3.6 Implied Forward Rates
Example I
You are given:
1. The investor will be indifferent as between the following two alternatives:
a) One-year Bill.
b) “Ride” a six-month Bill and buy another six-month at undetermined price with undetermined rate, i.e. “forward” rate – f.
2. The Maturity price on the Bill is Par.
Solution I
[latex]\frac{100}{(1+y_{2})^{2}}= \frac{100}{(1+y_{1})(1+f)}[/latex]
Where:
y1 is ½ the BEY of the theoretical 6-month Spot Rate and
y2 is ½ the BEY of the theoretical 1-year Spot Rate
[latex]\frac{100}{100}= \frac{(1+y_{2})^{2}}{(1+y_{1})(1+f)}[/latex]
[latex]1= \frac{(1+y_{2})^{2}}{(1+y_{1})(1+f)}[/latex]
[latex]f= \frac{(1+y_{2})^{2}-1}{(1+y_{1})}[/latex]
Note
Have you noticed that this is the same formula, which was used for the Modified Internal Rate of Return (MIRR) and Realized Compound Yield (RCY)?
Example II
You are given:
6-month Spot Rate = .0800
1-year Spot Rate = .0830
Therefore:
y1 = .0400
y2 = .0415
[latex]f= \frac{(1.0415)^{2}}{(1.0400)}-1=.043[/latex]
[latex]BEY f= (2)(.043)= .086[/latex]
Notation:
The above “f” is really: “1f1”
The first “1” is for one period hence, that is, starting one period from now; the second “1” is the rate for one period – beginning one period hence.
From this, we can derive the general formula:
nft = [(1 + yn+t) n+t ÷ (1+yn)n]1/t – 1
Where:
n is the number of periods hence and
t is the number of forward periods
Note
Once again this is the same formula, slightly modified, which was used for the MIRR and RCY. Do you recognize it? If not, conduct a review of those formulae.
Example III
You are given:
3-year Spot Rate = .09787
5-year Spot Rate = .11021
Find:
The Forward Rate on a 2-year Treasury, starting 3 years from now. (Remember: Treasuries are semis.)
Solution:
- We need to find: 6f4
- Using the formula: 6f4 = [(1 + y10)10 ÷ (1 + y6)6]1/4 − 1
-
y6 = (.09787) ÷ (2) = .048935 and y10 = (.11021) ÷ (2) = .055105 6f4 = [(1 + .055105)10 ÷ (1 +.048935)6]1/4 -1 = .064428 BEY 6f4 = (2) (.0644) = .1288
Note
“6f4“ means starting six (half-year) periods from now – at the beginning of the sixth period, for the length of four (half-year) subsequent periods.
