Deriving the Borrowing Rate in a Sterling Bond Arbitrage Example

Summary

The document presents a currency and bond arbitrage exercise involving a sterling bond, a dollar loan, spot conversion, and a forward sale of sterling. The question asks why the loan repayment is written as 1.68 times the borrowed dollar amount when the stated dollar investment grows from 100 to 105 over the period. The short answer corrects the interpretation: the 0.08 increment is measured against 1.60, so it corresponds to a 5% rate, not an 8% rate.

This clarification identifies the arithmetic behind the apparent borrowing rate and links it to the given accumulation factor. It does not complete the bond-pricing calculation, derive the no-arbitrage price, or explain how the negative carrying cost affects the forward quote. Readers should therefore treat it as a narrow correction to the rate interpretation, not as a full solution to the arbitrage strategy or a general guide to currency-adjusted bond pricing.

Key ideas

• The dollar accumulation factor in the problem is 1.05 over the period.
• An increase of 0.08 on a starting amount of 1.60 represents a 5% rate.
• The stated repayment amount uses the dollar loan principal multiplied by the accumulation factor.
• The response corrects the rate interpretation but does not derive the sterling bond's price.

Tags

# Interest rate on loan for purchasing Sterling bond


# Interest rate on loan for purchasing Sterling bond












I am struggling trying to find out where they get the $8$% interest rate for the loan you make to purchase the Sterling Bond in the following strategy:

Problem:

Suppose that $A(0)$ = $100$ and $A(1)$ = $105$ dollars, the present price of pound sterling is $S(0)$ = $1.6$ dollars, and the forward price is $F = 1.50$ dollars to a pound with delivery date $1$. How much should a sterling bond cost today if it promises to pay $£100$ at time $1$? Hint: The forward contract is based on an asset involving negative carrying costs (the interest earned by investing in sterling bonds).

Strategy:

Suppose that a sterling bond promising to pay $£100$ at time $1$ is selling for $x$ pounds at time $0$. To find $x$ consider the following strategy.

At time $0$:

• Borrow $1.6x$ dollars and change the sum into $x$ pounds.

• Purchase a sterling bond for $x$ pounds.

• Take a short forward position to sell $£100$ for $\$1.50$ to a pound with delivery date $1$.

Then, at time $1$:

• Cash the bond, collecting $£100$.

• Close the short forward position by selling $£100$ for $\$150$.

• Repay the cash loan with interest, that is, $1.68x$ dollars in total.

## Answer by Degustaf (score 1, accepted)

https://quant.stackexchange.com/a/14940

First, it's not a 8% loan. The .08 interest on 1.6 is 5%. It appears that it is coming from the $A (1) = 105$.

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