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Class 12 &quot;and 15&quot; Completed

# An interest rate risk premium k always increasing

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– An interest rate risk premium K Always increasing with time to maturity. K That is because longer-term bonds are always riskier than shorter- term bonds. – An inflation premium K The inflation premium can increase or decrease with time to maturity. – If investors believe that inflation will be higher in the future, the yield curve slopes upwards. – If investors believe that inflation will decrease in the future, the yield curve will be downward sloping.

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Upward-Sloping Yield Curve 14
Downward-Sloping Yield Curve 15

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16 The Yield Curve and Discount Rates K The term structure can be used to compute the present and future values of a risk-free cash flow over different investment horizons. K Present value of one cash flow: Where r n is the EAR for valuing cash-flows that occur in period n. K Present value of a cash flow stream using the Term Structure of Interest Rates: n n n r CF PV ) 1 ( 0 + = n n n r CF r CF r CF PV ) 1 ( ... ) 1 ( 1 2 2 2 1 1 0 + + + + + + =
17 iClicker Example: Discounting using the Term Structure K Compute the present value of a risk-free three-year annuity of \$500 per year, given the following term structure: a) \$ 1,358.56 b) \$ 1,357.81 c) \$ 1,427,12 d) \$ 1,500.00 e) None of the above Term (Years) Rate 1 r 1 5.06% 2 r 2 5.11% 3 r 3 5.15%

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18 iClicker Example: Discounting using the Term Structure K Compute the present value of a risk-free three-year annuity of \$500 per year, given the following term structure: Term (Years) Rate 1 5.06% 2 5.11% 3 5.15% 56 . 358 , 1 \$ ) 0515 . 1 ( 500 ) 0511 . 1 ( 500 0506 . 1 500 3 2 0 = + + = PV
19 The Term Structure and our Annuity Formula K If the term structure is not flat, can we still use our annuity formula? K Consider our previous example: K Then if we can find an interest rate r for which the following holds, we can still use our annuity formula. K By the way: Such an r , that is the interest rate that sets the present value of a series of cash flows equal to its price is called the Internal Rate of Return (IRR) 56 . 358 , 1 \$ ) 0515 . 1 ( 500 ) 0511 . 1 ( 500 0506 . 1 500 3 2 0 = + + = PV 56 . 358 , 1 \$ ) 1 ( 500 ) 1 ( 500 1 500 3 2 = + + + + + r r r

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20 Reviving our Annuity Formula K Using a spreadsheet or a financial calculator, the IRR for below equation is equal to 5.12%. K This means that for r = 5.12%, we can price the annuity of cash flows using our annuity formula: – where the difference of 1 cent is due to rounding. K Note that the IRR always lies between the lowest and the highest discount rate in the term structure used to initially calculate the PV of the annuity. 56 . 358 , 1 \$ ) 1 ( 500 ) 1 ( 500 1 500 3 2 = + + + + + r r r 57 . 358 , 1 \$ 0512 . 0 ) 0512 . 1 ( 1 500 0512 . 1 500 0512 . 1 500 0512 . 1 500 3 3 2 0 = ⎡ − = + + = PV
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An interest rate risk premium K Always increasing with time...

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