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Yield rates Net Present Value method= simply compare present values Second method= Compare internal rates of return Dollar weighted: Calculate i, assuming simple interest Time weighted: Multiply together growth factors, find equivalent for one year, keep careful track of deposits and withdrawals, do not include them in the factors Also, approximation assuming all transactions occur at mid year, where I= interest earned, A= beginning value, B= end value

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Example Dollar/Time Weighted Find the time weighted and dollar weighted yields if the original deposit of 100,000 dropped to 90,000 at mid-year but the deposit made at that point was 10,000 and the final amount in the fund was 110,000. Ans: Time: -1% Dollar: 0%

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Portfolio Methods Portfolio method Everybody receives same interest every year Just read down the table Investment year method Interest rates are based on investment for a few years, then pool under the portfolio rate Read across the row and then down the table

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Calendar Year of Original InvestmentInvestment Year Rates (in %) Portfolio Rates (in%) Calendar Year of Portfolio Rate yi1i1 i2i2 i3i3 i4i4 i5i

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Rates Spot rate- yield rate for zero coupon bond bought now Forward rates- yield rate for bond bought in the future Get comfortable deriving these rates from each other Formula Inflation Consider it a negative interest rate Just divide by (inflation rate)

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Duration Duration is a measure of time until cash flows, can be used to measure price sensitivity to changes in interest rates Macaulay duration, or just duration Weight times using PV of cashflow (current price) at those times Also the relative change in price due to changes in force of interest Modified duration Simply v times the Macaulay duration Relative change in price due to changes in i

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Convexity Convexity Relative second derivative of price, with respect to interest rate Second order approximations using convexity

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Immunization Immunization- protecting from changes in interest rates Cash-flow matching/ exact matching/ dedication Match liabilities exactly with assets, one for one Redington immunization PV(assets)=PV(liabilities) ModD(assets)=ModD(liabilities), or first order derivatives equal Convexity(assets)>Convexity(liabilities), or second order derivative greater Protects against small changes in i Full immunization PV(assets)=PV(liabilities) ModD(assets)=ModD(liabilities), or first order derivatives equal One asset cash inflow before and after liability cash outflow

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Problem 1 You are given this information about the activity in two different investment accounts. During 1999, the dollar weighted return for investment account K equals the time weighted return for investment account L, which equals i. Calculate i.ASM p.273 Answer: 15% Account K Activity DateFund Value Before ActivityDepositWithdrawal January 1, July 1, X October 1, X December 31, Account L Activity DateFund Value Before ActivityDepositWithdrawal January 1, July 1, X December 1,

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Calendar Year of Original InvestmentInvestment Year Rates (in %) Portfolio Rates (in%) Calendar Year of Portfolio Rate yi1i1 i2i2 i3i3 i4i4 i5i Problem 2 A person deposits 1000 on January 1, Let the following be the accumulated value of the 1000 on January 1, 2000: P: under the investment year method Q: under the portfolio yield method R: where the balance is withdrawn at the end of every year and is reinvested at the new money rate Determine the ranking of P, Q, and RASM p.284 Answer: R>P>Q

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Problem 3 The one-year forward rate for year 2 is 4%. The four-year spot rate is 10%. The expected spot rate at the end of year two on a zero-coupon bond maturing at the end of year 4 is 7%. Determine the one-year spot rate.ASM p.435 Answer: 22.96%

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Problem 4 The real rate of interest is 4%. The expected annual inflation rate over the next two years is 5%. What is the net present value of the following cash flows? ASM p.433 Year012 Cash Flow Answer:

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Problem 5 A bond with 7.5% annual coupons will mature at par on June 30, Determine the duration of the bond on December 31, 2004 if the effective rate of interest is 5.5% per annum. ASM p.452 Answer: 1.43

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Problem 6 A $100 par value bond with 7% annual coupons and maturing at par in 4 years sells at a price to yield 6%. Determine the modified duration of the bond. ASM p.452 Answer: 3.43

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Problem 7 An annuity-immediate has payments of $1,000, $3,000, and $7,000 at the end of one, two and three years, respectively. Determine the convexity of the payments evaluated at i=10%. ASM p.472 Answer: 7.63

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Problem 8 A company must pay liabilities of $1,000 due one year from now and another $2,000 due two years from now. There are two available investments: one year zero coupon bonds and two-year bonds with 10% annual coupons maturing at par. The one year spot rate is 8% and the one-year forwarrd rate is 9%. What is the company’s total cost of the bonds required to exactly (absolutely) match the liabilities? ASM p.472 Answer: 2,625

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Problem 9 A company must pay a benefit of $1,000 to a customer in two years. To provide for this benefit, the company will buy a one-year and three-year zero-coupon bonds. The one-year and three-year spot rates are 8% and 10%, respectively. The company wants to immunize itself from small changes in the interest rates on either side of 10% (Redington immunization). What amount should it invest in the one-year bonds? ASM p.472 Answer: 420

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