calibrated to satisfy the equality. Note that if the price of a bond traded on the market goes up, its calibrated YtM will fall, and vice versa.




Example: a United States Treasury (UST) has 2 years remaining until maturity, pays bi-annual 3% coupons and returns $100 at maturity. It is currently trading at $102.50. What is its YtM (assume it has already paid out its most recent coupon)?

This bond generates four payments of $1.5 in 6m, 12m 18m and 24m, and additionally returns $100 notional at 24m. The YtM is chosen to satisfy this equality:

    \[\$101.50 = \frac{\$1.50}{(1 + r_y)^{0.5}} + \frac{\$1.50}{(1 + r_y)} + \frac{\$1.50}{(1 + r_y)^{1.5}} + \frac{\$101.50}{(1 + r_y)^2}\]

Plugging this in to a root solver (eg. via excel) we find that this price implies a r_y of 1.731%. Note that, if this bond was trading at par (ie. $100) its YtM would equal its coupon rate of 3%. As the price is above par, the calculated YtM is below this.

When trading govvies, rather than quoting the price of a particular bond, dealers will usually quote the price in terms of its YtM, so that the return expected from a bond is more easily compared to other securities. Typically bonds with longer maturities will trade with a higher YtM, and for each currency we can use the liquidly-traded govvies to create a yield curve for this currency.

Because of this relationship, one way a portfolio manager can increase the yield of her portfolio is to increase its maturity profile by trading shorter-dated bonds for longer-dated bonds which will have a higher yield (although as we will see later, this also increases the interest rate risk of the portfolio).

An example Yield Curve is shown below for USD as of 22nd December 2017 from the US Treasury website:

Yield curves for USTs in early and late December 2017, built from US Treasury data

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