Forward Rates: Implied Forward Rate Calculation
Suppose you recently received an inheritance from your uncle in the amount of $100,000. You want to save this for a down payment on a house you expect to buy in ten years. After much research, you have narrowed your investment choices down to just two:
1. Purchasing a 10-year bond with a yield to maturity of 6.5%, and
2. Purchasing two consecutive bonds, with the first bond having a yield to maturity of 6.2% with seven years to maturity, and the second bond being a three-year issue with a yield that will be determined seven years from now.
Of course, your expectations for future interest rates will determine which choice you prefer. If you believe that rates will not rise above 7% in the next 10 years, what would that suggest?
In summary:
[[summary]]
That interest rate or yield that makes you indifferent between the two bond choices is known as the __forward rate__, and it can be implied from current market yields. How might you state the "forward" idea at work here?
Finally, you might also find that you need to consider semi-annual bonds, since they are most common. Try a forward rate calculation for this same scenario using semi-annual periods instead, just doubling the semi-annual rate found in the end. It would look like this:
$$\displaystyle (1 + 0.031)^{14} \times (1 + IFR_{14},_{20-14})^{20-14} = (1 + 0.0325)^{20} $$.
The annual example leads more accurately to 7.2033% while this semi-annual adjustment leads to 7.2017%, so the difference isn't usually a large one. But it sure could make a difference on a high-stakes exam!
The question becomes, then, to determine what future return you expect to earn during that second period to make you indifferent between your two options. What do you know about the second three-year bond yield that would make you indifferent?
Actually, you do know something about this yield. Think about having this choice, and what that second bond would have to look like in order for you to consider the second choice of two bonds.
That wouldn't follow. If this was the case, the second choice of two bonds would be clearly the better offer.
Absolutely!
It's kind of like driving in a car. You can travel at a speed of 6.5 for 10 hours, or you can travel for the first seven hours at a speed of 6.2. But if you do that, you'll definitely need a speed greater than 6.5 for those last three hours to reach your destination in the same time frame.
No, that's a different spot rate that's not actually used here.
That's it!
The spot rate for $$A$$ periods is $$z_A$$, and then the implied forward rate (IFR) connects this rate and the longer-term spot rate $$z_B$$ for $$B$$ periods. The general formula looks like this:
$$\displaystyle (1+z_A)^A \times (1+IFR_A,_{B-A})^{B-A} = (1+z_B)^B $$.
No, that's some rate beyond the time frame of this situation.
Continuing with this example, what is the value of $$B-A$$ in this equation?
Right.
You can see this equation as just "the seven-year compounded rate times the three-year compounded forward rate equals the 10-year compounded rate." Substituting in what you know, this leads to:
$$\displaystyle (1 + 0.062)^7 \times (1 + IFR_7,_{10-7})^{10-7} = (1 + 0.065)^{10} $$.
And finally, some algebra leads to the implied forward rate:
$$\displaystyle IFR_{10} = \left[ \frac{(1 + 0.065)^{10}}{(1 + 0.062)^7} \right]^{1/3} - 1 \approx 0.072 = 7.2 \% $$.
Not quite. That's $$A$$ here.
Not quite. That's $$B$$ here.
No, the calculation connects the two rates perfectly. But now this is a question of reality and what choice is best.
Yes.
In some instances, these forward rates can be observed or computed from derivatives markets. The forward rate is interpreted as the marginal rate of return for extending an investment one period, and because it occurs at the point of an investor's indifference, it is also considered to be the breakeven rate.
Not really.
Note that if rates really don't rise over 7%, then you won't get enough "speed" to your future investment to make up for accepting just 6.2% today.
The calculation is wrong
It's the 3-year rate going forward from today
Continue
Nothing
That it has to be between 6.2% and 6.5%
That it has to be more than 6.5%
It's the 3-year rate going forward and starting 7 years from now
It's the 1-year rate that takes things forward at the end of the 10-year period
3
7
10
The 10-year bond is the better deal
The seven-year bond and reinvestment plan is a better deal
