The yield offered by a bond with 18 months remaining to maturity is 5%. The coupon is 3%, paid semi-annually, and there are two more coupon payments to go in addition to the interest payment made at maturity. What is the bond's price?
Correct Answer: D
Explanation This is a straightforward bond valuation problem where all variables required are known. The coupon payments will be $1.50 in 6 months and 1 year from now, and a final paymento of $101.50 will be received in an year's time. This can be discounted at the yield provided as follows, and summed together to get the bond price.
Question 7
An investor enters into a 4 year interest rate swap with a bank, agreeing to pay a fixed rate of 4% on a notional of $100m in return for receiving LIBOR. What is the value of the swap to the investor two years hence, immediately after the net interest payments are exchanged? Assume the current zero coupon bond yields for 1, 2 and 3 years are 5%, 6% and 7% respectively. Also assume that the yield curve stays the same after two years (ie, at the end of year two, the rates for the following three years are 5%, 6%, and 7% respectively).
Correct Answer: C
Explanation The swap can be valued by valuing the two individual components of the swap. The fixed rate bond equivalent in the swap is valued at =4/1.05 + 104/(1.06^2) = $96,369,154. The FRN component will be valued at par as we are at a point where the rate has just been reset, ie $100m. The investor is paying the fixed rate, and is therefore short the bond. He/she is receiving LIBOR, and is therefore long the FRN. The value of the swap to the investor therefore is +$100,000,000-$96,369,154 = $3,630,846 Detailed explanation: An Interest Rate Swap exchanges fixed interest flows for floating rate flows. The floating rate leg is tied to some reference rate, such as LIBOR. The parties exchange net cash flows periodically. Conceptually, an interest rate swap is the combination of a fixed coupon bond and a floating rate note. The party receiving the fixed rate is long the fixed coupon bond and short the FRN, and the party receiving the floating rate is long the FRN and short the fixed coupon bond. An interest rate swap can be valued as the difference between the two hypothetical bonds. FRNs sell for par at issue time as they pay whatever the current rate is, subject to periodic resets. Therefore immediately after a payment is made on a swap, the value of the FRN component is equal to its par value. The bond can be valued by discounting its cash flows. The difference between the two represents the value of the swap. When the swap is entered into, the fixed rate leg is set in such a way that the value of the hypothetical bond is equal to that of the FRN, and therefore the swap is valued at zero. The rate at which the fixed rate leg is set is called the swap rate. Over its life, market rates change and the value of the fixed coupon bond equivalent in our swap diverges from par (whereas the FRN stays at par - at least right after payments are exchanged and the new floating rate is set for the next period). Thus the swap acquires a non-zero value. There are two ways to value a swap. If interest rates for the future are known, the bond and the FRN can be valued and their difference will be equal to the value of the swap. Sometimes, the current swap rates are known. In such a case, the swap can be valued by imagining entering into an opposite swap at the new swap rate, which will leave a residual fixed cash flow for the remaining life of the swap. This residual cash flow can be valued and that represents the value of the swap. For example, if a 4 year swap was entered into exchanging an annual fixed 5% payment on a notional of $100m for a floating payment equal to LIBOR, and at the end of year 1 the swap rate is 6%, then the party paying fixed can choose to enter into a new swap to receive 6% and pay LIBOR. All cash flows between the old and the new swap will offset each other except a net receipt of 1% for the next 3 years. This cash flow can be valued using the current yield curve and represents the value of the swap.
Question 8
What would be the expected return on a stock with a beta of 1.2, when the risk free rate is 3% and the broad market index is expected to earn 8%?
Correct Answer: C
Explanation The stock has a beta of 1.2, therefore intuitively it can be expected to earn more than the broad market index. It will earn the risk free rate, ie 3%, and 1.2 times the equity risk premium of 5% (8% - 3%). The expected returns from the stock therefore are 3% + (8% - 3%)*1.2 = 9%
Question 9
A short position in a 3 x 6 FRA is equivalent to which of the following?
Correct Answer: D
Explanation The buyer of an FRA is considered 'long', and a 3 x 6 FRA allows him or her to borrow funds at the agreed rate starting at the end of month 3 till the end of month 6. (ie, 3 x 6 indicates that the borrowing period commences at 3 months, and ends at 6 months, for a period of 6 - 3 = 3 months). The seller has exactly the opposite position, ie he or she is committed to lend funds at the agreed rate for 3 months starting at the end of 3 months from today (for a 3 x 6 FRA). [Note that the obligation to borrow or lend in and FRA does not mean that either parties will actually do that, instead they will just exchange cash flows to get to get to an identical economic situation.] Since the seller, or the short, is committed to lending in the future starting 3 months from now, it is akin to borrowing now for 3 months, and investing the borrowed amount for 6 months. During the first 3 months, the amounts borrowed and lent 'cancel' out (conceptually) and after the 3 months the short returns the amount borrowed, and is left with just the net amount lent, ie the FRA. Thus the correct answer is Choice 'd'. Choice 'b' describes the position of the FRA buyer. Choice 'a' and Choice 'c' are nonsensical.
Question 10
If the 3 month interest rate is 5%, and the 6 month interest rate is 6%, what would be the contract rate applicable to a 3 x 6 FRA?
Correct Answer: B
Explanation The correct answer is Choice 'b', as this question is merely asking for the forward rate based on known spot rates. The forward rate applicable to the three month period commencing in 3 months time is given by [(1 + 6%*6/12)/(1 + 5%*3/12) - 1]*4 = 6.91%. Thus Choice 'b' is the correct answer. Here is a step by step way to think about it: $1 invested now at 6% for 6 months grows to (1 + 6%*6/12)=1.03. At the same time, using the 3 month rate, $1 invested now at 5% for 3 months grows to (1 + 5%*3/12)=1.0125. Effectively, this means that the 1.0125 at the end of 3 months grow to 1.03 at the end of 6 months, implying the rate of interest during the 3 months from 3 to 6 months is (1.03/1.0125 - 1)*4 = 6.91%.