Question

4. Alpha Insurance has investment horizon of 3 years. If it invests in a 5 year, 6% annual coupon bond with YTM of 8%, what will be its realized rate of return

1. If interest rates don’t change
2. If interest rate increases by 100 bps immediately after buying the bond
3. If interest rate decreases by 100 bps immediately after buying the bond
4. Is the difference between (a) and (b) the same as the difference between (a) and (c)? why or why not?

Answer 1

No of periods = 5 years

Coupon per period = (Coupon rate / No of coupon payments per year) * Face value

Coupon per period = (6% / 1) * $1000

Coupon per period = $60

a)

Bond Price = Coupon / (1 + YTM)^period + Face value / (1 + YTM)^period

Bond Price = $60 / (1 + 8%)¹ + $60 / (1 + 8%)² + ...+ $60 / (1 + 8%)⁵ + $1000 / (1 + 8%)⁵

Using PVIFA = ((1 - (1 + Interest rate)^{- no of periods}) / interest rate) to value coupons

Bond Price = $60 * (1 - (1 + 8%)^-5) / (8%) + $1000 / (1 + 8%)⁵

Bond Price = $239.56 + $680.58

Bond Price = $920.15

Bond price after 3 years

Bond Price = Coupon / (1 + YTM)^period + Face value / (1 + YTM)^period

Bond Price = $60 / (1 + 8%)¹ + $60 / (1 + 8%)² + $1000 / (1 + 8%)²

Using PVIFA = ((1 - (1 + Interest rate)^{- no of periods}) / interest rate) to value coupons

Bond Price = $60 * (1 - (1 + 8%)^-2) / (8%) + $1000 / (1 + 8%)²

Bond Price = $107.00 + $857.34

Bond Price after 3 years = $964.33

Reinvested coupon amounts = $60 * ((1 + YTM)^{investment horizon} - 1) / YTM

Reinvested coupon amounts = $60 * ((1 + 8%)³ - 1) / 8%

Reinvested coupon amounts = 194.78

Realized yield = ((Bond Price after 3 years + Reinvested coupon amounts) / Bond price)^{(1 / Investment horizon)} - 1

Realized yield = (($964.33 + 194.78) / $920.15)^{(1 / 3)} - 1

Realized yield = 8%

b)

Bond price after 3 years and YTM increases by 100 bps to 9%

Bond Price = Coupon / (1 + YTM)^period + Face value / (1 + YTM)^period

Bond Price = $60 / (1 + 9%)¹ + $60 / (1 + 9%)² + $1000 / (1 + 9%)²

Using PVIFA = ((1 - (1 + Interest rate)^{- no of periods}) / interest rate) to value coupons

Bond Price = $60 * (1 - (1 + 9%)^-2) / (9%) + $1000 / (1 + 9%)²

Bond Price = $105.55 + $841.68

Bond Price after 3 years = $947.23

Reinvested coupon amounts = $60 * ((1 + YTM)^{investment horizon} - 1) / YTM

Reinvested coupon amounts = $60 * ((1 + 9%)³ - 1) / 9%

Reinvested coupon amounts = 196.69

Realized yield = ((Bond Price after 3 years + Reinvested coupon amounts) / Bond price)^{(1 / Investment horizon)} - 1

Realized yield = (($947.23+ 196.69) / $920.15)^{(1 / 3)} - 1

Realized yield = 6.67%

c)

Bond price after 3 years and YTM decreases by 100 bps to 7%

Bond Price = Coupon / (1 + YTM)^period + Face value / (1 + YTM)^period

Bond Price = $60 / (1 + 7%)¹ + $60 / (1 + 7%)² + $1000 / (1 + 7%)²

Using PVIFA = ((1 - (1 + Interest rate)^{- no of periods}) / interest rate) to value coupons

Bond Price = $60 * (1 - (1 + 7%)^-2) / (7%) + $1000 / (1 + 7%)²

Bond Price = $108.48 + $873.44

Bond Price after 3 years = $981.92

Reinvested coupon amounts = $60 * ((1 + YTM)^{investment horizon} - 1) / YTM

Reinvested coupon amounts = $60 * ((1 + 7%)³ - 1) / 7%

Reinvested coupon amounts = 192.89

Realized yield = ((Bond Price after 3 years + Reinvested coupon amounts) / Bond price)^{(1 / Investment horizon)} - 1

Realized yield = (($981.92 + 192.89) / $920.15)^{(1 / 3)} - 1

Realized yield = 8.48%

d)

The difference between a & b i.e. YTM of 8% & YTM of 9% is not the same as a & c i.e. YTM of 8% & YTM of 7% since the investment horizon is of a short duration the market price risk takes precedence over reinvestment risk. The Realized yield is more volatile due to changes market price due to interest rates than changes in Reinvestment income due to changes in interest rates.