Question

10. (Lecture Note 2) The current spot price of trader, can borrow money at 6% per year wit 3% per year with continuous compounding. The cu ne current spot price of silver is $16.75 per troy ounce. Sam Chatwick, a metals w money at 6% per year with continuous compounding and lend money (risk-free) at with continuous compounding. The current six-month forward price of silver is $17.13. Does Jarl have any arbitrage opportunities? If one does exist. use an arbitrare table to demonstrate now you can make a riskless arbitrage profit. The arbitrage table should have the following column titles: Transaction", "Payoff (now)", and "Payoff (6 months)". For simplicity, you can assume there are no storage costs and no convenience yield on silver.

Answer 1

Values provided in the question are

• Current Spot rate of Silver = USD 16.75 per troy ounce
• Current Six month forward price = USD 17.13 per troy ounce
• Interest rate to Borrow Money = 6% per year with continuous compounding
• Interest rate to Lend Money = 3% per year with continuous compounding

Formula of Continuous compounding interest is as follows

FV = PV * e ^ (I * t)

Wherein

FY = Future Value

PV = Present Value

e = mathematical constant with approximate value of 2.7183

I = Interest rate

T = Time period in Year (In current case the time period is 6 months i.e. 0.5 years)

Sam Chatwik (SAM) can i) either borrow money at 6% per year with continuous compounding ii) Lend Money at 3% per year with continuous compounding and accordingly invest in Spot and Future Silver

Case I: Sam Borrowers Money at 6% per year with continuous compounding

SAM can borrow USD 16.75 at start of the period for six month and utilize the proceeds from the borrowed money to buy one troy ounce of silver at USD 16.75 and parallelly sell one troy ounce of silver at six-month forward price of USD 17.13

Using Continuous Compounding formula, we can calculate the money SAM needs to repay at end of the period i.e. 6 months from date SAM borrowed USD 16.75

Hence,

PV = Present Value = USD 26.75

E = mathematical constant = 2.7183

I = Interest rate = 6%

T = Time Period = 6 months = 0.5 years

As per Formula

FV = PV * e ^ (I * T)

FV = 26.75 * 2.7183 ^ (6% * 0.5) = USD 17.26

Hence SAM needs to repay USD 17.26 at end of the period however; SAM would only receive USD 17.13 from sale of one troy ounce of silver at six month forward. In view of the same it can be established that Arbitrage opportunity is not present when SAM Borrows money

CASE II: SAM Lends money at 3% continuous compounding

SAM can sell the one troy ounce of silver at current spot rate for which SAM will receive USD 16.75 and at the same time SAM can buy one troy ounce of silver at Six month forward Price of USD 17.13. SAM can lend the USD 16.75 at 3% continuous compounding for period of Six Month.

Using Continuous Compounding formula, we can calculate the money SAM will receive at end of the period i.e. 6 months from date SAM lent USD 16.75

Hence,

PV = Present Value = USD 26.75

E = mathematical constant = 2.7183

I = Interest rate = 3%

T = Time Period = 6 months = 0.5 years

As per Formula

FV = PV * e ^ (I * T)

FV = 26.75 * 2.7183 ^ (3% * 0.5) = USD 17.00

Hence SAM will receive USD 17.00 at end of the period, however SAM would need to pay USD 17.13 for buying one troy ounce of silver at six month forward. In view of the same it can be established that Arbitrage opportunity is not present when SAM Lends money.

Summary for both the cases can be represented below

CASE	Transaction	Payoff (now)	PayOff (6 months)
CASE I	Borrow Money at 6% annually continuous compound interest for six month	Receive USD 16.75	Pay USD 17.26
	Buy one troy ounce silver at current spot price and sell one troy ounce silver at Six Month Forward Price	Pay USD 16.75	Receive USD 17.13
CASE II	Lend Money at 3% annually continuous compound interest for six month	Pay USD 16.75	Receive USD 17.00
	Sell one troy ounce silver at current spot price and Buy one troy ounce silver at Six Month Forward Price	Receive USD 16.75	Pay USD 17.13

It can be established that Arbitrage opportunity is not present for either of the cases.