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EAR/APR Time Value of Money

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Returns and Compounding Returns are often stated in annual terms Interest is paid (accrues) within the year Example: Savings accounts: interest accrues every month How do Banks report interest rates? Unfortunately, they do not report annual growth rates

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APR Banks report Annual Percentage Yields or APRs Let N be the number of times a bank pays interest per year Example: if the bank pays interest annually N=12 Let r m be the effective monthly interest rate Then the APR is simply r m *12

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APR Example: A bank pays 5% semi-annually. What is APR? APR=0.05*2 = 10% What is the annual growth rate on the account? 1.05 2 -1=10.25%

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Effective Annual Rates The effective annual rate or EAR, is simply the annual growth rate. For the example above, the effective annual growth rate is 10.25%

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You cant move directly from EAR to APR or vice-versa without finding r first.

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Effective Annual Rate Effective Annual Rate: The effective return on an investment assuming all interest payments are reinvested at the same rate. Example: Account pays 1% per month In 1 year, a $1 investment has grown to Effective annual return is 13%

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Effective Annual Rates What is EAR on UCCU savings account if APR is 12%? Assume interest accrues monthly.

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Effective Monthly Rates The EAR is 15% What is effective 1-month rate? What is the APR? Assume interest accrues monthly.

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APR and EAR APRs are always defined for 1 year But we could have Effective 6 month rates Effective 3 year rates Effective 2-month rates

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Effective rates If the effective annual rate is 12%, what is the effective 6-month rate? 3 year rate? We just need to find the 6-month and 3-month growth rates.

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Effective Rates Effective 6-month rate is given by Effective 3-year rates is given by

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Pricing Bonds Zero-Coupon Bond Assume an investment of similar risk pays 8% per year. Face Value of Bond =1000 Bond Matures in 10 years Price? Just find present value

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Pricing Bonds How about a coupon paying bond? Assume an investment of similar risk pays 8% per year. Face Value of Bond =1000 Bond Matures in 3 years Bond pays a coupon of 50 at end of each year (5% of face value)

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Pricing Bonds Cash flows from bond 50 at end of year 1 50 at end of year 2 1050 at end of year 3 We can split up each of these cash flows and think of each of them as a separate zero-coupon bond.

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Pricing Bonds Cash Flow 1: 50 at end of year 1 Price of this cash flow: 50/1.08=46.30 Cash flow 2: 50 at end of year 2 Price of this cash flow: 50/1.08 2 = 42.87 Cash flow at end of year 3 Price of this cash flow: 1050/1.08 3 =900.21

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Pricing Bonds The price of this bond is just the sum of the PVs of each of the pieces Price= 46.3 + 42.87 + 900.21=989.38

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Pricing Semi-Annual Bonds Most Bonds pay coupons every six months Example: Face value: 1000 Coupon: 30 every six months Matures: 1.5 years Investment of similar risk: pays 2% every six months

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Pricing Semi-Annual Bonds Cash flows 30 at end of first 6 months 30 at end of first year 1030 when bond matures PV of cash flows 30/1.02 = 29.41 30/1.02 2 = 28.84 1030/1.02 3 = 970.59 Price of bond = 1028.84

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Yield-to-Maturity Suppose we know bond price, and cash flows. Yield-to-Maturity comes from the interest rate that makes the price equal to the sum of the PV of all cash flows. Notation: y: the rate that makes the price equal to the sum of discounted cash flows YTM: yield to maturity

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Yield-to-Maturity Example: Zero coupon bond Face value=1000 Price=600 Matures in 5 years For a zero-coupon bond, YTM=y

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Yield-to-Maturity Economic Interpretation: For a zero-coupon bond, YTM is the effective annual return. From example above: 5-year return: 1000/600-1=0.67

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Yield-to-Maturity Coupon paying bonds Example: Bond matures in 3 years Pays 4% coupon annually FV=1000 Price=$987

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Yield-to-Maturity In this case, it is too difficult to solve for y algebraically Use Financial Calculator N=3 pmt=40 Price=-987 FV=1000 Rate=? For an annual coupon paying bond, YTM = y

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Yield-to-Maturity Economic Interpretation For annual coupon paying bonds, YTM is the effective annual return assuming all coupons are reinvested at the same rate. From previous example Suppose we invest all coupons at 4.473% What do we get in 3 years?

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Yield-to-Maturity Future value of first coupon: 40(1.0473) 2 =43.66 Future value of second coupon 40(1.0473)=41.79 Future value of last cash flow 1040 Total future value =43.66+ 41.79+1040 = 1125.45 Total 3-yr return = 1125.45/987-1=14.027%

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Yield-to-Maturity Effective Annual Return:

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Yield-to-Maturity Semi-Annual Coupon paying bonds Example: Bond matures in 1 year Pays $20 coupon semi-annually FV=1000 Price=$990

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Yield-to-Maturity Again, in this case, it is too difficult to solve for y algebraically Use Financial Calculator N=2 pmt=20 Price=-990 FV=1000 Rate=?

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Yield-to-Maturity Economic Interpretation For semiannual coupon paying bonds, y is the effective six-month return assuming all coupons are reinvested at the same rate. From previous example Suppose we invest all coupons at 2.519% What do we get in 1 year?

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Yield-to-Maturity Future value of first coupon: 20(1.02519)=20.50 Future value of last cashflow 1020 Total future value =20.50 + 1020 = 1040.50 Total 1-yr return = 1040.50/990-1=5.10%

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Yield-to-Maturity Effective six-month return

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Yield-to-Maturity For semi-annual bonds, the YTM is always quoted as an APR: For semi-annual bonds, the coupon paid is always quoted as a percentage of face value times two.

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Example Semi annual bond Matures in 2 years Coupon rate=8% Face=1000 Price=955 What is y? What is YTM?

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Example Finding y: N=4 pmt =40 FV=1000 PV=-955 Rate?

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Example Find bond price Semi annual bond YTM=6% Coupon rate=9% Matures 3 years FV=1000

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Example Finding Price: N=6 FV=1000 r = 3% pmt= 45 Price=?

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Example Find Bond Price Annual bond YTM=6% Coupon rate = 9% Matures 3 years FV=1000

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Example Finding Price: N=3 (this is an annual bond) FV=1000 r = 6% pmt= 90 Price=?

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