-Could you please this statement:
"Any forward rate can be locked in today by buying one unit of the n-year zero at price Pn= 100/(1+sn)^n and by short selling Pn/Pm units of the m year at price Pm= 100/(1+sm)^m (such a weighting requires no net investment today because both the Cash inflow and outflow amount to Pn)
-I don’t understand
i. Why do we need to shortsell Pn/Pm units?
ii. I don’t understand this :“such a weighting requires no net investment today because both the Cash inflow and outflow amount to Pn”
Such arrangement requires no net investment because you buy one unit of the n-year zero at price Pn meanwhile short selling Pn/Pm units of the m year at price Pm. (You pay Pn and receive Pn/Pm*Pm)
I suggest you plug the numbers in excel to see what is happening.
I have tried to exaplain here but it is much easier when you see the formuals is excel.
1 year zero coupon sells at $0.95238 (5%) = Pn
2 year zero cooupn sells at $0.82645 (10%) = Pm
1yr fwd irate n 1 year time equals = 15.238%
1 year zcb in 1 years time should sell at = 1/1.15238 = $0.86777
from initital ZCB we have
price 2 year / price 1 year =
0.82645/0.95238 = $0.86777 (same as above)
Buy $1 par of 1 year zero cost $0.95238
short 2 year zero not $1 par but 0.95238/0.82645 = $1.15238 par
For this we would receive = $1.15238 x 0.82645 = $0.95238
The same as our purchase of the 1 year zero so no cost to position.
But remember we are short $1.15238 par of teh 2 year zero
In 1 years time will will receive the $1 par from our long (which could invest in what we like),
But still short $1.15238 par of the 2 year (which now has 1 year remainIng)
So we know we will have a 5% gain our long over 1 year
A 21% cost on your short ($1.5238/$0.95238) over 2 years
We have therefore locked in a implicit loss (borrowing cost) of 15.238% in the second year