Based on the follow-up queries in the same Q&A thread below (i.e.,

the answer to this practice question), I just wanted to now add an iteration on the same idea but now based on semi-annual compounding (the FRM assigned Tuckman uses semi-annual compounding throughout).

*We can always expect compound frequency to make a difference; it is the question's job to specify a compound frequency. The only real exception is when compound frequency is already implied by the instrument; e.g., a bond that pays a semi-annual compound is, by default, priced with semi-annual compound frequency; a 90-day Eurodollar futures contract is implicitly compounded quarterly. *
To illustrate how we can get a different six-month forward rate, compound frequency depending, say we have two bonds and their prices are:

P(0.5) = $99.00, and P(1.0) = $97.00

If the assumption is semi-annual compounding, prices are a function of spot rates discounted semi-annually::

P(0.5) = F/(1+R[0.5]/2)

P(1.0) = F/(1 + R[1.0]/2)^2

The no-arbitrage assumption (ex ante, neglecting liquidity preference, we have the same expectation for, on the left, investing for six months and then rolling over at the six month forward, as we do, on the right, for just investing directly for one year) tells us:

(1 + R[0.5]/2)*(1+F[0.5,1.0]/2) = (1+R[1.0]/2)^2, and solving for the forward:

F[0.5,1.0] = [(1+R[1.0]/2)^2 / (1 + R[0.5]/2) - 1] * 2, then substituting prices:

**F[0.5,1.0] = [P(0.5)/P(1.0) - 1] * 2; i.e., the six-month forward rate under semi-annual compounding**
Using the prices as an example, if the question is,

*what is the six-month forward rate, F[0.5, 1.0]*:

- Under semi-annual compounding (2 compound periods per year): F[0.5,1.0] = (99/97 - 1)*2 = 4.124%
- Now compare this, instead, to the annual compounding (
**illustrated in my prior post** above): F[0.5,1.0] = (99/97)^2 - 1 = 4.166%

And, let's test it:

**under semi-annual compounding:**
- spot rate s(0.5) = (100/99 - 1)*2 = 2.020%,
- s(1.0) = [SQRT(100/97) - 1] * 2 = 3.069%, such that we should have an equality:

(1 + 2.020%/2)*(1 + **4.124%**/2) = (1+3.069%/2)^2

**under annual compounding:**
- spot rate s(0.5) = (100/99)^(1/0.5)-1 = 2.0304%,
- spot rate s(1.0) = (100/97) - 1 = 3.0928%,such that we should have an equality:
- (1 + 2.0304%)^0.5 * (1 +
**4.166%**)^0.5 = 1 + 3.0928%

Both are valid six-month forward rates (the "six months" is the forward time period, in no way does it imply a six-month compound frequency). I hope that is helpful, thanks, David[

]

## Stay connected