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Interest Rates: Calculating the Yield of an Asset - Source: Mishkin and Serletis, Ch. 4- Interest rate: - a summary measure of the return on a financial asset, - importance: a measure of the benefit to the lender and a measure of the cost to the borrower.- We need a measure of this ‘Interest Rate’:- economists use a measure of the ‘internal rate of return’ on an investment. - this is also called the ‘Yield-to-Maturity’ (referred to as ‘Yield’ below).Yield (Internal Rate of Return):- Technically: the interest rate that equates the present value of cash flow from an asset with its cost. cost? could be the price of the asset; could be the present value of payments made to purchase the asset. - It measures the average annual return on the amount invested that gives a payment stream of the same value as the asset actually pays.- Yield is what economists mean by an “interest rate”.- Why is a measure of yield needed?- allows comparison of the generosity of very different payment streams. e.g. see text examples of the simple loan, discount bond, coupon bond.1 Calculating the Yield using Present Values: - Say an asset provides a stream of payments:A1 , A2 , A3 ,.... AN-1 , AN where Aj is the payment received in period j (think of a period as a year)- Dollars received in different years are not of equivalent value.$1 in the future is worth less than $1 now.Why? $1 received now could be invested at a rate of return "i", giving the person: $1 x (1+i) > $1 in one year e.g. i=.05 (5%) gives $1.05 in 1 year.How much is $1 in one-year worth now? i.e. what is its present valueAnswer: why? investing this amount at a return i now would give $1 in one year(1+i) x = 1Similarly, the values now (present values) of $1 received in: two years: (investing this for 2 years at i gives $1 in 2 years) three years: N years: - So the value in present dollars of the stream of payments: A1 , A2 , A3 ,.... AN-1 , AN i.e., its Present value is:2 - Another interpretation of the last expression? - Each term shows how much has to be invested now at a rate of return "i" to receive the payment Ak in 'k' years. - sum of the terms: total investment now that would give this stream of returns if the return (yield) was "i" in every period.- so "i" is the ‘average annual yield’ (average: since it is the same over the period 1 to N).- Define “D” as the cost of (or the amount deposited in) the asset.- Then if we know D and A1 , A2 , A3 .... AN you can calculate the yield from:(1) i.e.. solve this for "i" (as before Aj is the payment from the asset in year j).‘i’ is the ‘internal rate of return’ or ‘average annual yield’ on the asset.3 Examples: Treasury Bill or Commerical Paper: - Text discussion: this is an example of a “discount bond”.- 1 year to maturity:- One payment of A is made at maturity (one year in the future).- D is the price of the Treasury bill (amount invested), - Of course: D<A- The yield is obtained by solving the following for i:D = e.g.,T-Bill auction December 18, 2012:A=100 (practice is to quote payment as 100)D = 98.899 (price paid -- stated as a discount on A)98.899 = so: i = .0111 ( i.e. 1.11%)4 A Fixed Payment for N periods - Text: Fixed-Payment Loan.- Assets with this kind of payment stream: annuity, amortized loan (repay the loan in equal installments), preferred share.- Involves a fixed payment in each time period (year) for some specified length of time (N years). Fixed: so A same across time- Then: or:-The terms in the brackets [.] are a geometric series of the form: i 2 3 N ia = a + a + a + ... + a  i means this is a sum of terms from i=1 to N) i 2 3 N i a = a + a + a + ... + a (series) i 2 3 N+1 a i a = a + a + ... + a (series times 'a') ______i N+1 (1-a) i a = a - a (difference: between series and series times 'a' )So: i N+1 i a = (a - a ) / (1-a)Use this last expression to replace [.] in the annuity where a = , this gives:5 or after multiplying the top and bottom of [.] by (1+i):D i.e., a much simpler expression.- If N is infinity (payments forever) this is a perpetuity or consol and the expression is very simple:D = (why? if N→∞ and i>0)- The expression for a perpetuity can also give a reasonable approximation for other long-lived assets. e.g., a house that will last 100 years (N large) it provides a return of A=$1000x12 per year (saved rent), and its price is D=$200,000200,000 = 12000/i i = .06 (6%)6 Bond (text: “coupon bond”)- N years to maturity- Annual coupon payments of A dollars (see below for non-annual)- M: payment received when it matures or is sold (face value or resale price if sold before maturity)D = A + A + A +....+ A + M (1+i) (1+i)2 (1+i)3 (1+i)N (1+i)NLike an annuity, except for the last term.- the average annual yield can then be obtained by solving for i.- No simple solution for i: solve numerically (financial calculator will do this, Solver in Excel) e.g. say the face value is $1000 (=M is held until maturity); say the coupon rate is 10% (so the coupon payments A=.1x$1000=$100) and N=8 (eight years to maturity) and the price is $889.20$889.20Solving for i gives: i=.1225 (12.25%)- Loans requiring a fixed payment per period and a final payment of different size (e.g.,mortgages) have the same type of payment stream as the bond.- yield can be calculated in the same way as the bond7 Asset Price and Yield:- Take the general expression:D = A1 + A2 + A3 +....+ AN (1+i) (1+i)2 (1+i)3 (1+i)N D = the amount invested, the cost or price of the asset,Aj= payment from the asset in period j.- Given the payment stream (Aj), i and D are inversely related:- higher D means lower i. i.e. the more you pay for a given stream of payments the lower your yield. e.g., 1-year Treasury bill (worth 100 at maturity) Price 98.899 Yield .0111 (see above) Price 95.0 Yield .0526- This works both ways.- if yields on other assets are rising - price on this asset must fall if it is to pay a competitive return.- this gives rise to “interest rate risk”: resale price of an asset will change with changes in interest rates. - An important point: given a value of ‘i’ and a payment stream (Aj) the equation tells us what the competitive price of the asset would be (D). - asset price is higher the higher are the Aj;- asset price is lower the higher is the yield on alternative assets.8 Interest Rates and Yields:- In economics the terms are used interchangeably. - But an asset sometimes has a stated “interest rate” that differs from its yield. e.g. bond paying an x% coupon (the coupon rate is sometimes referred to as the interest rate on the bond)- yield will depend both on this coupon rate and the difference between the price of the bond and the size of the final payment (face value). Real and Nominal Yields - The yield calculations above ignore the effects of inflation on the value of money payments in the future.- Inflation involves a transfer of purchasing power from the lender to the borrower- Lending and borrowing decisions will likely be made on the basis then of inflation-adjusted yields, i.e., real yields.- Future rates of inflation are unknown at the time a financial asset is issued. - lenders and borrowers must use an estimate of the expected rate of future inflation.- Upon maturity, the actual real yield can be calculated.9 Calculating real yields- Say that  is the expected annual rate of inflation over the relevant time period.The stream of future payments:A1 , A2 , A3 ,.... AN-1 , AN will (in real terms) be worth: 2 3 N-1 N A1/(1+) , A2 /(1+) , A3 /(1+) ,... AN-1 /(1+) , AN /(1+) - where (1+) , (1+)2, (1+)3 discount each payment for inflation. - form of the inflation discount factor is just as with present value calculations. i.e. a payment of A1/(1+) now will be worth the same as A1= (1+) x A1/(1+) in one year. The real yield (ir) can be obtained by solving:D = A1 + A2 + ... + AN (2) 2 N (1+)(1+ir) {(1+) (1+ir)} {(1+)(1+ir)} i.e., just add an additional discount factor to each term to adjust for inflation.- Compare this to the yield equation used to calculate the nominal yield (i):D = A1 + A2 + A3 +....+ AN (1+i) (1+i)2 (1+i)3 (1+i)N 10 so, it must be that:(1+) (1+ir) = (1+i) or (1+ir)= (1+i)/ (1+) ir= { (1+i)/ (1+) } –1 (see text footnote 7, p. 83)- This can be used to obtain the real yield when information on the nominal yield and rate of inflation is known.- A common rule-of-thumb approximation is:Real yield = Nominal yield - expected inflation rate= i -  ( why? The nominal yield can also be written as: i = ir +  +  ir if the last term is small then the approximation will be quite good)- Data? See example and Figure 4-1.11 - “Real return bonds” or “indexed bonds”: - stream of payments is adjusted for inflation so inflation does not change the purchasing power of the payments. - Interesting implication of real return bonds?- two bonds: identical except one is indexed and the other isn’t.- difference in yields reflects expectations of future inflation over the term to maturity of the bonds. (ASIDE: Calculations of the real yield above assume inflation rate is constant. e.g. 2% (=.02) over the relevant period.- if it is expected that the inflation rate could differ over time e.g. ,,.. are expected inflation rates. then each term (1+)t in equation (2) would need to be replaced with :(1+)(1+)(1+)...(1+t) then solve for ir )12 An Extension: Yields on Assets that Make Multiple Payment per Year- What if payments received are are made "m" times per year?- A possibility? Use the same idea as above to compute an average yield for a "1/m year" period. e.g., m=2 calculate an “average semi-annual yield”. m=4 calculate an “average quarter-annual yield”.- This involves taking the equation above, but now the N year asset makes N·m payments:D = A1 + A2 + A3 +....+ ANm (3) 2 3 N·m (1+im) (1+im) (1+im) (1+im) solving for im gives the average 1/m period yield (N·m = is the total number of payments received e.g. an asset making semi-annual payments (m=2) over 10 years (N=10) makes 20 payments (10·2).13 How to annualize im?- We would like to be able to compare im to average annual yields from above.- The most common convention is to use iA as the measure of the annual yield where iA is defined: iA = m∙ im so: (1+im) = (1+iA/m) substitute this into (3) to get (4).For the N-year asset with m annual payments:D = A1 + A2 + A3 + . . . + ANm (4) 2 3 Nm (1+ iA/m) (1+ iA/m) (1+ iA/m) (1+ iA/m)Examples:- Bill or Paper with 3-month term to maturity:D = A (1+ iA/m) m=365/(days to maturity)Solving for iA: iA = m x (100-D)/D e.g. December 18, 2012 auction (maturity in 98 days): D= 99.738 , m = 365/98 , so iA = .00980 (0.980%)14 (Latest results: http://www.bankofcanada.ca/markets/government-securities-auctions/ )- Yield on a N -year bond that makes “m” equal payments per year and then its face value (M) at maturity can be calculated as: D = A + A + A + . . . + A + M 2 3 Nm Nm (1+ iA/m) (1+ iA/m) (1+ iA/m) (1+iA/m) (1+ iA/m) solve this for iA. - Common for bonds to make coupon payments every 6 months.- Example: 10 year Government of Canada bond issued in 2012, matures June 2022. Coupon: 2.75 (so A=2.75/2 every 6 months), M=100, D=108.9, iA=.01765 (1.765% ).15 Appendix: An alternative way to annualize im- Could calculate im as above then solve the following for i: m (1+im) = (1+i) where i is the average annual yield. i.e., the first term shows what $1 invested at the rate im for m periods (one year) pays at the end of m periods1/m so: (1+im) = (1+i) substitute this into (3) to get (5).- So the average annual yield when payments are made m times a year could be calculated from:D = A1 + A2 + A3 +....+ ANm (5) (1+i)1/m (1+i)2/m (1+i)3/m (1+i)N 16

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