Price Sensitivity to the Interest Rate and Maturity
Þ In general, the longer the term to maturity, the greater the sensitivity to interest rate changes
ü Example: Suppose the zero coupon yield curve is flat at 12%. Bond A pays $1762.34 in five years. Bond B pays $3105.85 in ten years, and both are currently priced at $1000.
Bond A: P = $1000 = $1762.34/(1.12)5
Bond B: P = $1000 = $3105.84/(1.12)10
Now suppose the interest rate increases by 1%.
Bond A: P = $1762.34/(1.13)5 = $956.53
Bond B: P = $3105.84/(1.13)10 = $914.94
ü The longer maturity bond has the greater drop in price because the payment is discounted a greater number of times.
Duration
Þ Duration is the weighted average time to maturity using the relative present values of the cash flows as weights.
D = Snt=1[Ct• t/(1+r)t]/ Snt=1 [Ct/(1+r)t]
Where
D = duration
t = number of periods in the future
Ct = cash flow to be delivered in t periods
n= term-to-maturity & r = yield to maturity (per period basis).
ü Combines the effects of differences in coupon rates and differences in maturity
ü Based on elasticity of bond price with respect to interest rate
ü Since the price of the bond must equal the present value of all its cash flows, we can state the duration formula another way:
D = Snt=1[t ´ (Present Value of Ct/Price)]
è For a zero coupon bond, duration equals maturity
è For all other bonds: duration < maturity
ü Computing duration: Consider a 2-year, 8% coupon bond, with a face value of $1,000 and yield-to-maturity of 12%. Coupons are paid semi-annually.
Therefore, each coupon payment is $40 and the per period YTM is (1/2) × 12% = 6%. Present value of each cash flow equals CFt ÷ (1+ 0.06)t where t is the period number
Two-year BondPar value = / $1,000
YTM = / 0.12
Time / Cash Flow / PV of CF
0.5 / $40 / $37.74
1 / $40 / $35.60
1.5 / $40 / $33.58
2 / $1040 / $823.78
Total = $930.70
D=0.5x(37.74/930.70)+1x(35.60/930.70)+1.5x(33.58/930.70)+2x(823.78/930.70)=1.883
Þ Features of duration
ü Duration and maturity: D increases with M, but at a decreasing rate.
ü Duration and yield-to-maturity: D decreases as yield increases.
ü Duration and coupon interest: D decreases as coupon increases
Þ Economic interpretation of duration
ü Duration is a measure of interest rate sensitivity or elasticity of a liability or asset:
[dP/P] ¸ [dR/(1+R)] = -D
Or dP/P = -D[dR/(1+R)] = -[D/(1+R)] × dR
ü To estimate the change in price, we can rewrite this as:
dP = -D[dR/(1+R)]P = -[(D/(1+R)] × (dR) × (P)
ü Classroom problem
Calculate the duration of a 2-year, $1,000 bond that pays an annual coupon of 10 percent and trades at a yield of 14 percent. What is the expected change in the price of the bond if interest rates decline by 0.50 percent (50 basis points)?
Two-year BondPar value = / $1,000 / Coupon = / 0.10 / Annual payments
YTM = / 0.14 / Maturity = / 2
Time / Cash Flow / PVIF / PV of CF / PV*CF*T
1 / $100.00 / 0.87719 / $87.72 / $87.72 / PVIF = 1/(1+YTM)^(Time)
2 / $1,100.00 / 0.76947 / $846.41 / $1,692.83
Price = / $934.13
Numerator = / $1,780.55 / Duration / = / 1.9061 / = Numerator/Price
Expected change in price = . This implies a new price of $941.94. The actual price using conventional bond price discounting would be $941.99. The difference of $0.05 is due to convexity, which was not considered in this solution.
Duration and Immunization
Þ Duration Gap:
ü Suppose the bond in the previous example is the only loan asset (L) of an FI, funded by a 2-year certificate of deposit (D).
ü Maturity gap: ML - MD = 2 -2 = 0
ü Duration Gap: DL - DD = 1.885 - 2.0 = -0.115
è Deposit has greater interest rate sensitivity than the loan
Þ From the balance sheet, E=A-L. Therefore, DE=DA-DL. In the same manner used to determine the change in bond prices, we can find the change in value of equity using duration.
DE = [-DAA + DLL] DR/(1+R)
Or DE = -[DA - DLk]A(DR/(1+R)), k = L/A
ü The formula shows 3 effects:
è Leverage adjusted D-Gap
è The size of the FI
è The size of the interest rate shock
ü An example:
Suppose DA = 5 years, DL = 3 years and rates are expected to rise from 10% to 11%. (Rates change by 1%). Also, A = 100, L = 90 and E = 10. Find change in E.
DE = -[DA - DLk]A[DR/(1+R)] = -[5 - 3(90/100)]100[.01/1.1] = - $2.09.
ü Methods of immunizing balance sheet
è Adjust DA , DL or k.
è To set DE = 0: DA = kDL