Mishkin ch.4: Interest Rates
Summary
1. Three key concepts:
Present Value
Yield to Maturity
Total Return
2. Know how to work with the key concepts:
Task in Exams: Problem solving
3. Applications:
Real versus nominal rates
Inflation-indexed bonds
PV of stocks: Discounted dividends
Tracking total returns
[Notes on Mishkin Ch.4 - P.1]
Fundamentals
O+F:G:BH06@I:
PVt =
Paymentt+1 Paymentt+2
Paymentt+N
+
+
...
+
(1+ i)
(1+ i)2
(1+ i)N
i = discount rate = interest rate used to discount future payments
O3>eld to Maturity CFG>AD@M3>:@9+6FH>8I@6F9>G8CIBHF6H:>H=6HGC@J:G
PB = PVt =
Paymentt+1 Paymentt+2
Paymentt+N
+
+...+
(1+ i)
(1+ i)2
(1+ i)N
One-D:F>C9:L6AD@:.F:6GIFM>@@DFCA>G:G
6HA6HIF>HM
PB =
10000
(1+ i)
=> >
10000
10000 − PB
=> i =
PB
PB
[Notes on Mishkin Ch.4 - P.2]
Application: Coupon Bonds
ODefined by:
OMarket data:
Maturity date
Coupon = C
Face Value = F
=> 3:6FGHCA6HIF>HM)
=> Coupon rate = C/F
Price = PB or PBt
=> CurrenH3>:@9+B
3>:@9HCA6HIF>HM = i or it
(Time subscript t used when timing matters.
OCommon problems: Compute yield from price. Compute price from yield.
Example
OBond quote from WSJ:
Date:
Security
Treasury note
7/17/2013
Maturity
7/31/2016
Coupon Rate
1.50%
Price
102.641
Question: How is the yield calculated?
[Notes on Mishkin Ch.4 - P.3]
Yield
0.622%
Example
Date:
Security
Treasury note
7/17/2013
Maturity
7/31/2016
Coupon Rate
1.50%
Price
102.641
Yield
0.622%
OTask #1: Set up the present value equation
About 3 years to maturity
=> N = 3
Price is per F=100 face value => CIDCB,6H: = 1.50
Payment at dates t+1,…,t+N-1:
Coupon = C
Payment at maturity date t+N:
Face value + final coupon = F+C
PV =
C
C
C + F 1.50 1.50 101.50
+
+
=
+
+
2
3
2
(1+ i) (1+ i) (1+ i) (1+ i) (1+ i) (1+ i)3
OTask #2: Solve the present value equation
Cubic equation – solve numerically by exploiting that PV is declining in i.
Knowing how this works is essential to understanding yield quotes.
[Notes on Mishkin Ch.4 - P.4]
Worksheet
try coupon:
try low:
try higher:
try again:
and again:
WSJ quote:
Discount
1.50%
0.00%
1.00%
0.50%
0.62%
0.62%
PV
Graph each pair
100.00 -> near par
104.50 -> high PV
101.47 -> yield too high
102.97 -> yield too low
102.61 -> close enough
102.64
[Notes on Mishkin Ch.4 - P.5]
Notes on Bond Quotes
O(CGHC;H=:H>A:
economists use published or online quotes to obtain yields.
O.F69:FGGCA:H>A:GKCF?K>H=M>:@9GK>H=CIHA:BH>CB>B<DF>8:G
Then PB = PV is implied. Everyone knows.
OCB9EICH:G=6J:HKCD6FHG
$9:BH>;MH=:G:8IF>HM issuer, maturity date, coupon = fixed data.
(6F?:H>B;CFA6H>CB price, yield, time of quote = changes over time.
O$AD@>:9>H:AGH=6H6@GC8=6B<:8IFF:BHM>:@9
H>A:HCA6HIF>HM
OGGIADH>CBGHC8=:8?
Prices are usually per $100 face value; but sometimes per $1000.
8 = 100.25
Prices are may be decimal or fractional (e.g. "100 : 8"= 100 32
Good sources should have legends or footnotes to confirm interpretation
O->AD@>;>86H>CBG;CFH=>G8@6GG
Disregard discounting over fractional periods; usually maturity = whole years
Disregard lumpiness in coupons; treat payments as smooth over the year
Disregard “accrued interest” – use prices are as quoted “clean”
[Notes on Mishkin Ch.4 - P.6]
The Total Return
(a.k.a,:HIFB
,6H:C;,:HIFB
ODefinition:
RET =
Payment + Pt+1 − Pt
Pt
O(:6GIF:9CJ:F6GD:8>;>8H>A: period t to t+1:
Pt = Price at the start; known.
Pt+1 = Price at the end; often unknown at time t.
Payment = current yield or other payout during the time period t to t+1; assumed known
OCan be cCADIH:9;CF)3;>B6B8>6@6GG:H– not only bonds
OCADonents:
IFF:BH3>:@9=
Payment
Pt
Pt+1-Pt Change in Price
Capital Gain = P
=
Initial Price
t
[Notes on Mishkin Ch.4 - P.7]
Application in Mishkin:
Total return on a coupon bond for one year:
C + PBt+1 − PBt
PBt
Mishkin’s formula is a special case: Payment = C. Period = one year
RET =
Remember the general principle:
RET =
Payment Pt+1 − Pt
+
Pt
Pt
e.g. if period = X days, then: Payment BBI6@>N:98IFF:BHM>:@92
if asset = real estate, then: Payment = Rental income minus expenses.
O6IH>CB3>:@9G6F:IGI6@@M6BBI6@>N:9IH8IFF:BHM>:@9>B,!.F:;:FGHCH=:
period over which RET is calculated – may require conversion.
[Notes on Mishkin Ch.4 - P.8]
Illustrations in Mishkin
1. Price – Yield Relation
Example: 10-year coupon bond
[Notes on Mishkin Ch.4 - P.9]
2. Key Linkages: Yield Change - Price Change - Return
Lesson: Price responses increase with maturity
Common measure of price-sensitivity: %∆P = - Duration * ∆i/(1+i)
Pure discount bonds have duration = maturity; for coupon bonds, duration < maturity.
Optional reading: Online Appendix to Ch.4
[Notes on Mishkin Ch.4 - P.10]
Real Returns and Real Yields
OReal interest rate = Nominal interest rate minus expected inflation
e
r=i-π
- Traditional measurement: Find nominal interest rates & estimate expected inflation
Problem: expectations are not directly observable.
- New approach: use yields on inflation-protected securities (in /-G>B8:
O Treasury Inflation-Protected Securities (TIPS):
- Face value is fixed in real terms:
Fr = $100
- Nominal face value varies with CPI:
F = Fr ∗ +$
- Nominal coupon varies with CPI:
CIDCBF6H:∗ F
- If Price = Face value, then: nominal return ≈CIDCBF6H:=6B<:>B+$
=>
Real yield to maturity = real return ≈ Coupon rate
=>
Interpret coupon rate as real interest rate.
- If Price <> Face value, real return differs from coupon rate & not easy to compute.
=>
O,:A:A7:F
,:@MCBDI7@>G=:9GCIF8:G;CFF:6@M>:@9GHCA6HIF>HM:<1-%
Quoted yields on TIPS are direct measures of REAL interest rates.
[Notes on Mishkin Ch.4 - P.11]
Other Applications
OPresent values of corporate stocks:
- Payment = Dividend. Example of infinitely lived asset
PVt =
Dividendt+1
1+i
+ Dividend2 t+2 + ...
(1+i)
- Return to stocks in Mishkin ch.7.
OBritish consols:
- Coupon bonds without repayment date.
C + C + ... = C
PB = 1+i
i
(1+i)
2
- Nice illustration of negative price-to-yield relationship.
[Notes on Mishkin Ch.4 - P.12]