Flow of Funds (SKIP

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 value

Answer:

why? investing this amount at a return i now would give $1 in one year

(1+i) x = 1

Similarly, 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:

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

- The yield is obtained by solving the following for i:

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:

-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):

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=$200,000

200,000 = 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)N

Like 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

$889.20

Solving 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 bond

7 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

(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 periods

1/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