Question

An investment account has a value of $3000 on 1/1/2011. On 6/1/2011, the value of the account has increased to $3100and a deposit of X is made. On 10/1/2011, the value of the account balance is $4651 and $630 is withdrawn. On 1/1/2012, the investment account is worth $3661.

Compute the time-weighted rate of return if the dollar-weighted rate of return is equal to 7.00%.

Answer 1

Dollar-weighted return = 7%

Dollar weighted return can be calculated by equating net present value (NPV) to zero

NPV = present value of cash in flows - present value of cash out flows

Here, cash out flows are $3000 on 1/1/2011 and X on 6/1/2011

cash in flows are $630 on 10/1/2011 and $3661 on 1/1/2012

Present value can be calculated for date 1/1/2011

Time period between 1/1/2011 and 6/1/2011 = (6/1/2011-1/1/2011))/365 = 0.4137

Time period between 1/1/2011 and 10/1/2011 = (10/1/2011-1/1/2011))/365 = 0.7479

Time period between 1/1/2011 and 1/1/2012 = (1/1/2012-1/1/2011))/365 = 1.0000

Present value of cash out flows = 3000 + X/(1+7%)^0.4137

Present value of cash in flows = 630/(1+7%)^0.7479 + 3661/(1+7%)¹

When NPV = 0, present value of cash out flows = present value of cash in flows

implies, 3000 + X/(1+7%)^0.4137 = 630/(1+7%)^0.7479 + 3661/(1+7%)¹

X/(1+7%)^0.4137 = 630/(1+7%)^0.7479 + 3661/(1+7%)¹ - 3000

X/1.028386 = 630/1.051907 + 3661/1.07 - 3000

X/1.028386 = 598.9121 + 3421.4953 - 3000 = 1020.4074

X = 1020.4074 * 1.028386 = $1049.3724

For calculating the time-weighted return we have to find the different holding intervals

Holding interval 1, HI₁ = 1/1/2011 and 6/1/2011

Holding interval 2, HI₂ = 6/1/2011 and 10/1/2011

Holding interval 3, HI₃ = 10/1/2011 and 1/1/2012

Holding Interval Return (HIR) = [Value at end of period - (value at beginning of period + addition during the period)] / (value at beginning of period + addition during the period)

Holding interval 1 return, HIR₁ = (3100-3000)/3000 = 100/3000 = 3.3333%

Holding interval 2 return, HIR₂ = (4651-(3100+X))/(3100+X) = (4651-(3100+1049.3724))/(3100+1049.3724)

= (4651-4149.4975)/4149.4975 = 501.5025/4149.4975 = 12.0859%

Holding interval 3 return, HIR₃ = (3661-(4651-630))/(4651-630) = (3661-4021)/4021 = -360/4021 = -9.8334%

Time weighted return = (1+HIR₁)*(1+HIR₂)*(1+HIR₃) - 1 = (1+3.3333%)*(1+12.0859%)*(1-9.8334%) - 1

= 1.0333*1.1209*0.9017 -1 = 1.044328 -1 = 0.044328 = 4.4328%

Time weighted return = 4.4328%