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We have seen in the article on what are bonds that the present value (PV) of a bond is given by its  discounted cash flow. We considered a 5 year treasury bond with  c =8% coupon rate, paid at the end of year and with a $1000 par value.

where r is our riskless rate of return.

Recall that the geometric sum can be expressed as

This allows us to further simplify the expression for PV - by avoiding the summation.

If in the above example r>c then the bond is selling at a discount with respect to the par value. This is of course because if I buy this bond for $1000 or more I would lose money with respect to putting this amount in a bank. So the PV<par value. If  c=r then  PV =and the bond is selling at the par value.  If c>r then the bond is selling at a premium since PV > par value.

Yield rate

All the above assumed a constant interest rate value. However, in reality this changes with time - so we cannot generally use the above equation. By knowing the time to maturity and the interest rate for each i year, we can find an effective rate called the yield rate y to maturity of the bond. In the example this is found by equating our geometric sum with constant yield y for the PV to the actual PV found by the discounted cash flow.

So what we find is that for different maturity dates we can compute the yield rate y. Note the yield is independent of the par value. A plot of yield y versus the maturity date is called the yield curve. Also note that since the bond must reach the par value at maturity, the higher the PV the lower the required yield rate.

Yield Curve

The yield curve is a valuable source of information to estimate the overall changes in interest rates. Under normal market conditions, the yield tends to  increase gently for longer maturities. The longer we fix our money in a bond, the larger the risk we assume in interest rate fluctuations. Therefore, bonds with long maturity dates require to give bigger yields in order to be attractive investments.

Typically, after an economic recession or at the beginning of an economic expansion the yield curve increases steeply with maturity date. Short term investors have the flexibility of trading their bonds in order to buy better yielding securities if the opportunity occurs.This flexibility increases the PV of the bond thus decreases the yield.Long term investor on the other hand require a larger yield to compensate from the cost in opportunity they are missing out on.

Conversely,a yield curve that falls with longer maturities is an indication of economic stagnation.There are less current opportunities in the market maybe due to a recession.Investors think it is best to be happy with lower longer term yield today than even lower yields tomorrow.

Duration

Assuming we know the yield ,the maturity date, coupon rate and the par value. At what time t=D will the coupon payments balance the future payments of the bond?  
(for physicists: The calculation of this time resembles those for finding center of masses.)

Financially, we are equally happy to receive a single total payment of V dollars in D years from now with present value PV as the bond coupon payments together with the par value at maturity.

D is called the duration of the bond. Note that in this calculation we assumed for all years i.

We also note that the duration measures the bond sensitivity to yield changes. We can see this by noting that:

This allows us to find the change in the bonds PV in relation to a yield change Δy assuming we know the PV the yield and duration. Durtion assumes a linear relationship between price and yield. However, this equation is not always accurate in predicting the change in value. This is because the PV does not generally fall linearly with increasing yield.

Convexity

When the yield is low the PV is very sensitive to any changes in the yield whilst for high yields this is not so. This is generally due to a number of reason. We can  reason this as follows. Interest rate fluctuations have a certain average response time associated to it. When the economy has a normal yield curve, low yielding bonds have a shorter maturity date. This means they are more vulnerable to interest rate fluctuations. To make these bonds attractive to investors - their price should increase more than just linearly if the interest rates drop. This drives their duration up. High yielding bonds have longer maturities and can average out these short term interest rate fluctuations and thus their duration is lower.

This second order response to yield change is called convexity C and is given by

Let PV(y) then by a simple Taylor expansion we have that

Then we define the effective duration and convexity as follows

A high convexity is desired since, as we have seen, if interest rates fall their price increases more than for other bonds. Whilst, if interest rates rise their price don't drop as much as the other bonds. If investors predict that interest rates are volatile they will prefer highly convex bonds. This will drive their demand up and thus their price.

Written by Raffaello Vardavas

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