What is the duration of a five$-$year, $$1,000$ Treasury bond with a $10$ percent semiannual coupon selling at par? Selling with a YTM of $12$ percent? $14$ percent? What can you conclude about the relationship between duration and yield to maturity? Plot the relationship. Why does this relationship exist?

Five$-$year Treasury Bond

Par value $=$ $$1,000$ Coupon $=0\mathrm{.}10$ Semiannual payments

YTM $=0\mathrm{.}10$ Maturity $=5$

Time Cash Flow PVIF PV of CF PV$*$CF$*$T

$0\mathrm{.}5$ $$50\mathrm{.}000\mathrm{.}95238$ $$47\mathrm{.}62$ $$23\mathrm{.}81$ PVIF $=1/(1+$YTM$/2)^($Time$*2)$

$1$ $$50\mathrm{.}000\mathrm{.}90703$ $$45\mathrm{.}35$ $$45\mathrm{.}35$

$1\mathrm{.}5$ $$50\mathrm{.}000\mathrm{.}86384$ $$43\mathrm{.}19$ $$64\mathrm{.}79$

$2$ $$50\mathrm{.}000\mathrm{.}8227$ $$41\mathrm{.}14$ $$82\mathrm{.}27$

$2\mathrm{.}5$ $$50\mathrm{.}000\mathrm{.}78353$ $$39\mathrm{.}18$ $$97\mathrm{.}94$

$3$ $$50\mathrm{.}000\mathrm{.}74622$ $$37\mathrm{.}31$ $$111\mathrm{.}93$

$3\mathrm{.}5$ $$50\mathrm{.}000\mathrm{.}71068$ $$35\mathrm{.}53$ $$124\mathrm{.}37$

$4$ $$50\mathrm{.}000\mathrm{.}67684$ $$33\mathrm{.}84$ $$135\mathrm{.}37$

$4\mathrm{.}5$ $$50\mathrm{.}000\mathrm{.}64461$ $$32\mathrm{.}23$ $$145\mathrm{.}04$

$5$ $$1,050\mathrm{.}000\mathrm{.}61391$ $$644\mathrm{.}61$ $$3,223\mathrm{.}04$

Price $=$ $$1,000\mathrm{.}00$

Numerator $=$ $$4,053\mathrm{.}91$ Duration $=4\mathrm{.}0539=$ Numerator$/$Price

$2.$ Consider three Treasury bonds which each have $10$ percent semiannual coupons and trade at par.

Calculate the duration for a bond that has a maturity of $4$ years, $3$ years, and $2$ years?

Please see the calculations on the next page.

a$.$ Four$-$year Treasury Bond

Par value $=$ $$1,000$ Coupon $=0\mathrm{.}10$ Semiannual payments

YTM $=0\mathrm{.}10$ Maturity $=4$

Time Cash Flow PVIF PV of CF PV$*$CF$*$T

$0\mathrm{.}5$ $$50\mathrm{.}000\mathrm{.}952381$ $$47\mathrm{.}62$ $$23\mathrm{.}81$ PVIF $=1/(1+$YTM$/2)^($Time$*2)$

$1$ $$50\mathrm{.}000\mathrm{.}907029$ $$45\mathrm{.}35$ $$45\mathrm{.}35$

$1\mathrm{.}5$ $$50\mathrm{.}000\mathrm{.}863838$ $$43\mathrm{.}19$ $$64\mathrm{.}79$

$2$ $$50\mathrm{.}000\mathrm{.}822702$ $$41\mathrm{.}14$ $$82\mathrm{.}27$

$2\mathrm{.}5$ $$50\mathrm{.}000\mathrm{.}783526$ $$39\mathrm{.}18$ $$97\mathrm{.}94$

$3$ $$50\mathrm{.}000\mathrm{.}746215$ $$37\mathrm{.}31$ $$111\mathrm{.}93$

$3\mathrm{.}5$ $$50\mathrm{.}000\mathrm{.}710681$ $$35\mathrm{.}53$ $$124\mathrm{.}37$

$4$ $$1,050\mathrm{.}000\mathrm{.}676839$ $$710\mathrm{.}68$ $$2,842\mathrm{.}73$

Price $=$ $$1,000\mathrm{.}00$

Numerator $=$ $$3,393\mathrm{.}19$ Duration $=3\mathrm{.}3932=$ Numerator$/$Price

b$.$ What conclusions can you reach about the relationship of duration and the time to maturity? Plot the relationship.

$3.$ You have discovered that the price of a bond rose from $$975$ to $$995$ when the YTM fell from $9\mathrm{.}75$ percent to $9\mathrm{.}25$ percent. What is the duration of the bond?

$4.$ 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 $0\mathrm{.}50$ percent $(50$ basis points$)?$

$5.$ The following balance sheet information is available $($amounts in $ thousands and duration in years$)$ for a financial institution:

Amount Duration

T$-$bills $$900\mathrm{.}50$

T$-$notes $550\mathrm{.}90$

T$-$bonds $176$ x

Loans $2,7247\mathrm{.}00$

Deposits $2,0921\mathrm{.}00$

Federal funds bought $2380\mathrm{.}01$

Equity $715$

Treasury bonds are $5-$year maturities paying $6$ percent semiannually and selling at par.

a$.$ What is the duration of the T$-$bond portfolio? $4\mathrm{.}393$ years as shown below.

Treasury Bond

Par value $=$ $$176$ Coupon $=0\mathrm{.}06$ Semiannual payments

YTM $=0\mathrm{.}06$ Maturity $=5$

Time Cash Flow PVIF PV of CF PV$*$CF$*$T

$0\mathrm{.}5$ $$5\mathrm{.}280\mathrm{.}97087$ $$5\mathrm{.}13$ $$2\mathrm{.}56$

$1$ $$5\mathrm{.}280\mathrm{.}94260$ $$4\mathrm{.}98$ $$4\mathrm{.}98$

$1\mathrm{.}5$ $$5\mathrm{.}280\mathrm{.}91514$ $$4\mathrm{.}83$ $$7\mathrm{.}25$

$2$ $$5\mathrm{.}280\mathrm{.}88849$ $$4\mathrm{.}69$ $$9\mathrm{.}38$

$2\mathrm{.}5$ $$5\mathrm{.}280\mathrm{.}86261$ $$4\mathrm{.}55$ $$11\mathrm{.}39$

$3$ $$5\mathrm{.}280\mathrm{.}83748$ $$4\mathrm{.}42$ $$13\mathrm{.}27$

$3\mathrm{.}5$ $$5\mathrm{.}280\mathrm{.}81309$ $$4\mathrm{.}29$ $$15\mathrm{.}03$

$4$ $$5\mathrm{.}280\mathrm{.}78941$ $$4\mathrm{.}17$ $$16\mathrm{.}67$

$4\mathrm{.}5$ $$5\mathrm{.}280\mathrm{.}76642$ $$4\mathrm{.}05$ $$18\mathrm{.}21$

$5$ $$181\mathrm{.}280\mathrm{.}74409$ $$134\mathrm{.}89$ $$674\mathrm{.}45.$

Price $=$ $$176\mathrm{.}00$

Numerator $=$ $$773\mathrm{.}18$ Duration $=4\mathrm{.}3931=$ Numerator$/$Price

b$.$ What is the average duration of all the assets?

c$.$ What is the average duration of all the liabilities?

d$.$ What is the leverage$-$adjusted duration gap? What is the interest rate risk exposure?

e$.$ What is the forecast impact on the market value of equity caused by a relative upward shift in the entire yield curve of $0\mathrm{.}5$ percent $[$i$.$e$.,$R$/(1+$R$)=0\mathrm{.}0050]?$

The market value of the equity will change by the following:

f$.$ If the yield curve shifted downward by $0\mathrm{.}25$ percent $($i$.$e$.,$R$/(1+$R$)=-0\mathrm{.}0025),$ what is the forecasted impact on the market value of equity?

g$.$ What variables are available to the financial institution to immunize the balance sheet? How much would each variable need to change to get DGAP equal to $0?$