Interest Rates and Interest Rates and Security Valuation
Financial Markets & Institutions
Chapter Three
Interest Rates and Security Valuation By Doc Brown
Doc Brown’s Trading University
http://investmenttrainingcourse.com
Various Interest Rate Measures
Coupon rate
Required rate of return (r)
rates used by individual market participants to calculate fair present values (PV)
Expected rate of return or E(r)
periodic cash flow a bond issuer contractually promises to pay a bond holder
rates participants would earn by buying securities at current market prices (P)
Realized rate of return ( r )
rate actually earned on investments
3-2
Required Rate of Return
The fair present value (PV) of a security is determined using the required rate of return (r) as the discount rate ~ ~ ~ ~ CF1 CF2 CF3 CFn PV = + + + ... + (1 + r )1 (1 + r )2 (1 + r )3 (1 + r )n CF1 = cash flow in period t (t = 1, …, n) ~ = indicates the projected cash flow is uncertain n = number of periods in the investment horizon
3-3
Expected Rate of Return
The current market price (P) of a security is determined using the expected rate of return or E(r) as the discount rate P=
~ CF1
1
(1 + E(r ))
+
~ CF2
2
(1 + E(r ))
+
~ CF3
3
(1 + E(r ))
+ ... +
~ CFn
(1 + E(r ))n
CF1 = cash flow in period t (t = 1, …, n) ~ = indicates the projected cash flow is uncertain n = number of periods in the investment horizon
3-4
Realized Rate of Return
The realized rate of return ( r ) is the discount rate that just equates the actual purchase price ( P) to the present value of the realized cash flows (RCFt) t (t = 1, …, n)
P=
RCF1
+
RCF2
(1 + r )1 (1 + r )2
+
RCF3 (1 + r )3
+ ... +
RCFn (1 + r )n
3-5
Bond Valuation
The present value of a bond (Vb) can be written as: t INT 2T 1 Par Vb = + ∑ 2 t =1 (1 + (r / 2)) (1 + (r / 2))2T
INT 1 − 1 (1 + (r 2))2T Par = + 2 (r 2) (1 + (r/2))2T
Par = the par or face value of the bond, usually $1,000 INT = the annual interest (or coupon) payment T = the number of years until the bond matures r = the annual interest rate (often called yield to maturity (ytm))
3-6
Bond Valuation
A premium bond has a coupon rate (INT) greater than the required rate of return (r)) and the fair present value of the bond (Vb) is greater than the face or par value (Par) Premium bond: If INT > r; then Vb > Par Discount bond: if INT < r, then Vb < Par Par bond: if INT = r, then Vb = Par
3-7
Equity Valuation
The present value of a stock (Pt) assuming zero growth in dividends can be written as:
Pt = D / rs D = dividend paid at end of every year Pt = the stock’s price at the end of year t rs = the interest rate used to discount future cash flows
3-8
Equity Valuation
The present value of a stock (Pt) assuming constant growth in dividends can be written as: t D0 (1 + g) Dt +1 Pt = = rs − g rs − g D0 = current value of dividends Dt = value of dividends at time t = 1, 2, …, ∞ g = the constant dividend growth rate
3-9
Equity Valuation
The return on a stock with zero dividend growth, if purchased at current price P0, can be written as:
rs = D / P0
The return on a stock with constant dividend growth, if purchased at price P0, can be written as: D (1 + g) D rs = 0 +g= 1+g P0 P0
3-10
Relation between Interest Rates and Bond Values Interest Rate 12%
10%
8%
874.50
1,000
1,152.47
Bond Value
3-11
Impact of Maturity on Price Volatility (a) Absolute Value of Percent Change in a Bond’s Price for a Given Change in Interest Rates
Time to Maturity
3-12
Impact of Maturity on Price Volatility (b) Coupon Par yield rate old yield rate change yield rate new
$
6.00% 1,000 7.00% 0.50% 7.50%
Absolute Price old Price new Rate of change 1 $ 990.65 $ 986.05 0.47% 5 $ 959.00 $ 939.31 2.05% 10 $ 929.76 $ 897.04 3.52% 15 $ 908.92 $ 867.59 4.55% 20 $ 894.06 $ 847.08 5.25% 25 $ 883.46 $ 832.80 5.74% 30 $ 875.91 $ 822.84 6.06% 35 $ 870.52 $ 815.91 6.27% 40 $ 866.68 $ 811.08 6.42% 45 $ 863.94 $ 807.72 6.51% 50 $ 861.99 $ 805.38 6.57% 55 $ 860.60 $ 803.75 6.61% 60 $ 859.61 $ 802.61 6.63% 65 $ 858.90 $ 801.82 6.65% 70 $ 858.40 $ 801.27 6.66% 75 $ 858.04 $ 800.88 6.66% 80 $ 857.78 $ 800.61 6.66% 85 $ 857.60 $ 800.43 6.67% 90 $ 857.47 $ 800.30 6.67% 95 $ 857.37 $ 800.21 6.67% 100 $ 857.31 $ 800.14 6.67% 105 $ 857.26 $ 800.10 6.67% Predicted limit price change = 1 - (r old / r new) 6.67% Maturity
IM Figure 3.1
3-13
Impact of Coupon Rates on Price Volatility Bond Value High-Coupon Bond
Low-Coupon Bond
Interest Rate
3-14
Impact of Coupon on Price Volatility (b) Coupon Par rate old rate change rate new
Varies $1,000 7.00% -0.50% 6.50%
Maturity 10 years
Absolute Rate of Coupon rate Price old Price new change 6.00% $ 929.76 $ 964.06 3.69% 5.50% $ 894.65 $ 928.11 3.74% 5.00% $ 859.53 $ 892.17 3.80% 4.50% $ 824.41 $ 856.22 3.86% 4.00% $ 789.29 $ 820.28 3.93% 3.50% $ 754.17 $ 784.34 4.00% 3.00% $ 719.06 $ 748.39 4.08% 2.50% $ 683.94 $ 712.45 4.17% 2.00% $ 648.82 $ 676.50 4.27% 1.50% $ 613.70 $ 640.56 4.38% 1.00% $ 578.59 $ 604.61 4.50% 0.50% $ 543.47 $ 568.67 4.64% 0.00% $ 508.35 $ 532.73 4.80% IM Figure 2 Coupons and Price Volatility
3-15
Impact of r on Price Volatility Bond Price
How does volatility change with interest rates?
Price volatility is inversely related to the level of the initial interest rate
Interest Rate
r
3-16
Duration
Duration is the weighted-average time to maturity (measured in years) on a financial security Duration measures the sensitivity (or elasticity) of a fixed-income security’s price to small interest rate changes Duration captures the coupon and maturity effects on volatility.
3-17
Duration
Duration (Dur) for a fixed-income security that pays interest annually can be written as: T
CF t ×t
T
∑ (1+ r )t ∑ PVt × t
Dur = t =1
P0
= t =1
P0
P0= Current price of the security t = 1 to T, the period in which a cash flow is received T = the number of years to maturity CFt = cash flow received at end of period t r = yield to maturity or required rate of return PVt = present value of cash flow received at end of period t
3-18
Duration and Volatility
9% Coupon, 4 year maturity annual payment bond with a 8% ytm T
CF t ×t
∑ (1 + r)t
Dur = t =1
P0
Year (T) 1 2 3 4 Totals
Cash Flow $ 90 90 90 $1090
PV @8% CFT/(1+r)T $ 83.33 77.16 71.45 $ 801.18 $1033.12
% of Value PV/Price 8.06% 7.47% 6.92% 77.55% 100.00%
Weighted % of Value (PV/Price)*T 0.0806 0.1494 0.2076 3.1020 3.5396
Duration = 3.5396 years 3-19
Duration
Duration (Dur) (measured in years) for a fixedincome security, in general, can be written as: T
∑
CF t ×t
mt ( 1 + r / m ) Dur = t =1/ m P0 m = the number of times per year interest is paid, the sum term is incremented in m units
3-20
Closed form duration equation: 1 − (1 + r) − N PVIFA r, N = r
INT × [N − ((1 + r) × PVIFA r,N )] Dur = N − (Po × r) • P0 = Price • INT= Periodic cash flow in dollars, normally the semiannual coupon on a bond or the periodic monthly payment on a loan. • r = periodic interest rate = APR / m, where m = # compounding periods per year • N = Number of compounding or payment periods (or the number of years * m) • Dur = Duration = # Compounding or payment periods; Durationperiod is what you actually get from the formula 3-21
Duration
Duration and coupon interest
Duration and yield to maturity
the higher the coupon payment, the lower the bond’s duration the higher the yield to maturity, the lower the bond’s duration
Duration and maturity
duration increases with maturity but at a decreasing rate
3-22
Duration and Modified Duration
Given an interest rate change, the estimated percentage change in a (annual coupon paying) bond’s price given by
∆P ∆r = −Dur P 1 + r
3-23
Duration and Modified Duration
Modified duration (DurMod) can be used to predict price changes for non-annual payment loans or securities: It is found as: where rperiod = APR/m
DurMod
DurAnnual = (1 + rperiod )
Using modified duration to predict price changes:
ΔP = −DurMod × Δrannual P
3-24
Duration Based Prediction Errors
3-25
Convexity
Convexity (CX) measures the change in slope of the price-yield curve around interest rate level R Convexity incorporates the curvature of the priceyield curve into the estimated percentage price change of a bond given an interest rate change:
∆P ∆r 1 2 = −Dur + CX ( ∆ r ) 2 P 1 + r
3-26
Practice Problem
P0 =
$C [ ] Dur = N − × N − ((1 + r) × PVIFA ) r,N 1 − 1.035 −10 $1000 (P × r) $30 × = $ 958 . 42 o + 0.035 1.03510
$30 Dursemi = 10 − × [10 − (1.035 × 8.316605) ] = 8 .7548 six month periods ($958.42 × 0.035) − Δrsemi ΔP − 0.0025 = Dursemi × = 8.7548 × = − 2.1147% P (1 + roldsemi ) 1.035 P1 Predicted = $958.42 × (1 + −0.021147) =
$ 938 . 15
Using Modified Duration DurMod =
DurAnnual 4.3774 (8.7548 / 2) = 4.2294 = = (1 + rperiod ) 1.035 1.035
Predicted Price Change Using Modified Duration ΔP = −DurMod × Δrannual = −4.2294 × 0.0050 = − 2.1147% P 3-27
Chapter Three
Interest Rates and Security Valuation By Doc Brown
Doc Brown’s Trading University
http://investmenttrainingcourse.com
Various Interest Rate Measures
Coupon rate
Required rate of return (r)
rates used by individual market participants to calculate fair present values (PV)
Expected rate of return or E(r)
periodic cash flow a bond issuer contractually promises to pay a bond holder
rates participants would earn by buying securities at current market prices (P)
Realized rate of return ( r )
rate actually earned on investments
3-2
Required Rate of Return
The fair present value (PV) of a security is determined using the required rate of return (r) as the discount rate ~ ~ ~ ~ CF1 CF2 CF3 CFn PV = + + + ... + (1 + r )1 (1 + r )2 (1 + r )3 (1 + r )n CF1 = cash flow in period t (t = 1, …, n) ~ = indicates the projected cash flow is uncertain n = number of periods in the investment horizon
3-3
Expected Rate of Return
The current market price (P) of a security is determined using the expected rate of return or E(r) as the discount rate P=
~ CF1
1
(1 + E(r ))
+
~ CF2
2
(1 + E(r ))
+
~ CF3
3
(1 + E(r ))
+ ... +
~ CFn
(1 + E(r ))n
CF1 = cash flow in period t (t = 1, …, n) ~ = indicates the projected cash flow is uncertain n = number of periods in the investment horizon
3-4
Realized Rate of Return
The realized rate of return ( r ) is the discount rate that just equates the actual purchase price ( P) to the present value of the realized cash flows (RCFt) t (t = 1, …, n)
P=
RCF1
+
RCF2
(1 + r )1 (1 + r )2
+
RCF3 (1 + r )3
+ ... +
RCFn (1 + r )n
3-5
Bond Valuation
The present value of a bond (Vb) can be written as: t INT 2T 1 Par Vb = + ∑ 2 t =1 (1 + (r / 2)) (1 + (r / 2))2T
INT 1 − 1 (1 + (r 2))2T Par = + 2 (r 2) (1 + (r/2))2T
Par = the par or face value of the bond, usually $1,000 INT = the annual interest (or coupon) payment T = the number of years until the bond matures r = the annual interest rate (often called yield to maturity (ytm))
3-6
Bond Valuation
A premium bond has a coupon rate (INT) greater than the required rate of return (r)) and the fair present value of the bond (Vb) is greater than the face or par value (Par) Premium bond: If INT > r; then Vb > Par Discount bond: if INT < r, then Vb < Par Par bond: if INT = r, then Vb = Par
3-7
Equity Valuation
The present value of a stock (Pt) assuming zero growth in dividends can be written as:
Pt = D / rs D = dividend paid at end of every year Pt = the stock’s price at the end of year t rs = the interest rate used to discount future cash flows
3-8
Equity Valuation
The present value of a stock (Pt) assuming constant growth in dividends can be written as: t D0 (1 + g) Dt +1 Pt = = rs − g rs − g D0 = current value of dividends Dt = value of dividends at time t = 1, 2, …, ∞ g = the constant dividend growth rate
3-9
Equity Valuation
The return on a stock with zero dividend growth, if purchased at current price P0, can be written as:
rs = D / P0
The return on a stock with constant dividend growth, if purchased at price P0, can be written as: D (1 + g) D rs = 0 +g= 1+g P0 P0
3-10
Relation between Interest Rates and Bond Values Interest Rate 12%
10%
8%
874.50
1,000
1,152.47
Bond Value
3-11
Impact of Maturity on Price Volatility (a) Absolute Value of Percent Change in a Bond’s Price for a Given Change in Interest Rates
Time to Maturity
3-12
Impact of Maturity on Price Volatility (b) Coupon Par yield rate old yield rate change yield rate new
$
6.00% 1,000 7.00% 0.50% 7.50%
Absolute Price old Price new Rate of change 1 $ 990.65 $ 986.05 0.47% 5 $ 959.00 $ 939.31 2.05% 10 $ 929.76 $ 897.04 3.52% 15 $ 908.92 $ 867.59 4.55% 20 $ 894.06 $ 847.08 5.25% 25 $ 883.46 $ 832.80 5.74% 30 $ 875.91 $ 822.84 6.06% 35 $ 870.52 $ 815.91 6.27% 40 $ 866.68 $ 811.08 6.42% 45 $ 863.94 $ 807.72 6.51% 50 $ 861.99 $ 805.38 6.57% 55 $ 860.60 $ 803.75 6.61% 60 $ 859.61 $ 802.61 6.63% 65 $ 858.90 $ 801.82 6.65% 70 $ 858.40 $ 801.27 6.66% 75 $ 858.04 $ 800.88 6.66% 80 $ 857.78 $ 800.61 6.66% 85 $ 857.60 $ 800.43 6.67% 90 $ 857.47 $ 800.30 6.67% 95 $ 857.37 $ 800.21 6.67% 100 $ 857.31 $ 800.14 6.67% 105 $ 857.26 $ 800.10 6.67% Predicted limit price change = 1 - (r old / r new) 6.67% Maturity
IM Figure 3.1
3-13
Impact of Coupon Rates on Price Volatility Bond Value High-Coupon Bond
Low-Coupon Bond
Interest Rate
3-14
Impact of Coupon on Price Volatility (b) Coupon Par rate old rate change rate new
Varies $1,000 7.00% -0.50% 6.50%
Maturity 10 years
Absolute Rate of Coupon rate Price old Price new change 6.00% $ 929.76 $ 964.06 3.69% 5.50% $ 894.65 $ 928.11 3.74% 5.00% $ 859.53 $ 892.17 3.80% 4.50% $ 824.41 $ 856.22 3.86% 4.00% $ 789.29 $ 820.28 3.93% 3.50% $ 754.17 $ 784.34 4.00% 3.00% $ 719.06 $ 748.39 4.08% 2.50% $ 683.94 $ 712.45 4.17% 2.00% $ 648.82 $ 676.50 4.27% 1.50% $ 613.70 $ 640.56 4.38% 1.00% $ 578.59 $ 604.61 4.50% 0.50% $ 543.47 $ 568.67 4.64% 0.00% $ 508.35 $ 532.73 4.80% IM Figure 2 Coupons and Price Volatility
3-15
Impact of r on Price Volatility Bond Price
How does volatility change with interest rates?
Price volatility is inversely related to the level of the initial interest rate
Interest Rate
r
3-16
Duration
Duration is the weighted-average time to maturity (measured in years) on a financial security Duration measures the sensitivity (or elasticity) of a fixed-income security’s price to small interest rate changes Duration captures the coupon and maturity effects on volatility.
3-17
Duration
Duration (Dur) for a fixed-income security that pays interest annually can be written as: T
CF t ×t
T
∑ (1+ r )t ∑ PVt × t
Dur = t =1
P0
= t =1
P0
P0= Current price of the security t = 1 to T, the period in which a cash flow is received T = the number of years to maturity CFt = cash flow received at end of period t r = yield to maturity or required rate of return PVt = present value of cash flow received at end of period t
3-18
Duration and Volatility
9% Coupon, 4 year maturity annual payment bond with a 8% ytm T
CF t ×t
∑ (1 + r)t
Dur = t =1
P0
Year (T) 1 2 3 4 Totals
Cash Flow $ 90 90 90 $1090
PV @8% CFT/(1+r)T $ 83.33 77.16 71.45 $ 801.18 $1033.12
% of Value PV/Price 8.06% 7.47% 6.92% 77.55% 100.00%
Weighted % of Value (PV/Price)*T 0.0806 0.1494 0.2076 3.1020 3.5396
Duration = 3.5396 years 3-19
Duration
Duration (Dur) (measured in years) for a fixedincome security, in general, can be written as: T
∑
CF t ×t
mt ( 1 + r / m ) Dur = t =1/ m P0 m = the number of times per year interest is paid, the sum term is incremented in m units
3-20
Closed form duration equation: 1 − (1 + r) − N PVIFA r, N = r
INT × [N − ((1 + r) × PVIFA r,N )] Dur = N − (Po × r) • P0 = Price • INT= Periodic cash flow in dollars, normally the semiannual coupon on a bond or the periodic monthly payment on a loan. • r = periodic interest rate = APR / m, where m = # compounding periods per year • N = Number of compounding or payment periods (or the number of years * m) • Dur = Duration = # Compounding or payment periods; Durationperiod is what you actually get from the formula 3-21
Duration
Duration and coupon interest
Duration and yield to maturity
the higher the coupon payment, the lower the bond’s duration the higher the yield to maturity, the lower the bond’s duration
Duration and maturity
duration increases with maturity but at a decreasing rate
3-22
Duration and Modified Duration
Given an interest rate change, the estimated percentage change in a (annual coupon paying) bond’s price given by
∆P ∆r = −Dur P 1 + r
3-23
Duration and Modified Duration
Modified duration (DurMod) can be used to predict price changes for non-annual payment loans or securities: It is found as: where rperiod = APR/m
DurMod
DurAnnual = (1 + rperiod )
Using modified duration to predict price changes:
ΔP = −DurMod × Δrannual P
3-24
Duration Based Prediction Errors
3-25
Convexity
Convexity (CX) measures the change in slope of the price-yield curve around interest rate level R Convexity incorporates the curvature of the priceyield curve into the estimated percentage price change of a bond given an interest rate change:
∆P ∆r 1 2 = −Dur + CX ( ∆ r ) 2 P 1 + r
3-26
Practice Problem
P0 =
$C [ ] Dur = N − × N − ((1 + r) × PVIFA ) r,N 1 − 1.035 −10 $1000 (P × r) $30 × = $ 958 . 42 o + 0.035 1.03510
$30 Dursemi = 10 − × [10 − (1.035 × 8.316605) ] = 8 .7548 six month periods ($958.42 × 0.035) − Δrsemi ΔP − 0.0025 = Dursemi × = 8.7548 × = − 2.1147% P (1 + roldsemi ) 1.035 P1 Predicted = $958.42 × (1 + −0.021147) =
$ 938 . 15
Using Modified Duration DurMod =
DurAnnual 4.3774 (8.7548 / 2) = 4.2294 = = (1 + rperiod ) 1.035 1.035
Predicted Price Change Using Modified Duration ΔP = −DurMod × Δrannual = −4.2294 × 0.0050 = − 2.1147% P 3-27
