Finding Macaulay Duration given a term structure of interest rates
How do you find Macaulay Duration when there are interest rate shocks throughout the horizon of the debt till maturity
For example
Semi-Annual compounding of interest
Par value $1000
Annual Nominal Coupon 6%
Years to Maturity 10
Nominal Annual Yield to Maturity (YTM): 4%, 7%, 6%, 5%, 7.5%, 6.5%, 4.5%, 5.5%, 8%, 7%
On the same lines how would you find a single investor's rate of return (YTM) on this investment given that she received a series of returns over the horizon of the debt till its maturity
Isn't Macaulay Duration an instantaneous measure, which means that, at every point in time, it's based on the current YTM?
As to your 2nd question, I don't really get it at all. If the investor purchases the bond at a given point in time at a particular YTM, it doesn't matter what happens to this bond subsequently. So this idea that there's a "serie of returns" is a little alien.
Yes you buy a bond at a particular yield but over the horizon yield does not stay fixed thus one would seek a single rate that reflects the overall return on the bond
For example a 3% annual coupon bond with par value of $1000 having 10 years to maturity has the following yields for each of the next 10 years
1%, 2%, 3%, 3%, 2%, 1%, 2%, 3%, 1%, and 2%
The price of the bond or market value of this bond is $1,090.27
The discounted cash flows are
DCF1 = 30 x (1+1.00%)^-1 30 x0.99010 29.70 DCF2 = 30 x (1+2.00%)^-2 30 x0.96117 28.84 DCF3 = 30 x (1+3.00%)^-3 30 x0.91514 27.45 DCF4 = 30 x (1+3.00%)^-4 30 x0.88849 26.65 DCF5 = 30 x (1+2.00%)^-5 30 x0.90573 27.17 DCF6 = 30 x (1+1.00%)^-6 30 x0.94205 28.26 DCF7 = 30 x (1+2.00%)^-7 30 x0.87056 26.12 DCF8 = 30 x (1+3.00%)^-8 30 x0.78941 23.68 DCF9 = 30 x (1+1.00%)^-9 30 x0.91434 27.43 DCF10 = 1030 x (1+2.00%)^-10 1030 x0.82035 844.96
There is a equivalent single yield at which the bond has the same price of $1,090.27
And that yield happens to be roughly 2%
The yield calculation
f(i) = 1000 + -1090.27 * (1+i)^10 + 30 [(1+i)^10 - 1]/i
f'(i) = 10 * -1090.27 * (1+i)^9 + 30 * (10 i (1 + i)^9 - (1 + i)^10 + 1) / (i^2)
i = 0.1 f(i) = -1349.7569 f'(i) = -23415.3806 i1 = 0.1 - -1349.7569/-23415.3806 = 0.042355972037435 Error Bound = 0.042355972037435 - 0.1 = 0.057644 > 0.000001
i1 = 0.042355972037435 f(i1) = -286.6577 f'(i1) = -14145.9346 i2 = 0.042355972037435 - -286.6577/-14145.9346 = 0.022091653206444 Error Bound = 0.022091653206444 - 0.042355972037435 = 0.020264 > 0.000001
i2 = 0.022091653206444 f(i2) = -24.8892 f'(i2) = -11753.5739 i3 = 0.022091653206444 - -24.8892/-11753.5739 = 0.019974070028028 Error Bound = 0.019974070028028 - 0.022091653206444 = 0.002118 > 0.000001
i3 = 0.019974070028028 f(i3) = -0.2425 f'(i3) = -11525.1552 i4 = 0.019974070028028 - -0.2425/-11525.1552 = 0.019953026199606 Error Bound = 0.019953026199606 - 0.019974070028028 = 2.1E-5 > 0.000001
i4 = 0.019953026199606 f(i4) = -0 f'(i4) = -11522.9046 i5 = 0.019953026199606 - -0/-11522.9046 = 0.019953024144546 Error Bound = 0.019953024144546 - 0.019953026199606 = 0 0.000001
Periodic YTM = 2% Nominal YTM = 2% Annual effective yield = 2%
Right, so you're trying to compute what's sometimes referred to as the "realized compound YTM" then? Specifically, you're making an explicit assumption that coupons will be reinvested not at the original YTM, but rather at the YTM's that prevail in the future.
If this is correct, you just need to do the math and come up with a number. I can't really offer you much insight, since the concept isn't something that's commonly used, AFAIK.
You have the coupon and price, you know that the yield is a function of those two things right?
Building cash flows for bonds is simple, I would suggest you do it for this example.. I'll set up the example below
Year 0 -10,903 (10,000 * Price / 1000) Year 1 300 (3% * 10,000) Year 2 300 Year 3 300 Year 4 300 Year 5 300 Year 6 300 Year 7 300 Year 8 300 Year 9 300 Year 10 10,300 (Interest + Principal)
What is the internal rate of return of this cash flow? 1.99%. What is the yield? 1.99%.
What are you doing above? I don't know.
What does "the yield does not stay fixed" mean? I don't know.
Very well said
My assumption was that all such yields are projected and have not materialized yet
Isn't this why they do hedging against the unexpected change in interest rate to immunize the debt
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