April 2012 Q4

[padasala]

Any suggestions on how to go about solving question II and III?

I went about finding the forward price of a futures contract drawn at time 0 and a futures contract drawn at time 4. I then subtracted the two to find the value of the futures contract drawn at 0 at time 9 and discounted it back to time 4 to obtain the value at time 4....however, the answer does not do that...can you explain how to solve this?

Also, I am clueless on how to solve part (iii). Can you guide me here?

Regards,
Sunil




(i) Explain what is meant by the “no arbitrage” assumption in financial
mathematics. [2]
An investor entered into a long forward contract for a security four years ago and the
contract is due to mature in five years’ time. The price of the security was £7.20 four
years ago and is now £10.45. The risk-free rate of interest can be assumed to be 2.5%
per annum effective throughout the nine-year period.
(ii) Calculate, assuming no arbitrage, the value of the contract now if the security
will pay dividends of £1.20 annually in arrear until maturity of the contract.
[3]
(iii) Calculate, assuming no arbitrage, the value of the contract now if the security
has paid and will continue to pay annually in arrear a dividend equal to 3% of
the market price of the security at the time of payment. [3]
[Total 8]

 

[padasala]

Update

Hi,

I found out where I was going wrong in part 1.

However, I am still confused about the methodology of solving the question in part ii.

In part 1, I assumed that the dividends are paid from the beginning of the bond...but the question wants us to assume that the dividends are paid out only from the fifth year

 

[PaperBeetle]

iii.
The dividend is not a fixed amount per share, but is linked to share price. This is unknown in the future, so we cannot know how much the dividend will earn if invested at the risk-free rate. But we can reinvest it as shares; that way the unknown share price will cancel itself out in each re-investment, and each time we end up with 1.03 times as many shares.

So, the two non-arbitrage scenarios:
A. Buy 1 share at t=0 for £7.20. Reinvest dividends as shares each year, to end up with 1.03^9 shares.
B. Invest £7.20 at the risk-free rate, to end up with £7.20*(1+i)^9. Spend that on 1.03^9 shares at the forward price K(0,9).
\[\therefore K_{(0,9)}=7.20\frac{(1+i)^{9}}{1.03^{9}}\]
Similarly for the forward price K(4,5)...
\[K_{(4,9)}=10.45\frac{(1+i)^{5}}{1.03^{5}}\]
And so the value of the long position K(0,9) at t=4 is:
\[(K_{(4,9)}-K_{(0,9)})v^{5}=\frac{10.45}{1.03^{5}}-\frac{7.20(1+i)^{4}}{1.03^{9}}=£2.92\]
Well, I don't have the 2012 answers, so I don't know if that's right, but that's how I'd go about it.

 

[Mark Mitchell]

I confirm that's the right answer.

 

[Janboyd]

Why in the examiner's report do they invest for 4 and 0 years instead of 9 and 5?


The current value of the forward price of the old contract is:

7.20(1.025)^4 (1.03)^-9 = 6.0911

whereas the current value of the forward price of a new contract is:

10.45(1.03)^-5 = 9.0143

⇒ current value of old forward contract is:

9.0143− 6.0911= £2.9232

 

[Mark Mitchell]

To calculate the original forward price, we would need to use an accumulation factor for 9 years from time 0 to time 9.
To calculate the second forward price, we would need to accumulate for 5 years from time 4 ("now" in the question) to time 9.

Then to calculate the value of the contract at time 4, the difference between these two forward prices is discounted back for 5 years to time 4. The examiners report cancels some of the accumulation with the discounting, so the accumulations appear to be for 4 years and 0 years respectively.